Planet Musings

July 14, 2025

John Baez — The Kepler Problem (Part 2)

I’ve been working on a math project involving the periodic table of elements and the Kepler problem—that is, the problem of a particle moving in an inverse square force law. I started it in 2021, but I just finished. I want to explain it. But I need some things to be fresh in your mind, so I’m going to republish some old blog articles, starting with this one. (For Part 1 go here, but you won’t need that for what’s to come.)

Let’s start with the ‘eccentricity vector’. This is a conserved quantity for the Kepler problem. It was named the ‘Runge–Lenz vector’ after Lenz used it in 1924 to study the hydrogen atom in the framework of the ‘old quantum mechanics’ of Bohr and Sommerfeld: Lenz cite Runge’s popular German textbook on vector analysis from 1919, which explains this vector. But Runge never claimed any originality: he attributed this vector to Gibbs, who wrote about it in his book on vector analysis in 1901!

Nowadays many people call it the ‘Laplace–Runge–Lenz vector’, honoring Laplace’s discussion of it in his famous treatise on celestial mechaics in 1799. But in fact this vector goes back at least to Jakob Hermann, who wrote about it in 1710, triggering further work on this topic by Johann Bernoulli in the same year.

Nobody has seen signs of this vector in work before Hermann. So, we might call it the Hermann–Bernoulli–Laplace–Gibbs–Runge–Lenz vector, or just the Hermann vector. But I prefer to call it the eccentricity vector, because for a particle in an inverse square law its magnitude is the eccentricity of that orbit!

Let’s suppose we have a particle whose position $\vec q \in \mathbb{R}^3$ obeys this version of the inverse square force law:

$\ddot{\vec q} = - \frac{\vec q}{q^3}$

where I remove the arrow from a vector when I want to talk about its magnitude. So, I’m setting the mass of this particle equal to 1, along with the constant saying the strength of the force. That’s because I want to keep the formulas clean! With these conventions, the momentum of the particle is

$\vec p = \dot{\vec q}$

For this system it’s well-known that the following energy is conserved:

$H = \frac{1}{2} p^2 - \frac{1}{q}$

as well as the angular momentum vector:

$\vec L = \vec q \times \vec p$

But the interesting thing for me today is the eccentricity vector:

$\vec e = \vec p \times \vec L - \frac{\vec q}{q}$

Let’s check that it’s conserved! Taking its time derivative,

$\dot{\vec e} = \dot{\vec p} \times \vec L + \vec p \times \dot{\vec L} - \frac{\vec p}{q} + \frac{\dot q}{q^2} \,\vec q$

But angular momentum is conserved so the second term vanishes, and

$\dot q = \frac{d}{dt} \sqrt{\vec q \cdot \vec q} = \frac{\vec p \cdot \vec q}{\sqrt{\vec q \cdot \vec q}} = \frac{\vec p \cdot \vec q}{q}$

so we get

$\dot{\vec e} = \dot{\vec p} \times \vec L - \frac{\vec p}{q} + \frac{\vec p \cdot \vec q}{q^2}\, \vec q$

But the inverse square force law says

$\dot{\vec p} = - \frac{\vec q}{q^3}$

$\dot{\vec e} = - \frac{1}{q^3} \, \vec q \times \vec L - \frac{\vec p}{q} + \frac{\vec p \cdot \vec q}{q^2}\, \vec q$

How can we see that this vanishes? Mind you, there are various geometrical ways to think about this, but today I’m in the mood for checking that my skills in vector algebra are sufficient for a brute-force proof—and I want to record this proof so I can see it later!

To get anywhere we need to deal with the cross product in the above formula:

$\vec q \times \vec L = \vec q \times (\vec q \times \vec p)$

There’s a nice identity for the vector triple product:

$\vec a \times (\vec b \times \vec c) = (\vec a \cdot \vec c) \vec b - (\vec a \cdot \vec b) \vec c$

I could have fun talking about why this is true, but I won’t now! I’ll just use it:

$\vec q \times \vec L = \vec q \times (\vec q \times \vec p) = (\vec q \cdot \vec p) \vec q - q^2 \, \vec p$

and plug this into our formula

$\dot{\vec e} = - \frac{1}{q^3} \, \vec q \times \vec L - \frac{\vec p}{q} + \frac{\vec p \cdot \vec q}{q^2}\, \vec q$

getting

$\dot{\vec e} = -\frac{1}{q^3} \Big((\vec q \cdot \vec p) \vec q - q^2 \vec p \Big) - \frac{\vec p}{q} + \frac{\vec p \cdot \vec q}{q^3}\, \vec q$

But look—everything cancels! So

$\dot{\vec e} = 0$

and the eccentricity vector is conserved!

So, it seems that the inverse square force law has 7 conserved quantities: the energy $H,$ the 3 components of the angular momentum $\vec L,$ and the 3 components of the eccentricity vector $\vec e$ . But they can’t all be independent, since the particle only has 6 degrees of freedom: 3 for position and 3 for momentum. There can be at most 5 independent conserved quantities, since something has to change. So there have to be at least two relations betwen the conserved quantities we’ve found.

The first of these relations is pretty obvious: $\vec e$ and $\vec L$ are at right angles, so

$\vec e \cdot \vec L = 0$

But wait, why are they at right angles? Because

$\vec e = \vec p \times \vec L - \frac{\vec q}{q}$

The first term is orthogonal to $\vec L$ because it’s a cross product of $\vec p$ and $\vec L;$ the second is orthogonal to $\vec L$ because $\vec L$ is a cross product of $\vec q$ and $\vec p$ .

The second relation is a lot less obvious, but also more interesting. Let’s take the dot product of $\vec e$ with itself:

$e^2 = \left(\vec p \times \vec L - \frac{\vec q}{q}\right) \cdot \left(\vec p \times \vec L - \frac{\vec q}{q}\right)$

or in other words,

$e^2 = (\vec p \times \vec L) \cdot (\vec p \times \vec L) - \frac{2}{q} \vec q \cdot (\vec p \times \vec L) + 1$

But remember this nice cross product identity:

$(\vec a \times \vec b) \cdot (\vec a \times \vec b) + (\vec a \cdot \vec b)^2 = a^2 b^2$

Since $\vec p$ and $L$ are at right angles this gives

$(\vec p \times \vec L) \cdot (\vec p \times \vec L) = p^2 L^2$

$e^2 = p^2 L^2 - \frac{2}{q} \vec q \cdot (\vec p \times \vec L) + 1$

Then we can use the cyclic identity for the scalar triple product:

$\vec a \cdot (\vec b \times \vec c) = \vec c \cdot (\vec a \times \vec b)$

to rewrite this as

$e^2 = p^2 L^2 - \frac{2}{q} \vec L \cdot (\vec q \times \vec p) + 1$

or simply

$e^2 = p^2 L^2 - \frac{2}{q} L^2 + 1$

or even better,

$e^2 = 2 \left(\frac{1}{2} p^2 - \frac{1}{q}\right) L^2 + 1$

But this means that

$e^2 = 2HL^2 + 1$

which is our second relation between conserved quantities for the Kepler problem!

This relation makes a lot of sense if you know that $e$ is the eccentricity of the orbit. Then it implies:

• if $H > 0$ then $e > 1$ and the orbit is a hyperbola.

• if $H = 0$ then $e = 1$ and the orbit is a parabola.

• if $H < 0$ then $0 < e < 1$ and the orbit is an ellipse (or circle).

But why is $e$ the eccentricity? And why does the particle move in a hyperbola, parabola or ellipse in the first place? We can show both of these things by taking the dot product of $\vec q$ and $\vec e:$

$\begin{array}{ccl} \vec q \cdot \vec e &=& \vec q \cdot \left(\vec p \times \vec L - \frac{\vec q}{q} \right) \\ \\ &=& \vec q \cdot (\vec p \times \vec L) - q \end{array}$

Using the cyclic property of the scalar triple product we can rewrite this as

$\begin{array}{ccl} \vec q \cdot \vec e &=& \vec L \cdot (\vec q \times \vec p) - q \\ \\ &=& L^2 - q \end{array}$

Now, we know that $\vec q$ moves in the plane orthogonal to $\vec L$ . In this plane, which contains the vector $\vec e,$ the equation $\vec q \cdot \vec e = L^2 - q$ defines a conic of eccentricity $e$ . I won’t show this from scratch, but it may seem more familiar if we rotate the whole situation so this plane is the $xy$ plane and $\vec e$ points in the $x$ direction. Then in polar coordinates this equation says

$er \cos \theta = L^2 - r$

$r = \frac{L^2}{1 + e \cos \theta}$

This is well-known, at least among students of physics who have solved the Kepler problem, to be the equation of a conic of eccentricity $e$ .

Another thing that’s good to do is define a rescaled eccentricity vector. In the case of elliptical orbits, where $H < 0,$ we define this by

$\vec M = \frac{\vec e}{\sqrt{-2H}}$

Then we can take our relation

$e^2 = 2HL^2 + 1$

and rewrite it as

$1 = e^2 - 2H L^2$

and then divide by $-2H$ getting

$- \frac{1}{2H} = \frac{e^2}{-2H} + L^2$

$- \frac{1}{2H} = L^2 + M^2$

This suggests an interesting similarity between $\vec L$ and $\vec M,$ which turns out to be very important in a deeper understanding of the Kepler problem. And with more work, you can use this idea to show that $-1/4H$ is the Hamiltonian for a free particle on the 3-sphere. But more about that some other time, I hope!

For now, you might try this:

• Wikipedia, Laplace–Runge–Lenz vector.

and of course this:

• The Kepler problem (part 1).

by John Baez at July 14, 2025 06:57 PM

Clifford Johnson — The Power of the String Equation

[More technical post follows.] I've been working on this project with Maciej on and off for a while now, but only in the last couple of weeks did I have the time to hunker down and help push the writing of the results to the finish. For your Sunday reading, it is already up on the arXiv here (it came out Thursday but I've been too busy to pause to post about it - partly because I've begun work on writing up the next paper in the backlog). The title is "Extended JT supergravity and random matrix models: The power of the string equation", and it is co-authored with a UCSB postdoc, Maciej Kolanowski.

In a way, it is a natural continuation of work I've described here from 2023 and 2024, described here and here. At a meeting at the Institute for Advanced Study in December 2023 I described in a talk (YouTube video here, look in particular from minute 35) something miraculous I'd discovered concerning capturing certain special supergravity (and black hole) behaviour using a random matrix model. The effective physics is [...] Click to continue reading this post →

The post The Power of the String Equation appeared first on Asymptotia.

by Clifford at July 14, 2025 05:44 AM

July 13, 2025

David Hogg — is it surprising that there are high-redshift supermassive black holes?

A nice talk at MPIA by Hanna Ũbler (MPE) about very high-redshift (redshifts 8 to 14 even) galaxies and their black-hole contents started some nice discussions in the audience and afterwards about the formation of black holes. Because of the Eddington limit (which is a limit on luminosity), black-hole growth is probably limited. The limit is on luminosity, not mass accretion rate, and the relationship between these is the radiative efficiency. As the efficiency goes down, the mass accretion rate of an Eddington accretor goes up. So the question: When can you first form a super-massive (10 million solar masses, say) black hole is a question simultaneously about seed black holes and about radiative efficiency. Anyway, this is all unfortunate, because if the efficiency couldn't get very low, then the black holes we find with NASA JWST would already be putting very strong pressure on fundamental physics in the early Universe.

by Hogg at July 13, 2025 01:48 PM

David Hogg — coherent oscillator injection and recovery

Coherent oscillators—astronomical sources that pulse or oscillate in a phase-stable way over long timescales—have been useful astrophysical tools. For two examples: Stably pulsing pulsars were used to discover gravitational radiation (and are being used to find the stochastic background). Delta-scuti star asteroseismic modes were used to find orbital companions. This led undergrad Nana Miller (NYU) and me and others to look for all the coherent modes we can find among all the stars in the NASA Kepler Mission data. We have have technology to find modes, and we have technology to test for coherence. Now we have to do injection–recovery tests to estimate our detection limits. Today I delivered a simple plan for doing injections.

Our plan is to produce a catalog, not of stars but of modes, every one of which has a coherence time that is longer than the lifetime (4 years) of the Kepler Mission. Then: What do we use them for?

by Hogg at July 13, 2025 11:44 AM

July 12, 2025

n-Category Café Potential Functions and the Magnitude of Functors 2

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Despite the “2” in the title, you can follow this post without having read part 1. The whole point is to sneak up on the metricky, analysisy stuff about potential functions from a categorical angle, by considering constructions that are categorically reasonable in their own right. Let’s go!

$MathML-enabled post (click for more details).$

Let $V$ be a monoidal category, which I’ll assume is symmetric, closed, complete or cocomplete as necessary. Take a pair of $V$ -categories $A$ and $B$ , and a $V$ -functor

$F: A \to B.$

Then we get a new functor

$N_F: B \to [A^{op}, V]$

defined by

$(N_F(b))(a) = B(F a, b).$

This is sometimes called the nerve functor induced by $F$ , because of the first example in the following list.

Examples In the first few examples, I’ll take the enriching category $V$ to be $Set$ .

Let $F$ be the inclusion $\Delta \hookrightarrow Cat$ , which takes a finite nonempty totally ordered set $[n] = \{0, \ldots, n\}$ and realizes it as a category in the usual way. Then $N_F: Cat \to [\Delta^{op}, Set]$ sends a small category $D$ to its nerve, which is the simplicial set whose $n$ -simplices are paths in $D$ of length $n$ .
Let $F$ be the functor $\Delta \to Top$ that sends $[n]$ to the topological $n$ -simplex $\Delta^n$ . Then $N_F: Top \to [\Delta^{op}, Set]$ sends a topological space $X$ to its “singular simplicial set”, whose $n$ -simplices are the continuous maps $\Delta^n \to X$ .
Let $F: A \hookrightarrow B$ be an inclusion of partially ordered sets, viewed as categories in the usual way. Then $N_F : B \to [A^{op}, Set]$ is given by

$(N_F(b))(a) = \begin{cases} 1 &\text{if}   b \leq a \\ 0 &\text{otherwise}. \end{cases}$

For example, let $B$ be the set $\mathbb{Z}$ of integers, ordered by divisibility. Let $A \subseteq \mathbb{Z}$ be the set of (positive) primes. Then $N_F$ is the functor $\mathbb{Z} \to Set^{\{primes\}}$ that sends $n \in \mathbb{Z}$ to

$([p | n])_{\text{primes}   p}$

where $[p|n]$ is Iverson notation: it’s $1$ if $p$ divides $n$ and $0$ otherwise.
If $F$ has a right adjoint $G$ , then $N_F(b) = A(-, G b): A^{op} \to V$ . In particular, $N_F(b)$ is representable for each $b \in B$ .
Take the inclusion $Field \hookrightarrow Ring$ , where $Ring$ is the category of commutative rings. Now take opposites throughout to get $F: Field^{op} \hookrightarrow Ring^{op}$ . The resulting nerve functor

$N_F: Ring^{op} \to [Field, Set]$

sends a ring $R$ to $Ring(R, -)|_{Field}$ (by which I mean the restriction of $Ring(R, -): Ring \to Set$ to the category of fields). More explicitly,

$N_F(R) = \sum_{p \in Spec(R)} Field(Frac(R/p), -).$

Here $Spec(R)$ is the set of prime ideals $p$ of $R$ , and $Frac(R/p)$ is the field of fractions of the integral domain $R/p$ . In particular, $N_F(b)$ is a coproduct of representables for each $b \in B$ . (In the terminology of Diers, one therefore says that $Field \hookrightarrow Ring$ is a “right multi-adjoint”. A closely related statement is that it’s a parametric right adjoint. But that’s all just jargon that won’t matter for this post.)

Now let’s consider $V = (\mathbb{R}^+, \geq)$ with its additive monoidal structure, so that $V$ -categories are generalized metric spaces and $V$ -functors are the functions $f$ between metric spaces that are variously called short, distance-decreasing, contractive, or 1-Lipschitz: $d(f(a), f(a')) \leq d(a, a')$ for all $a, a'$ . I’ll just call them “maps” of metric spaces.

Take a map $f: A \to B$ of metric spaces. The induced map $N_f: B \to [A^{op}, \mathbb{R}^+]$ is given by

$(N_f(b))(a) = d(f(a), b).$

Unlike in the first post, I’m not going to assume here that our metric spaces are symmetric. so the “op” on the $A$ matters, and $d(a, f(b)) \neq d(f(b), a)$ in general.

We’ll be particularly interested in the case where the $V$ -functor $F$ is full and faithful. Most of the examples above have this property. In the metric case, $V = \mathbb{R}^+$ , being full and faithful means being distance-preserving, or in other words, an inclusion of a metric subspace. In that case, it’s normal to drop the $f$ . So we’d usually then write

$(N_f(b))(a) = d(a, b).$

Next I’ll define the potential function of a $V$ -functor. For this we need some of the apparatus of magnitude. Roughly speaking, this means we have a notion of the “size” of each object of our base monoidal category $V$ .

More exactly, we take a field $k$ and a function $|\cdot|: ob(V) \to k$ , to be thought of as assigning to each object $S$ of $V$ its size ${|S|} \in k$ . And to make everything work, we assume our size function $|\cdot|$ has reasonable properties:

$S \cong T \implies |S| = |T|, \quad |S \otimes T| = |S| \cdot |T|, \quad |I| = 1,$

where $\otimes$ is the monoidal product on $V$ and $I$ is its unit.

The basic definitions are these. A weighting on a finite $V$ -category $A$ is a function $w_A : ob(A) \to k$ such that

$\sum_{b \in A} |A(a, b)| w_A(b) = 1$

for all $a \in A$ , and a coweighting $w^A$ on $A$ is a weighting on $A^{op}$ . Although it’s not true that every finite $V$ -category has a unique weighting, it’s a harmless enough assumption that I’ll make it here. The magnitude of a finite $V$ -category $A$ is

$|A| = \sum_{a \in A} w_A(a) = \sum_{a \in A} w^A(a) \in k.$

It’s a tiny lemma that the total weight $\sum_{a \in A} w_A(a)$ is equal to the total coweight $\sum_{a \in A} w^A(a)$ , so you can equivalently define the magnitude to be one or the other.

There’s a whole lot to say about why this is a worthwhile definition, and I’ve said it in detail many times here before. But here I’ll just say two short things:

Taking $V = FinSet$ , $k = \mathbb{Q}$ , and $|\cdot|$ to be cardinality, the magnitude of a finite category is equal to the Euler characteristic of its classifying space, under suitable hypotheses guaranteeing that the latter is defined.
Take $V = \mathbb{R}^+$ , $k = \mathbb{R}$ , and $|\cdot|$ to be $s \mapsto e^{-s}$ (because that’s just about the only function converting the tensor product of $V$ into addition, as our size functions are contractually obliged to do). Then we get a notion of the magnitude of a metric space, a real-valued invariant that turns out to be extremely interesting.

That’s the magnitude of enriched categories. But we can also talk about the magnitude of enriched presheaves. Take a finite $V$ -category $A$ and a $V$ -presheaf $X: A^{op} \to V$ . Its magnitude is

$|X| = \sum_{a \in A} w^A(a) |X(a)| \in k.$

When I wrote about the magnitude of functors before, I concentrated on covariant functors $A \to V$ , which meant that the weights in the weighted sum that defines $|X|$ were, well, the weights $w_A(a)$ . But since we’ve now changed $A$ to $A^{op}$ , the weights have become coweights.

Let me briefly recall why the magnitude of presheaves is interesting, at least in the case $V = FinSet$ :

If our presheaf $X$ is a coproduct of representables then $|X|$ is $|colim(X)|$ , the cardinality of the colimit of $X$ .
The magnitude of presheaves generalizes the magnitude of categories. The magnitude of a category $A$ is equal to the magnitude of the presheaf $A^{op} \to Set$ with constant value $1$ .
In the other direction, the magnitude of categories generalizes the magnitude of presheaves: $|X| = |\mathbb{E}(X)|$ for all presheaves $X$ . Here $\mathbb{E}(X)$ means the category of elements of $X$ , also called the (domain of the) Grothendieck construction.

It’s enlightening to think of this result as follows. If we extend the codomain of $X$ from $Set$ to $Cat$ then $\mathbb{E}(X)$ is the colax colimit of $X: A^{op} \to Cat$ . So $|X| = |colaxcolim(X)|$ . You can compare this to the more limited result that $|X| = |colim(X)|$ , which only holds when $X$ has the special property that it’s a coproduct of representables.

There’s also another closely related result, due to Ponto and Shulman: under mild further hypotheses, $|X| = \chi(hocolim(X))$ . Mike and I had a little discussion about the relationship here.

Now here’s something funny. Which do you think is a more refined invariant of a presheaf $X: A^{op} \to FinSet$ : the magnitude $|X|$ , which is equal to $|colaxcolim(X)|$ and $\chi(hocolim(X))$ , or the cardinality $|colim(X)|$ of the strict colimit?

From one perspective, it’s the magnitude. After all, we usually think of lax or pseudo 2-dimensional things as more subtle and revealing than strict 1-dimensional things. But from another, it’s the cardinality of the strict colimit. For instance, if $A$ is the one-object category corresponding to a group $G$ then $X$ is a right $G$ -set, the cardinality of the colimit is the number of orbits (an interesting quantity), but the magnitude is always just the cardinality of the set $X$ divided by the order of $G$ (relatively boring and crude).

In this post I’ll keep comparing these two quantities. Whichever seems “more refined”, the comparison is interesting in itself.

(I stuck to the case $V = FinSet$ here, because for an arbitrary $V$ , there isn’t a notion of “colimit” as such: we’d need to think about weighted colimits. It’s also harder to say what the category of elements or homotopy colimit of an enriched presheaf should be.)

Now here’s something new. The potential function of a $V$ -functor $F: A \to B$ is the function

$h_F: \ob(B) \to k$

defined by

$h_F(b) = |N_F(b)|$

for all $b \in B$ . Expanding out the definitions, this means that

$h_F(b) = |A(F-, b)| = \sum_{a \in A} w^A(a) |B(F a, b)|,$

where $w^A$ is the notation I introduced for the coweighting on $A$ . If $F$ is full and faithful then $h_F \equiv 1$ on the set of objects in the image of $F$ .

Examples The definition I just gave implicitly assumes that $\ob(A)$ is finite. But I’ll relax that assumption in some of these examples, at the cost of some things getting less than rigorous.

Again, I’ll begin with some examples where $V = FinSet$ .

What’s the potential function of the inclusion $\Delta \hookrightarrow Cat$ ? To make sense of this, we need a coweighting on $\Delta$ , and I’m pretty sure there isn’t one. So let’s abandon $\Delta$ and instead use its subcategory $\Delta_{inj}$ , consisting of all the finite nonempty totally ordered sets $[n] = \{0, \ldots, n\}$ but only the injective order-preserving maps between them. This amounts to considering only face maps, not degeneracies.

The coweighting on $\Delta_{inj}$ is $[n] \mapsto (-1)^n$ . So, the potential function $h$ of $\Delta_{inj} \hookrightarrow Cat$ is given, on a category $D$ , by

$h(D) = \sum_{n \geq 0} (-1)^n |Cat([n], D)|.$

Here $Cat([n], D)$ is the set of paths of length $n$ in $D$ , which we often write as $D_n$ . So

$h(D) = \sum_{n \geq 0} (-1)^n |D_n|.$

Even if the category $D$ is finite, this is a divergent series. But morally, $h(D)$ is the alternating sum of the number of $n$ -cells of $D$ .

In other words, the potential function of $\Delta_{inj} \to Cat$ is morally Euler characteristic.

As I hinted above, it’s interesting to compare the potential function with what you get if you take the cardinality of the colimit of $B(F-, b)$ instead of its magnitude. I’ll call this the “strict potential function”. In this case, it’s

$D \mapsto |colim_{[n] \in \Delta_{inj}} D_n| = |\pi_0(D)|$

— the cardinality of the set of connected-components of $D$ . So while the potential function gives Euler characteristic, the strict potential function gives the number of connected components.
Similarly, the potential function of the usual functor $\Delta_{inj} \hookrightarrow Top$ , sending $n$ to $\Delta^n$ , is given by

$D \mapsto \sum_{n \geq 0} (-1)^n |Top(\Delta^n, D)|$

for topological spaces $D$ . Again, this formally looks like Euler characteristic: the alternating sum of the number of cells in each dimension. And much as for categories, the strict potential function gives the number of path-components.
For the inclusion $\{primes\} \hookrightarrow (\mathbb{Z}, |)$ that we saw earlier, the potential function $h: \mathbb{Z} \to \mathbb{Q}$ is what’s often denoted by $\omega$ , sending an integer $n$ to the number of distinct prime factors of $n$ . The strict potential function is the same.
An easy example that works for any $V$ : if $F: A \to B$ has a right adjoint then since $A(F-, b)$ is representable, its magnitude is equal to the cardinality of its colimit, which is $1$ . So $h_F: \ob(B) \to k$ has constant value $1$ , as does the strict potential function.

More generally, if $A(F-, b)$ is a coproduct of $n$ representables then $h_F(b) = n$ , and the same is true for the strict potential function.
For the inclusion $F: Field^{op} \hookrightarrow Ring^{op}$ , we saw that

$N_F(R) = \sum_{p \in Spec(R)} Field(Frac(R/p), -)$

for every ring $R$ . This is a coproduct of representables, so by the previous example, $|N_F(R)| = |Spec(R)|$ . That is, the potential function is given by

$h_F: R \mapsto |Spec(R)|$

(as is the strict potential function). Usually $Spec(R)$ is infinite, so this calculation is dodgy, but if we restrict ourselves to finite fields and rings then everything is above board.
For an opfibration $F: A \to B$ , one can show that the potential function $h_F$ is given by

$h_F(b) = |F^{-1}(b)|.$

I should explain the notation on the right-hand side. The category $F^{-1}(b)$ is the fibre over $b \in B$ , consisting of the objects of $A$ that $F$ sends to $b$ and the maps in $A$ that $F$ sends to $1_b$ . We can take its magnitude (at least if it’s finite and lightning doesn’t strike), which is what I’ve denoted by $|F^{-1}(b)|$ .

I won’t include the proof of this result, but I want to emphasize that it doesn’t involve finding a coweighting on $A$ . You might think it would, because the definition of $h_F(b)$ involves the coweighting on $A$ . But it turns out that it’s enough to just assume it exists.
Now take $V = \mathbb{R}^+$ , so that $V$ -categories are generalized metric spaces. The potential function of a map $f: A \to B$ of metric spaces is the function $h_f : B \to \mathbb{R}$ given by

$h_f(b) = \sum_{a \in A} w^A(a) e^{-d(f(a), b)}.$

In particular, if $f$ is the inclusion of a subspace $A \subseteq B$ , then it’s natural to write $h_A$ instead of $h_f$ , and we have

$h_A(b) = \sum_{a \in A} w^A(a) e^{-d(a, b)}.$

This is equal to $1$ on $A$ but can take other values on $B \setminus A$ . In suitable infinite contexts, sums become integrals and $w^A$ becomes something like a measure or distribution, in which case the formula becomes

$h_A(b) = \int_A e^{-d(a, b)} \, d w^A(a).$

This is exactly the formula for the potential function in Part 1, with one difference: there, I used weightings on $A$ , and here, I’m using coweightings. It’s coweightings that one should use. In the previous post, I assumed that all metrics were symmetric, which means that weightings and coweightings are the same. So there’s no inconsistency.

(Of course, one could take duals throughout and use weightings on $A$ instead. But we’ll see that whichever choice you make, you end up having to consider weightings on one of $A$ and $B$ and coweightings on the other.)

What about the strict potential function, sending a point $b \in B$ to the “cardinality of the colimit of $B(f-, b): A \to \mathbb{R}^+$ ”? Well, I put that phrase in inverted commas because we’re now in an enriched context, so it needs a bit of interpretation. “Cardinality” is okay: it becomes the size function $|\cdot| = e^{-(\cdot)}: \mathbb{R}^+ \to \mathbb{R}$ . “Colimit” wouldn’t usually make sense in an enriched world, but we’re saved by the fact that the monoidal unit of $V = \mathbb{R}^+$ (namely, $0$ ) is terminal. The colimit of a $V$ -functor into $V$ is just its infimum. So the strict potential function of a map $f: A \to B$ of metric spaces is just

$b \mapsto e^{-\inf_{a \in A} d(f(a), b)} = e^{-d(f A, B)}.$

I showed you an example of the difference between the potential function and the strict potential function in Part 1, although not with those words. If we take $f$ to be the inclusion $\{-1, 1\} \hookrightarrow \mathbb{R}$ then the potential function is

whereas the strict potential function is

These two functions are identical on $(-\infty, -1] \cup [1, \infty)$ , but different on the interval $(-1, 1)$ . If you’ve ever wondered what the difference is between strict and lax colimits, here’s an example!

By definition, the potential function of an enriched functor $F: A \to B$ is given by

$h_F(b) = \sum_{a \in A} w^A(a) |B(F(a), b)|,$

but a slightly different viewpoint is sometimes helpful. We can take the function $w^A: ob(A) \to k$ and push it forward forward along $F$ to obtain a new function

$\begin{array}{cccc} F_* w^A: &ob(B) &\to &k, \\ &b &\mapsto &\sum_{a \in F^{-1}(b)} w^A(a). \end{array}$

Really it’s best to think of $w^A$ as not a function but a measure taking values in $k$ (although these are the same thing on a finite set). Then this is just the usual pushforward measure construction. In any case, the formula for the potential function now becomes

$h_F(b) = \sum_{b' \in B} (F_* w^A)(b') |B(b', b)|,$

which has the advantage that everything is taking place in $B$ rather than $A$ . In the case where $f: A \to B$ is an embedding of a subspace of a metric space, $f_* w^A$ is just the extension of the measure $w^A$ on $A$ to all of $B$ , which we’d usually just write as $w^A$ (with a wink to the audience). In integral notation, the last formula becomes

$h_F(b) = \int_B e^{-d(x, b)} \, d w^A(x),$

which is what we had in the last post.

I’ve now explained what potential functions are. But what are they good for?

Last time, I explained that they’re very good indeed for helping us to calculate the magnitude of metric spaces. The key was that

$|A| = \int_B h_A \, d w_B$

for a subspace $A$ of a metric space $B$ . And as I recounted, that key unlocks the door to the world of PDE methods.

So you might hope something similar is true for enriched categories in general: that there’s a formula for the magnitude in terms of the potential function. And there is! For any $V$ -functor $F: A \to B$ , it’s a theorem that

$|A| = \sum_{b \in B} h_F(b) w_B(b).$

This is an easy calculation from the definitions.

(If you’re really paying attention, you’ll notice that we used the coweighting on $A$ to define the potential function, and now we’re using the weighting on $B$ . That’s just how it turns out. One is a weighting, and the other is a coweighting.)

Examples

For a $V$ -functor $F: A \to B$ that has a right adjoint, we saw that the potential function $h_F$ has constant value $1$ , so this formula tells us that

$|A| = \sum_{b \in B} w_B(b).$

But the right-hand side is by definition the magnitude of $B$ , so what this formula is saying is that

$|A| = |B|.$

In other words, if there’s an adjunction between two categories, their magnitudes are equal!

This has been known forever (Proposition 2.4), and is also intuitive from a homotopical viewpoint. But it’s nice that it just pops out.
Less trivially, we saw above that for an opfibration $F: A \to B$ , the potential function is $h_F: b \mapsto |F^{-1}(b)|$ . So

$|A| = \sum_{b \in B} w_B(b) |F^{-1}(b)|.$

In other words, the magnitude of the “total” category $A$ is the weighted sum over the base $B$ of the magnitudes of the fibres. (Special case: the magnitude of a product is the product of the magnitudes.)

Now this has been known forever too (Proposition 2.8), but I want to emphasize that the proof is fundamentally different from the one I just linked. That proof constructs the weighting on $A$ from the weightings on the base $B$ and the fibres. Now that’s an easy and informative proof, but what we’ve just done is different, because it didn’t involve figuring out the weighting or coweighting on $A$ . So although the result isn’t new or difficult, it’s perhaps grounds for optimism that the method of potential functions will let us prove new things about enriched categories other than metric spaces.

What new things? I don’t know! This is where I’ve got to now. Maybe there are applications in the metric world in which $f: A \to B$ is nothing like an inclusion. Maybe there are applications to graphs, replacing the PDE methods used for subspaces of $\mathbb{R}^n$ by discrete analogues. Maybe the potential function method can be used to shed light on the tricky result that the magnitude of graphs is invariant under certain Whitney twists (Theorem 5.2), and more generally under the sycamore twists introduced by Emily Roff (Theorem 6.5). Let’s find out!

n-Category Café Potential Functions and the Magnitude of Functors 1

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Next: Part 2

In the beginning, there were hardly any spaces whose magnitude we knew. Line segments were about the best we could do. Then Mark Meckes introduced the technique of potential functions for calculating magnitude, which was shown to be very powerful. For instance, Juan Antonio Barceló and Tony Carbery used it to compute the magnitude of odd-dimensional Euclidean balls, which turn out to be rational functions of the radius. Using potential functions allows you to tap into the vast repository of knowledge of PDEs.

In this post and the next, I’ll explain this technique from a categorical viewpoint, saying almost nothing about the analytic details. This is category theory as an organizational tool, used to help us understand how the various ideas fit together. Specifically, I’ll explain potential functions in terms of the magnitude of functors, which I wrote about here a few weeks ago.

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Before I can describe this categorical viewpoint on potential functions, I have to explain what potential functions are in the magnitude context, and why they’re very useful. That’s what I’ll do today.

This part of the story is about metric spaces. For now I’ll assume they satisfy all the classical axioms, including symmetry of the metric, meaning that $d(a, a') = d(a', a)$ for all points $a$ and $a'$ . When metric spaces are viewed as enriched categories, symmetry isn’t automatic — but we’ll come to that next time.

A weighting on a finite metric space $A$ is a function $w: A \to \mathbb{R}$ such that for all $a \in A$ ,

$\sum_{a' \in A} e^{-d(a, a')} w(a') = 1.$

Everyone who sees this formula for the first time asks where the exponential comes from. Ultimately it’s because of the enriched category viewpoint (which again we’ll come to next time), but the short story is that exponential is essentially the only reasonable function that converts addition into multiplication.

For simplicity, I’ll assume here that every finite metric space $A$ has a unique weighting, which I’ll call $w_A$ . Since the definition of weighting involves the same number of equations as unknowns, this is generically true (and it’s always true for subspaces of $\mathbb{R}^n$ ), even though there are exceptions.

The magnitude of $A$ is

$|A| = \sum_{a \in A} w_A(a) \in \mathbb{R}.$

That’s for finite metric spaces. To extend the definition to compact metric spaces, there are various ideas you might try. You could define the magnitude of a compact space as the supremum of all the magnitudes of its finite subspaces. Or, you could take an ever-denser sequence of finite subsets of your compact space, then define its magnitude to be the limit of the magnitudes of the approximating subsets. Or, you could try replacing the sums in the formulas above by integrals, somehow.

Mark Meckes showed that all these approaches are equivalent. They all give the same definition of the magnitude of a compact space. (At least, this is true subject to a condition called “positive definiteness” which I won’t discuss and which always holds for subspaces of $\mathbb{R}^n$ .)

How do we actually calculate magnitude, say for compact subspaces $A$ of $\mathbb{R}^n$ ?

In principle, you already know how do it. You run through all the finite subsets of $A$ , calculate the magnitude of each using the definition above, and take the sup. The trouble is, this procedure is incredibly hard to work with. It’s completely impractical.

A slightly more practical approach is to look for a “weight measure”, that is, a signed measure $w$ on $A$ such that for all $a \in A$ ,

$\int_A e^{-d(a, a')}\, d w(a') = 1.$

As Mark showed, if $w$ is a weight measure then the magnitude of $A$ is given by $|A| = w(A)$ . This is the analogue of the formula $|A| = \sum_a w(a)$ for finite spaces.

Example Take an interval $[c, d]$ in $\mathbb{R}$ . It’s an elementary exercise to check that

$\tfrac{1}{2}(\delta_c + \lambda_{[c, d]} + \delta_d)$

is a weight measure on $[c, d]$ , where $\delta_c$ and $\delta_d$ denote the Dirac deltas at $c$ and $d$ , and $\lambda_{[c, d]}$ is Lebesgue measure on $[c, d]$ . It follows that $[c, d]$ is the total mass of this measure, which is

$1 + \tfrac{1}{2}(d - c).$

In other words, it’s $1$ plus half the length of the interval.

The trouble with weight measures is that very few spaces have them. Even Euclidean balls don’t, beyond dimension one. It turns out that we need something more general than a measure, something more like a distribution.

In another paper, Mark worked out exactly what kind of measure-like or distribution-like thing a weighting should be. The answer is very nice, but I won’t explain it here, because I want to highlight the formal aspects above all.

So I’ll simply write the weighting on a metric space $A$ as $w_A$ , without worrying too much about what kind of thing $w_A$ actually is. All that matters is that it behaves something like a measure: we can pair it with “nice enough” functions $\phi: A \to \mathbb{R}$ to get a real number, which I’ll write as

$\int_A \phi(a) \, d w_A(a)$

or simply

$\int_A \phi \, d w_A.$

And I’ll assume that all the spaces $A$ that we discuss do have a unique weighting $w_A$ .

That’s the background. But back to the question: how do we actually calculate magnitude?

First main idea Don’t look at metric spaces $A$ in isolation. Instead, consider spaces embedded in some big space $B$ that we understand well.

Think of the big space $B$ as fixed. The typical choice is $\mathbb{R}^n$ . One sense in which we “understand” $\mathbb{R}^n$ well is that we have a weight measure on it: it’s just Lebesgue measure divided by a constant factor $c_n$ . This follows easily from the fact that $\mathbb{R}^n$ is homogeneous. (It doesn’t matter for anything I’m going to say, but the constant $c_n$ is $\int_{\mathbb{R}^n} e^{-\|x\|}\, d x$ , for which there’s a standard formula involving $\pi$ s and factorials.)

The potential function of a subspace $A \subseteq B$ is the function $h_A: B \to \mathbb{R}$ defined by

$h_A(b) = \int_A e^{-d(a, b)} \, d w_A(a).$

By definition of weighting, $h_A$ has constant value $1$ on $A$ . But outside $A$ , it could be anything, and it turns out that its behaviour outside $A$ is very informative.

Although $w_A$ is a measure-like thing on the subspace $A$ of $B$ , we typically extend it by $0$ to all of $B$ , and then the definition of the potential function becomes

$h_A(b) = \int_B e^{-d(x, b)} \, d w_A(x)$

Examples In all these examples, I’ll take the big space $B$ to be $\mathbb{R}$ .

Let $A = \{0\}$ . Then the weighting $w_A$ on $A$ is the Dirac delta at $0$ , and the potential function $h_A: \mathbb{R} \to \mathbb{R}$ is given by

$h(b) = e^{-|b|}$

( $b \in \mathbb{R}$ ), whose graph looks like this:
Let $A = [-2, 2]$ . As we’ve already seen,

$w_A = \tfrac{1}{2}(\delta_{-2} + \lambda_{[-2, 2]} + \delta_2),$

and a little elementary work then reveals that the potential function $h_A: \mathbb{R} \to \mathbb{R}$ is given by

$h_A(b) = \begin{cases} e^{-(-2 - b)} &\text{if}   b \leq -2, \\ 1 &\text{if}   -2 \leq b \leq 2, \\ e^{-(b - 2)} &\text{if}   b \geq 2, \end{cases}$

which looks like this:
In the two examples we just did, the potential function of $A$ is just the negative exponential of the distance between $A$ and the argument $b$ . That’s not exactly coincidence: as we’ll see in the next part, the function just described corresponds to a strict colimit, whereas the potential function corresponds to a lax colimit. So it’s not surprising that they coincide in simple cases.

But this only happens in the very simplest cases. It doesn’t happen for Euclidean balls above dimension $1$ , or even for two-point spaces. For example, taking the subset $A = \{-1, 1\}$ of $B = \mathbb{R}$ , we have

$h_A(x) = \frac{e^{-|x + 1|} + e^{-|x - 1|}}{1 + e^{-2}},$

which looks like this:

whereas $b \mapsto e^{-d(A, b)}$ looks like this:

Similar, but different!

And now we come to the:

Second main idea Work with potential functions instead of weightings.

To explain what this means, I need to tell you three good things about potential functions.

The potential function determines the magnitude:

$|A| = \int_B h_A \, d w_B.$

At the formal level, proving this is a one-line calculation: substitute the definition of $h_A$ into the right-hand side and follow your nose.

For example, we saw that $\mathbb{R}^n$ has weighting $\lambda/c_n$ , where $\lambda$ is Lebesgue measure and $c_n$ is a known constant. So

$|A| = \frac{1}{c_n} \int_{\mathbb{R}^n} h_A(x) \, d x$

for compact $A \subseteq \mathbb{R}^n$ . Here $d x$ refers to ordinary Lebesgue integration.

(For $\mathbb{R}^n$ , and in fact a bit more generally, this result appears as Theorem 4.16 here.)
You can recover the weighting from the potential function. So, you don’t lose any information by working with one rather than the other.

How do you recover it? Maybe it’s easiest to explain in the case when the spaces are finite. If we write $Z_B$ for the $B \times B$ matrix $(e^{-d(b', b)})_{b', b \in B}$ then the definition of the potential function can be expressed very succinctly:

$h_A = Z_B w_A.$

Here $h_A$ and $w_A$ are viewed as column vectors with entries indexed by the points of $B$ . (For $w_A$ , those entries are $0$ for points not in $A$ .) Assuming $Z_B$ is invertible, this means we recover $w_A$ from $h_A$ as $Z_B^{-1} h_A$ . And something similar is true in the non-finite setting.

However, what really makes the technique of potential functions sing is that when $B = \mathbb{R}^n$ , there’s a much more explicit way to recover the weighting from the potential function:

$w_A = (I - \Delta)^{(n + 1)/2} h_A$

(up to a constant factor that I’ll ignore). Here $I$ is the identity and $\Delta$ is the Laplace operator, $\sum_{i = 1}^n \frac{\partial^2}{\partial x_i^2}$ . This is Proposition 5.9 of Mark’s paper.

How much more of this bullet point you’ll want to read depends on how interested you are in the analysis. The fundamental point is simply that $(I - \Delta)^{(n + 1)/2}$ is some kind of differential operator. But for those who want a bit more:
- To make sense of everything, you need to interpret it all in a distributional sense. In particular, this allows one to make sense of the power $(n + 1)/2$ , which is not an integer if $n$ is even.
- Maybe you wondered why someone might have proved a result on the magnitude of odd-dimensional Euclidean balls only, as I mentioned at the start of the post. What could cause the odd- and even-dimensional cases to become separated? It’s because whether or not $(n + 1)/2$ is an integer depends on whether $n$ is odd or even. When $n$ is odd, it’s an integer, which makes $(I - \Delta)^{(n + 1)/2}$ a differential rather than pseudodifferential operator. Heiko Gimperlein, Magnus Goffeng and Nikoletta Louca later worked out lots about the even-dimensional case, but I won’t talk about that here.
- Finally, where does the operator $(I - \Delta)^{(n + 1)/2}$ come from? Sadly, I don’t have an intuitive explanation. Ultimately it comes down to the fact that the Fourier transform of $e^{-\|\cdot\|}$ is $\xi \mapsto (1 + \|\xi\|)^{-(n + 1)/2}$ (up to a constant). But that itself is a calculation that’s really quite tricky (for me), and it’s hard to see anything beyond “it is what it is”.
The third good thing about potential functions is that they satisfy a differential equation, in the situation where our big space $B$ is $\mathbb{R}^n$ . Specifically:

$\begin{cases} h_A \equiv 1 &\text{on}   A \\ (I - \Delta)^{(n + 1)/2} h_A \equiv 0&\text{on}   \mathbb{R}^n \setminus A. \end{cases}$

Indeed, the definition of weighting implies that $h_A \equiv 1$ on $A$ , and the “second good thing” together with the fact that $w_A$ is supported on $A$ give the second clause.

Not only do we have a differential equation for $h_A$ , we also have boundary conditions. There are boundary conditions at the boundary of $A$ , because of something I’ve been entirely vague about: the functions we’re dealing with are meant to be suitably smooth. There are also boundary conditions at $\infty$ , because our functions are also meant to decay suitably fast.

Maybe Mark will read this and correct me if I’m wrong, but I believe there are exactly the right number of boundary conditions to guarantee that there’s (typically? always?) a unique solution. In any case, the following example — also taken from Mark’s paper — illustrates the situation.

Example Let’s calculate the magnitude of a real interval $A = [c, d]$ using the potential function method.

Its potential function $h_A$ is a function $\mathbb{R} \to \mathbb{R}$ such that $h_A \equiv 1$ on $[c, d]$ and $h_A - h''_A = 0$ on the rest of the real line. That differential equation comes from taking $(I - \Delta)^{-(n + 1)/2}$ in the case $n = 1$ .

The functions $f$ satisfying $f'' = f$ are those of the form $k e^{\pm x}$ for some constant $k$ , and in our case we’re free to choose the constant and the $\pm$ sign differently on the two connected components of $\mathbb{R} \setminus A$ . So there are lots of solutions. But $h_A$ is required to be continuous and to converge to $0$ at $\pm \infty$ , and that pins it down uniquely: the one and only solution is

$h_A(x) = \begin{cases} e^{-(c - x)} &\text{if}   x \leq c, \\ 1 &\text{if}   c \leq x \leq d, \\ e^{-(x - d)} &\text{if}   d \leq x. \end{cases}$

I showed you the graph of this potential function above, in the case where $A = [-2, 2]$ .

So the magnitude of $A = [c, d]$ is

$\frac{1}{c_1} \int_\mathbb{R} h_A(x)\, d x,$

where $c_1$ is the constant $\int_\mathbb{R} e^{-|x|} \, d x = 2$ . This gives the answer:

$|A| = 1 + \tfrac{1}{2}(d - c) = 1 + \tfrac{1}{2}length(A).$

The crucial point is that in this example, we didn’t have to come up with a weighting on $A$ . The procedure was quite mechanical. And that’s the attraction of the method of potential functions.

Next time, I’ll put all this into a categorical context using the notion of the magnitude of a functor, which I introduced here.

John Baez — Stone–Wales Transformations

Buckminsterfullerene is a molecule shaped like a soccer ball, made of 60 carbon atoms. If one of the bonds between two hexagons rotates, we get a weird mutant version of this molecule:

This is an example of a Stone-Wales transformation: a 90° rotation in a so-called ‘π bond’ between carbon atoms. Here’s how it works in graphene:

Graphene is a sheet of carbon molecules arranged in hexagons. When they undergo a Stone–Wales transformation, we get a Stone–Wales defect with two pentagons and two heptagons, like this:

This picture is from Jacopo Bertolotti, based on a picture by Torbjörn Björkman which appeared in this paper:

• Torbjörn Björkman, Simon Kurasch, Ossi Lehtinen, Jani Kotakoski, Oleg V. Yazyev, Anchal Srivastava, Viera Skakalova, Jurgen H. Smet, Ute Kaiser and Arkady V. Krasheninnikov, Defects in bilayer silica and graphene: common trends in diverse hexagonal two-dimensional systems, Scientific Reports 3 (2013), 3482.

This paper also shows other interesting defects, and electron microscope pictures of how they actually look in graphene and hexagonal bilayer silica.

You’ll notice that in buckminsterfullerne the Stone–Wales transformation turned 2 hexagons and 2 pentagons into two pentagons and 2 hexagons, while in graphene it turned 4 hexagons into 2 pentagons and 2 heptagons. The general pattern is this:

I think it’s cool how a simple topological transformation akin to the Pachner moves shows up in chemistry!

I got the above image from here:

• Wei-Wei Wang, Jing-Shuang Dang, Jia Zheng, Xiang Zhao, Shigeru Nagase, Selective growth of fullerenes from C₆₀ to C₇₀: inherent geometrical connectivity hidden in discrete experimental evidence, The Journal of Physical Chemistry C 117 (2013), 2349–2357.

I got the image of a Stone–Wales transformation in buckminsterfullerene from here:

• L. A. Openov and Mikhail Maslov, On the vineyard formula for the pre-exponential factor in the Arrhenius law, Physics of the Solid State 56 6 (2014), 1239–1244.

The Arrhenius equation is a simple rule of thumb for the rates of chemical reactions. A lot of statistical mechanics gives simple laws for equilibrium behavior. For example, a state of energy $E$ will show up with probability proportional to $\exp(-E/kT)$ in equilibrium at temperature $T,$ where $k$ is Boltzmann’s constant. But dynamics is much harder, so the Arrhenius equation for the rates of transitions between states is precious, even though only approximate. I would like to understand this law better, and its range of approximate validity. The above paper digs into that. As an example, it studies the rate at which Stone–Wales transformations happen in buckminsterfullerene!

There should be a nice theory of topological transformations like Stone–Wales transformations or Pachner moves occurring randomly, following the laws of statistical mechanics. I guess the study of matrix models moves in this direction, but there’s no Arrhenius equation there, I don’t think!

by John Baez at July 12, 2025 01:16 PM

July 11, 2025

n-Category Café Category Theory 2025

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Guest post by John Bourke.

The next International Category Theory Conference CT2025 will take place at Masaryk University (Brno, Czech Republic) from Sunday, July 13 and will end on Saturday, July 19, 2025.

Brno is a beautiful city surrounded by nature with a long tradition in category theory. If you are interested in attending, please read on!

Important dates

April 2: talk submission
April 18: early registration deadline
May 7: notification of speakers
May 23: registration deadline
July 13-19: conference

In addition to 25 minute contributed talks, there will be speed talks replacing poster sessions, and we hope to accommodate as many talks as possible.

The invited speakers are:

Clark Barwick (University of Edinburgh)
Maria Manuel Clementino (University of Coimbra)
Simon Henry (University of Ottawa)
Jean-Simon Lemay (Macquarie University)
Wendy Lowen (University of Antwerp)
Maru Sarazola (University of Minnesota)

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A couple of extra notes:

We recommend that people book accommodation early as possible as there will be other events taking place in town that week, such as the Moto GP. We have booked a limited number of rooms in advance.
In order to promote environmentally friendly travel within Europe, there will be a small prize for the person who travels the furthest from their starting point to the conference by train or bus. Obviously this will not be practical for most attendees, but if you are the sort of person who fancies an adventurous train trip, this could be your chance!

If you would like more details on any aspect of the conference, have a look at the website or send an email to the organizers at ct2025@math.muni.cz

We look forward to seeing as many of you in Brno as possible!

Matt von Hippel — Did the South Pole Telescope Just Rule Out Neutrino Masses? Not Exactly, Followed by My Speculations

Recently, the South Pole Telescope’s SPT-3G collaboration released new measurements of the cosmic microwave background, the leftover light from the formation of the first atoms. By measuring this light, cosmologists can infer the early universe’s “shape”: how it rippled on different scales as it expanded into the universe we know today. They compare this shape to mathematical models, equations and simulations which tie together everything we know about gravity and matter, and try to see what it implies for those models’ biggest unknowns.

Some of the most interesting such unknowns are neutrino masses. We know that neutrinos have mass because they transform as they move, from one type of neutrino to another. Those transformations let physicists measure the differences between neutrino masses, but but themselves, they don’t say what the actual masses are. All we know from particle physics, at this point, is a minimum: in order for the neutrinos to differ in mass enough to transform in the way they do, the total mass of the three flavors of neutrino must be at least 0.06 electron-Volts.

(Divided by the speed of light squared to get the right units, if you’re picky about that sort of thing. Physicists aren’t.)

Neutrinos also influenced the early universe, shaping it in a noticeably different way than heavier particles that bind together into atoms, like electrons and protons, did. That effect, observed in the cosmic microwave background and in the distribution of galaxies in the universe today, lets cosmologists calculate a maximum: if neutrinos are more massive than a certain threshold, they could not have the effects cosmologists observe.

Over time as measurements improved, this maximum has decreased. Now, the South Pole Telescope has added more data to the pool, and combining it with prior measurements…well, I’ll quote their paper:

Ok, it’s probably pretty hard to understand what that means if you’re not a physicist. To explain:

There are two different hypotheses for how neutrino masses work, called “hierarchies”. In the “normal” hierarchy, the neutrinos go in the same order as the particles they interact with with the weak nuclear force: electron-neutrinos are lighter than muon neutrinos, which are lighter than tau neutrinos. In the “inverted” hierarchy, they come in the opposite order, and the electron neutrino is the heaviest. Both of these are consistent with the particle-physics data.
Confidence is a statistics thing, which could take a lot of unpacking to define correctly. To give a short but likely tortured-sounding explanation: when you rule out a hypothesis with a certain confidence level, you’re saying that, if that hypothesis was true, there would only be a 100%-minus-that-chance chance that you would see what you actually observed.

So, what are the folks at the South Pole Telescope saying? They’re saying that if you put all the evidence together (that’s roughly what that pile of acroynms at the beginning means), then the result would be incredibly uncharacteristic for either hypothesis for neutrino masses. If you had “normal” neutrino masses, you’d only see these cosmological observations 2.1% of the time. And if you had inverted neutrino masses instead, you’d only see these observations 0.01% of the time!

That sure makes it sound like neither hypothesis is correct, right? Does it actually mean that?

I mean, it could! But I don’t think so. Here I’ll start speculating on the possibilities, from least likely in my opinion to most likely. This is mostly my bias talking, and shouldn’t be taken too seriously.

5. Neutrinos are actually massless

This one is really unlikely. The evidence from particle physics isn’t just quantitative, but qualitative. I don’t know if it’s possible to write down a model that reproduces the results of neutrino oscillation experiments without massive neutrinos, and if it is it would be a very bizarre model that would almost certainly break something else. This is essentially a non-starter.

4. This is a sign of interesting new physics

I mean, it would be nice, right? I’m sure there are many proposals at this point, tweaks that add a few extra fields with some hard-to-notice effects to explain the inconsistency. I can’t rule this out, and unlike the last point there isn’t anything about it that seems impossible. But we’ve had a lot of odd observations, and so far this hasn’t happened.

3. Someone did statistics wrong

This happens more often. Any argument like this is a statistical argument, and while physicists keep getting better at statistics, they’re not professional statisticians. Sometimes there’s a genuine misunderstanding that goes in to testing a model, and once it gets resolved the problem goes away.

2. The issue will go away with more data

The problem could also just…go away. 97.9% confidence sounds huge…but in physics, the standards are higher: you need 99.99994% to announce a new discovery. Physicists do a lot of experiments and observations, and sometimes, they see weird things! As the measurements get more precise, we may well see the disagreement melt away, and cosmology and particle physics both point to the same range for neutrino masses. It’s happened to many other measurements before.

1. We’re reaching the limits of our current approach to cosmology

This is probably not actually the most likely possibility, but it’s my list, what are you going to do?

There are basic assumptions behind how most theoretical physicists do cosmology. These assumptions are reasonably plausible, and seem to be needed to do anything at all. But they can be relaxed. Our universe looks like it’s homogeneous on the largest scales: the same density on average, in every direction you look. But the way that gets enforced in the mathematical models is very direct, and it may be that a different, more indirect, approach has more flexibility. I’ll probably be writing about this more in future, hopefully somewhere journalistic. But there are some very cool ideas floating around, gradually getting fleshed out more and more. It may be that the answer to many of the mysteries of cosmology right now is not new physics, but new mathematics: a new approach to modeling the universe.

by 4gravitons at July 11, 2025 04:00 PM

Justin Wilson — Two Dimensional Materials have gone crazy!

There are a ton of two-dimensional materials these days. You’ve probably heard of graphene, a single layer of carbon atoms arranged in a hexagonal grid.

a close up of a woven surface — In graphene, carbon atoms sit at the vertices of these hexagons. Photo by Andrew Draper on Unsplash

In 2018, everything changed when two layers of graphene were twisted to reveal superconductivity! The twist itself is interesting (I briefly discussed it in a previous post), but the key takeaway is that these materials now come with an extra knob for accessing new phases of matter. It’s remarkable. We can first think of these materials like Lego blocks:

blue, red, and white artwork — Photo by Omar Flores on Unsplash

Each layer is a different material: mix and match, and you might discover an exotic new phase. This “Lego” idea had already been in the air before 2018, but the physics since then has shown that it’s not just about stacking—we can twist too, creating not just patterns, but new ways for electrons to move.

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Two hexagonal layers twisted on top of each other, creating a moiré pattern.

We knew these patterns would occur, but we didn’t realize we could make it superconduct. Now we can stack and twist to great effect. Of course, twisted bilayer graphene isn’t about to revolutionize high-speed trains (it goes superconducting at only 4K1), but the way it goes superconducting is eerily reminiscent of higher-temperature superconductors. That means it might help us understand those other materials better.

And once people started twisting, they didn’t stop. We now have twisted multilayers of graphene, transition-metal dichalcogenide (TMD) bilayers2, and more. But it doesn’t end there; you can also apply magnetic fields, electric fields, and pattern the lattice in sophisticated ways. With all that in mind, here’s a short and incomplete survey of some of the exotic phases in these materials:

“Fractional… what now?”

All of these phases are exceptionally hard to understand and model. Some of the best minds in the field are actively working on them. One particularly exciting phase is the fractional Chern insulator, which could be useful for quantum computing.

But even setting aside applications, what’s astonishing is that all of these phenomena come from nothing more than electrons moving on a lattice and experiencing a few fields. Nature seems to treat electrons like Play-Doh, shaping them into wildly different quantum phases.

This is a deep and fundamental question: What can be accomplished using electrons alone?

That’s -452.47 degrees Fahrenheit.

To this day, I still can’t say the full name, so I just say “TMD.”

Doug Natelson — US science funding - now time to push on the House appropriators

Some not-actively-discouraging news out of Washington DC yesterday: The Senate appropriations committee is doing its markups of the various funding bills (which all technically originated in the House), and it appears that they have pushed to keep the funding for NASA and NSF (which are bundled in the same bill with the Department of Justice for no obvious reason) at FY24 levels. See here as well.

This is not yet a done deal within the Senate, but it's better than many alternatives. If you are a US citizen or permanent resident and one of your senators is on the appropriations committee, please consider calling them to reinforce how devastating massive budget cuts to these agencies would be. I am told that feedback to any other senators is also valuable, but appropriators are particularly important here.

The House appropriations committee has not yet met to mark up their versions. They had been scheduled to do so earlier this week but punted it for an unknown time. Their relevant subcommittee membership is here. Again, if you are a constituent of one of these representatives, your calls would be particularly important, though it doesn't hurt for anyone to make their views heard to their representative. If the House version aligns with the presidential budget request, then a compromise between the two might still lead to 30% cuts to NSF and NASA, which would (IMO) still be catastrophic for the agencies and US science and competitiveness.

This is a marathon, not a sprint. There are still many looming difficulties - staffing cuts are well underway. Spending of already appropriated funds at agencies like NSF is way down, leading to the possibility that the executive branch may just order (or not-order-but-effectively-order) agencies not to spend and then claw back the funds. This year and in future years they could decide to underspend appropriations knowing that any legal resistance will take years and cost a fortune to work its way through the courts. This appropriations battle is also an annual affair - even if the cuts are forestalled for now (it is unlikely that the executive would veto all the spending bills over science agency cuts), this would have to happen again next year, and so on.

Still, right now, there is an opportunity to push against funding cuts. Failing to try would be a surrender.

(Obligatory notice: yes, I know that there are large-scale budgetary challenges facing the US; I don't think destroying government investment in science and engineering research is an intelligent set of spending cuts.)

by Douglas Natelson at July 11, 2025 01:29 PM

n-Category Café Categorical Linguistics in Quanta

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Quanta magazine has just published a feature on Tai-Danae Bradley and her work, entitled

Where Does Meaning Live in a Sentence? Math Might Tell Us.

The mathematician Tai-Danae Bradley is using category theory to try to understand both human and AI-generated language.

It’s a nicely set up Q&A, with questions like “What’s something category theory lets you see that you can’t otherwise?” and “How do you use category theory to understand language?”

Particularly interesting for me is the part towards the end where Bradley describes her work with Juan Pablo Vigneaux on magnitude of enriched categories of texts.

John Baez — Neanderthal Bone Grease Factory

Today I learned about ‘rabbit starvation’ and how Neanderthals avoided it.

When you’re a hunter-gatherer and it’s winter, you may try to survive by eating only meat—like rabbits, but also deer and other game. But this gives you too much protein and not enough carbohydrates and fat: most of this meat is very lean. If you eat enough lean meat to get all the calories you need, you can die from an overdose of protein! It’s called ‘protein toxicity’.

Hunter-gatherers in this situation sometimes throw away the ‘steaks’ and ‘roasts’—the thighs and shoulders of the animals they kill—or feed them to their dogs. They need fat to survive! So they focus on eating the fatty parts, including bone marrow.

(Now I better understand how we domesticated wolves. In the winter we didn’t just give them scraps. We gave them steaks!)

In some cultures, while the men are out hunting, the women spend time making bone grease. This takes a lot of work. They take bones and break them into small pieces with a stone hammer. They boil them for several hours. The fat floats to the top. Then they let the water cool and skim off the fat.

There’s been evidence for people doing this as far back as 28,000 BC. But now some scientists have found a Neanderthal ‘bone grease factory’ that’s 125,000 years old!

This was during the last interglacial, the Eemian, in Germany. In a site near a lake, called Neumark-Nord, Neanderthals killed a lot of bison, horses and deer and crushed their bones, leaving behind tens of thousands of small bone fragments.

• Lutz Kindler, Sabine Gaudzinski-Windheuser, Fulco Scherjon, Alejandro Garcia-Moreno, Geoff M. Smith, Eduard Pop, John D. Speth, and Wil Roebroeks, Large-scale processing of within-bone nutrients by Neanderthals, 125,000 years ago, Science Advances 11 27 (2025), eadv1257.

by John Baez at July 11, 2025 07:52 AM

David Hogg — likelihood ratios not posteriors, please

There is an informal meeting at MPIA every Wednesday regarding binary stars, with a bit of a focus on massive binaries. Today there was a very nice presentation by Jakob Stegmann (MPA) about some anomalies among the (only six) black-hole–neutron-star binaries discovered by NSF LIGO. He showed the example of GW 200105, which shows a large eccentricity (0.15-ish). This eccentricity is very hard to explain, given how the inspirals evolve as they radiate. But the analysis of the eccentricity (from perhaps this paper) is Bayesian, so it isn't clear how much the eccentricity result is forced by the data and how much is forced by the prior over nuisance parameters. That's one of the main points of my forthcoming paper on measurement. I think maybe I should just re-analyze this one with a profile likelihood. I hope the data and code are public!

by Hogg at July 11, 2025 06:55 AM

July 10, 2025

n-Category Café Position in Stellenbosch

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guest post by Bruce Bartlett

Stellenbosch University is hiring!

$MathML-enabled post (click for more details).$

The Mathematics Division at Stellenbosch University in South Africa is looking to hire a new permanent appointment at Lecturer / Senior Lecturer level (other levels may be considered too under the appropriate circumstances).

Preference will be given to candidates working in number theory or a related area, but those working in other areas of mathematics will definitely also be considered.

The closing date for applications is 30 April 2025. For more details, kindly see the official advertisement.

Consider a wonderful career in the winelands area of South Africa!

Scott Aaronson Trump and Iran, by popular request

I posted this on my Facebook, but several friends asked me to share more widely, so here goes:

I voted against Trump three times, and donated thousands to his opponents. I’d still vote against him today, seeing him as a once-in-a-lifetime threat to American democracy and even to the Enlightenment itself.

But last night I was also grateful to him for overruling the isolationists and even open antisemites in his orbit, striking a blow against the most evil regime on the planet, and making it harder for that regime to build nuclear weapons. I acknowledge that his opponents, who I voted for, would’ve probably settled for a deal that would’ve resulted in Iran eventually getting nuclear weapons, and at any rate getting a flow of money to redirect to Hamas, Hezbollah, and the Houthis.

May last night’s events lead to the downfall of the murderous ayatollah regime altogether, and to the liberation of the Iranian people from 46 years of oppression. To my many, many Iranian friends: I hope all your loved ones stay safe, and I hope your great people soon sees better days. I say this as someone whose wife and 8-year-old son are right now in Tel Aviv, sheltering every night from Iranian missiles.

Fundamentally, I believe not only that evil exists in the world, but that it’s important to calibrate evil on a logarithmic scale. Trump (as I’ve written on this blog for a decade) terrifies me, infuriates me, and embarrasses me, and through his evisceration of American science and universities, has made my life noticeably worse. On the other hand, he won’t hang me from a crane for apostasy, nor will he send a ballistic missile to kill my wife and son and then praise God for delivering them into his hands.

Update: I received the following comment on this post, which filled me with hope, and demonstrated more moral courage than perhaps every other anonymous comment in this blog’s 20-year history combined. To this commenter and their friends and family, I wish safety and eventually, liberation from tyranny.

I will keep my name private for clear reasons. Thank you for your concern for Iranians’ safety and for wishing the mullah regime’s swift collapse. I have fled Tehran and I’m physically safe but mentally, I’m devastated by the war and the internet blackout (the pretext is that Israeli drones are using our internet). Speaking of what the mullahs have done, especially outrageous was the attack on the Weizmann Institute. I hope your wife and son remain safe from the missiles of the regime whose thugs have chased me and my friends in the streets and imprisoned my friends for simple dissent. All’s well that ends well, and I hope this all ends well.

by Scott at July 10, 2025 02:37 PM

July 09, 2025

David Hogg — robust dimensionality reductions

Dimensionality reduction (the basic being PCA) is very sensitive to outliers: A single bad pixel can dominate most objectives and thus create a spurious dimension. One of the best and most classic solutions to this is the robust PCA method, which is presented in a (very long) paper with impressive math and beautiful results. Yesterday Hans-Walter Rix (MPIA) and I coded it up and applied it to ESA Gaia RVS spectra, with extensive (and impressive) help from Claude. It looks very promising, especially in capturing oddities in hot stars. Today I worked out that there should be something similar that takes into account data weights (inverses of squared uncertainties), and I wrote down the algorithm (on paper). We'll see.

by Hogg at July 09, 2025 02:17 PM

David Hogg — how did the Solar System form?

I saw a very nice talk today by Philippine Griveaud (MPIA) about how the Solar System formed. The idea is that the giant planets formed in an accretion disk. Their formation opened gaps and caused migration (first Type I and then Type II, if you must know :). That migration pulled them into a resonant chain. That is, if the giant planets formed the way we think they formed, they must have been in a resonant chain. But they aren't in such a chain now; what gives?

The idea is that when the gas is expended (or blown out by winds), the remaining planetestimals (think: asteroids, comets, Kuiper Belt objects) interact with the planets such that they get moved from orbit to orbit and eventually ejected. These dynamical interactions break the resonant chain, migrate the giant planets to their current locations, and scatter rocks and ice balls into the interstellar regions.

It was a great talk, but also led to a lot of interesting questions, such as: How does this all fit in with the formation of the rocky planets? And how does this square with our observations (growing rapidly, apparently) of interstellar asteroids? Oh and: How does all this connect to observations of debris disks, which I now (officially) love.

by Hogg at July 09, 2025 02:10 PM

Doug Natelson — New updates + tetrahedra, tunneling times, and more

Here are a number of items from the past week or so that I think readers of this blog might find interesting:

Essentially all the news pertaining to the US federal funding of science continues to be awful. This article from Science summarizes the situation well, as does this from The Guardian and this editorial in the Washington Post. I do like the idea of a science fair of cancelled grants as a way to try to get alleged bipartisan appropriator notice of just how bad the consequences would be of the proposed cuts.
On a more uplifting note, mathematicians have empirically demonstrated a conjecture originally made by John Conway, that it is possible to make a tetrahedral pyramid that, under gravity, has only one stable orientation. Quanta has a nice piece on this with a cool animated gif, and here is the actual preprint about it. It's all about mass distributions and moments of inertia about edges. As others have pointed out including the authors, this could be quite useful for situations like recent lunar lander attempts that seem to have a difficult time not topping over.
A paper last week in Nature uses photons and a microcavity to try to test how long it takes photons to tunnel through a classically forbidden region. In this setup, it is mathematically legit to model the photons as if they have an effective mass, and one can model the barrier they need to traverse in terms of an effective potential energy. Classically, if the kinetic energy of the particle of interest is less than the potential energy of the barrier, the particle is forbidden inside the barrier. I've posted about the issue of tunneling time repeatedly over the years (see here for a 2020 post containing links), because I think it's a fascinating problem both conceptually and as a puzzle for experimentalists (how does one truly do a fair test of this?). The take-away from this paper is, the more classically forbidden the motion, the faster the deduced tunneling time. This has been seen in other experiments testing this idea. A key element of novelty in the new paper is the claim that the present experiment seems (according to the authors) to not be reasonably modeled by Bohmian mechanics. I'd need to read this in more depth to better understand it, as I had thought that Bohmian mechanics applied to problems like this is generally indistinguishable in predictions from conventional quantum mechanics, basically by design.
In other non-condensed matter news, there is an interstellar comet transiting the solar system right now. This is very cool - it's only the third such object detected by humans, but to be fair we've only really been looking for a few years. This suggests that moderately sized hunks of material are likely passing through from interstellar space all the time, and the Vera C. Rubin Observatory will detect a boatload of them. My inner science fiction fan is hoping that the object changes its orbit at perihelion by mysterious means.

This week is crunch time for a final push on US congressional appropriators to try to influence science agency budgets in FY26. I urge you to reach out if this matters to you. Likewise, I think it's more than reasonable to ask congress why the NSF is getting kicked out of its headquarters with no plan for an alternative agency location, so that the HUD secretary can have a palatial second home in that building.

by Douglas Natelson at July 09, 2025 03:18 AM

July 08, 2025

John Baez — CatColab for Model Building

guest post by Nathaniel D. Osgood

Together with 4 students from our Computational Epidemiology and Public Health Informatics Lab (CEPHIL), I spent my Friday at one of our community group model building event, this one focused on drivers for homelessness in our city (Saskatoon, Canada).

Although our behavioural ethics review board stipulated that the group should not include people who are currently homeless, the participants were people with lived experience of homelessness, with most having personally experienced homelessness within recent years.

Building on facilitated discussion, the focus of the day consisted of a group model building session. To allow for some diversity of thought and exploration, the participants divided into two teams. The causal loop diagrams resulting from about 1.5-2 hours of work on the part of each team were very thoughtful, and the diagrams captured many important insights and perspectives, and lived experiences. It bears emphasis that while some of those from our lab helped facilitate the discussions, the identification of variables and the existence, directionality and polarity of the links between such variables came firmly from the participants with lived experience themselves. Although they are not yet suitable for public distribution, I thought that I would provide a glimpse of the work products.

For the next stages of this work, CatColab will be a key tool—and arguably the single most important tool in our toolbox to secure substantive insights and value from these diagrams. Building on the strong applied category theory experience of 3 of the 4 students involved, we will be using CatColab to find feedback loops in these diagrams individually, and then when combined. This ability to find feedbacks in this fashion will be a tremendous asset for learning from these diagrams. It will also be used to visualize and explore the diagrams, although it will not be the only tool to serve in this capacity. When the ability to compose causal loop diagrams is added to CatColab, we plan to make central use of that feature as well.

Friday’s event is the first in a series focusing on this pressing problem in our community through tapping the deep and grounded knowledge of those with lived experience. It is a great testimonial to the power of CatColab that it will play such a central role in the value delivery from such events. We hope to contribute to the development of CatColab to further its ability to deliver insights and benefits not only to our research team, but also to community members themselves. I wish to express my—indeed, our—gratitude to the core CatColab team for their delivery of such a valuable tool for insight into complex social issues such as homelessness (together with cognate issues such as mental health, domestic violence and substance use, lack of affordable housing), and CEPHIL’s committment to contributing to the development of that tool to further develop its potential in this sphere.

by John Baez at July 08, 2025 08:04 PM

Terence Tao — Salem Prize now accepting nominations for 2025

The Salem prize was established in 1968 and named in honor of Raphaël Salem (1898-1963), a mathematician famous notably for his deep study of the links between Fourier series and number theory and for pioneering applications of probabilistic methods to these fields. It was not awarded from 2019-2022, due to both the COVID pandemic and the death of Jean Bourgain who had been almost single-handedly administering the prize, but is now active again, being administered by Akshay Ventakesh and the IAS. I chair the scientific committee for this prize, whose other members are Guy David and Mikhail Sodin. Last year, the prize was awarded to Miguel Walsh and Yilin Wang.

Nominations for the 2025 Salem Prize are now open until September 15th. Nominations should include a CV of the nominee and a nomination letter explaining the significance of the nominee’s work. Supplementary documentation, such as supporting letters of recommendation or key publications, can additionally be provided, but are not required.

Nominees may be individuals from any country or institution. Preference will be given to nominees who have received their PhD in the last ten years, although this rule may be relaxed if there are mitigating personal circumstances, or if there have been few Salem prize winners in recent years. Self-nominations will not be considered, nor are past Prize winners or Scientific Committee members eligible.

The prize does not come with a direct monetary award, but winners will be invited to visit the IAS and to give a lecture associated with the award of the prize.

See also the previous year’s announcement of the Salem prize nomination period.

by Terence Tao at July 08, 2025 03:30 PM

July 07, 2025

Matt Strassler Extreme and Dumb Cuts to US Science

As many of you are no doubt aware, in the past few days the US Congress voted to make major cuts to scientific research, and the president signed the bill. The government’s National Science Foundation has been cut by more than half, which means that its actual science budget has been cut by much more than that after you account for fixed costs. So vast, sudden and draconian are these cuts that it will take a long time for me and others in the field to figure out what has actually happened.

The reductions seem extreme, quite arbitrary and very poorly thought out. As an example, half of the LIGO observatory (the Laser Interferometer Gravitational-Wave Observatory, whose amazing discoveries, such as this one and this one, earned the United States a Nobel Prize in 2017) is being hit hard. There are currently two interferometers, one in Washington state and one in Lousiana, but one has been largely defunded in this bill, if I understand correctly.

I can see the logic: the scientists have two interferometers, but in tough times they ought to be able to get along with just one, right?

Well, that’s like cutting off one of a runner’s legs. Two were built because two were needed.

With just one, the signal from most gravitational wave events is so weak that you can’t distinguish it from noise. Other interferometers around the world just aren’t working well enough to make up for throwing away one of LIGOs. (And besides, you need three or four interferometers around the world to be able to know precisely in the sky where the waves are coming from, knowledge which can make other major discoveries possible.)

According to Science magazine, “In a two-sentence email to Science, an NSF spokesperson said the plan reflects `a strategic alignment of resources in a constrained fiscal environment.’ “

This is not strategic. This is stupid. The amount of money saved, less than 10 cents per year per US citizen, is very small compared to what we as a nation have already spent on this wonderful facility, and cutting LIGO in half makes it dramatically less than half as good — so this is actually a big waste of money both past and future. The decision to make this cut in this way is nothing short of ridiculous and incompetent.

[Not to mention that “constrained fiscal environment” is quite a phrase when you’re increasing the budget deficit rather than shrinking it.]

I fear there are many other similar examples to be found.

by Matt Strassler at July 07, 2025 02:05 PM

July 06, 2025

John Baez — David Suzuki on Climate Change

David Suzuki is an 89-year-old Canadian geneticist, science broadcaster and environmental activist. In this interview he says some things that I’ve come to agree with.

• ‘It’s too late’: David Suzuki says the fight against climate change is lost, iPolitics, 2 July 2025.

Q: It’s clear you haven’t lost your passion for a lot of the issues that you care about, but do you ever feel like you’re banging your head against the wall? If you look at public opinion data, climate change is often well down the list of priorities for most Canadians.

When you see that, where do you find the motivation to continue speaking to the values you believe are important?

A: I believe an informed public will do the right thing. Public concern in the late 1980s was right at the top and we had the first international conference on the atmosphere in 1988, where there were 300 people, over 40 governments, environmentalists, scientists, private sector people, you name it.

At the end of that conference, they said global warming represented a threat to humanity, second only to global nuclear war. If the world had followed the conclusions from that conference, we would not have the problem we face today and we would have saved trillions of dollars and millions of lives.

Now, it is too late.

I’ve never said this before to the media, but it’s too late. I say that because I go by science and Johan Rockström, the Swedish scientist who heads the Potsdam Institute, has defined nine planetary boundaries. These are constraints on how we live. As long as humans, like any other animal, live within those nine constraints, we can do it forever, and that includes the amount of carbon in the atmosphere, the pH of the oceans, the amount of available fresh water, the nitrogen cycle, etc.

There are nine planetary boundaries and we’ve only dealt with one of them—the ozone layer—and we think we’ve saved ourselves from that threat. But we passed the seventh boundary this year, and we’re in the extreme danger zone. Rockström says we have five years to get out of the danger zone.

If we pass one boundary, we should be shitting our pants. We’ve passed seven!

And, if you look at those boundaries, like the amount of carbon in the atmosphere, we’ve had 28 COP meetings on climate change and we haven’t been able to cap emissions.

We’re on our way to more than a three-degree temperature rise by the end of this century, and scientists agree we shouldn’t rise above one and half degrees.

Q: You say we’re too late to address climate change? That’s a pretty stark quote. Does that mean you’re giving up on the fight?

A: I’m not giving up on the immediate years, but the focus on politics, economics, and law are all destined to fail because they are based around humans. They’re designed to guide humans, but we’ve left out the foundation of our existence, which is nature, clean air, pure water, rich soil, food, and sunlight. That’s the foundation of the way we live and, when we construct legal, economic and political systems, they have to be built around protecting those very things, but they’re not.

Q: You’ve been quoted many times over the last couple decades saying it’s not too late to tackle climate change, so when did you come to this realization that the battle is lost?

A: It’s been coming all along.

We had previously said that the choice with climate change was mitigation and adaptation, and people began saying 20 years ago that we had to talk about adaptation. Other people said we can’t talk about adaptation because that acknowledges that climate change is real and impacting people. Well, we’re way past the time when we should have been thinking about adaptation.

Look, I’m not giving up in the sense of not doing anything, but Trump’s election was the dagger in my heart. Trump’s win was the triumph of capitalism and neoliberalism, and he’s going to wreak havoc. There’s nothing we can do about that, except maybe incremental changes. That’s not what we need. We need revolution. Can you have a peaceful revolution? I don’t know.

But I’m saying, as an environmentalist, we have failed to shift the narrative and we are still caught up in the same legal, economic and political systems.

For me, what we’ve got to do now is hunker down. The units of survival are going to be local communities, so I’m urging local communities to get together. Finland is offering a great example because the Finnish government has sent a letter to all of their citizens warning of future emergencies, whether they’re earthquakes, floods, droughts, or storms. They’re going to come and they’re going to be more urgent and prolonged.

Governments will not be able to respond on the scale or speed that is needed for these emergencies, so Finland is telling their citizens that they’re going to be at the front line of whatever hits and better be sure you’re ready to meet it. Find out who on your block can’t walk because you’re going to have to deal with that. Who has wheelchairs? Who has fire extinguishers? Where is the available water? Do you have batteries or generators? Start assessing the roots of escape. You’re going to have to inventory your community, and that’s really what we have to start doing now.

It’s hard to know what to do with my life if these things are true, because my life was based on other assumptions. I also believe there’s a decent chance I can eke out a comfortable existence for a couple more decades without dramatically changing my behavior (apart from moving to Scotland). But it would be silly not to update my assumptions, state this publicly, and take it from there.

It sounds like Suzuki, too, was reluctant to say publicly what he’s now come to believe. But I’m glad he’s come out and said it. If the basic outlines are correct, there’s a lot to do, and no point in staying quiet about it, even if we don’t yet have the courage and energy to change our lives as the situation calls for.

by John Baez at July 06, 2025 10:17 AM

Tommaso Dorigo — Highlights From MODE And EUCAIF

After a month of intense travel, which among other things included attendance to the MODE Workshop in Crete and the EUCAIF conference in Sardinia, I am back to northern Sweden. Besides significantly improving my well-being, given the horrible heat wave that hit Southern and Central Europe in the past few weeks, the move north allows me to finally give a relaxed look back at the most relevant information I gathered at those events, and other relevant things.

by Tommaso Dorigo at July 06, 2025 09:37 AM

July 05, 2025

Jordan Ellenberg — A small thing

Good advice I got from a colleague. If you have advisees who don’t have US citizenship, as many of us do, tell them directly that you are their advisor, no matter what, wherever they are located, and that no governmental entity has authority over that relationship.

by JSE at July 05, 2025 06:47 PM

July 04, 2025

Matt von Hippel — Bonus Info on the LHC and Beyond

Three of my science journalism pieces went up last week!

(This is a total coincidence. One piece was a general explainer “held in reserve” for a nice slot in the schedule, one was a piece I drafted in February, while the third I worked on in May. In journalism, things take as long as they take.)

The shortest piece, at Quanta Magazine, was an explainer about the two types of particles in physics: bosons, and fermions.

I don’t have a ton of bonus info here, because of how tidy the topic is, so just two quick observations.

First, I have the vague impression that Bose, bosons’ namesake, is “claimed” by both modern-day Bangladesh and India. I had friends in grad school who were proud of their fellow physicist from Bangladesh, but while he did his most famous work in Dhaka, he was born and died in Calcutta. Since both were under British India for most of his life, these things likely get complicated.

Second, at the end of the piece I mention a “world on a wire” where fermions and bosons are the same. One example of such a “wire” is a string, like in string theory. One thing all young string theorists learn is “bosonization”: the idea that, in a 1+1-dimensional world like a string, you can re-write any theory with fermions as a theory with bosons, as well as vice versa. This has important implications for how string theory is set up.

Next, in Ars Technica, I had a piece about how LHC physicists are using machine learning to untangle the implications of quantum interference.

As a journalist, it’s really easy to fall into a trap where you give the main person you interview too much credit: after all, you’re approaching the story from their perspective. I tried to be cautious about this, only to be stymied when literally everyone else I interviewed praised Aishik Ghosh to the skies and credited him with being the core motivating force behind the project. So I shrugged my shoulders and followed suit. My understanding is that he has been appropriately rewarded and will soon be a professor at Georgia Tech.

I didn’t list the inventors of the NSBI method that Ghosh and co. used, but names like Kyle Cranmer and Johann Brehmer tend to get bandied about. It’s a method that was originally explored for a more general goal, trying to characterize what the Standard Model might be missing, while the work I talk about in the piece takes it in a new direction, closer to the typical things the ATLAS collaboration looks for.

I also did not say nearly as much as I was tempted to about how the ATLAS collaboration publishes papers, which was honestly one of the most intriguing parts of the story for me. There is a huge amount of review that goes on inside ATLAS before one of their papers reaches the outside world, way more than there ever is in a journal’s peer review process. This is especially true for “physics papers”, where ATLAS is announcing a new conclusion about the physical world, as ATLAS’s reputation stands on those conclusions being reliable. That means starting with an “internal note” that’s hundreds of pages long (and sometimes over a thousand), an editorial board that manages the editing process, disseminating the paper to the entire collaboration for comment, and getting specific experts and institute groups within the collaboration to read through the paper in detail. The process is a bit less onerous for “technical papers”, which describe a new method, not a new conclusion about the world. Still, it’s cumbersome enough that for those papers, often scientists don’t publish them “within ATLAS” at all, instead releasing them independently. The results I reported on are special because they involved a physics paper and a technical paper, both within the ATLAS collaboration process. Instead of just working with partial or simplified data, they wanted to demonstrate the method on a “full analysis”, with all the computation and human coordination that requires. Normally, ATLAS wouldn’t go through the whole process of publishing a physics paper without basing it on new data, but this was different: the method had the potential to be so powerful that the more precise results would be worth stating as physics results alone.

(Also, for the people in the comments worried about training a model on old data: that’s not what they did. In physics, they don’t try to train a neural network model to predict the results of colliders, such a model wouldn’t tell us anything useful. They run colliders to tell us whether what they see matches the analytic, Standard, model. The neural network is trained to predict not what the experiment will say, but what the Standard Model will say, as we can usually only figure that out through time-consuming simulations. So it’s trained on (new) simulations, not on experimental data.)

Finally, on Friday I had a piece in Physics Today about the European Strategy for Particle Physics (or ESPP), and in particular, plans for the next big collider.

Before I even started working on this piece, I saw a thread by Patrick Koppenburg on some of the 263 documents submitted for the ESPP update. While my piece ended up mostly focused on the big circular collider plan that most of the field is converging on (the future circular collider, or FCC), Koppenburg’s thread was more wide-ranging, meant to illustrate the breadth of ideas under discussion. Some of that discussion is about the LHC’s current plans, like its “high-luminosity” upgrade that will see it gather data at much higher rates up until 2040. Some of it is assessing broader concerns, which it may surprise some of you to learn includes sustainability: yes, there are more or less sustainable ways to build giant colliders.

The most fun part of the discussion, though, concerns all of the other collider proposals.

Some report progress on new technologies. Muon colliders are the most famous of these, but there are other proposals that would specifically help with a linear collider. I never did end up understanding what Cooled Copper Colliders are all about, beyond that they let you get more energy in a smaller machine without super-cooling. If you know about them, chime in in the comments! Meanwhile, plasma wakefield acceleration could accelerate electrons on a wave of plasma. This has the disadvantage that you want to collide electrons and positrons, and if you try to stick a positron in plasma it will happily annihilate with the first electron it meets. So what do you do? You go half-and-half, with the HALHF project: speed up the electron with a plasma wakefield, accelerate the positron normally, and have them meet in the middle.

Others are backup plans, or “budget options”, where CERN could get a bit better measurements on some parameters if they can’t stir up the funding to measure the things they really want. They could put electrons and positrons into the LHC tunnel instead of building a new one, for a weaker machine that could still study the Higgs boson to some extent. They could use a similar experiment to produce Z bosons instead, which could serve as a bridge to a different collider project. Or, they could collider the LHC’s proton beam with an electron beam, for an experiment that mixes advantages and disadvantages of some of the other approaches.

While working on the piece, one resource I found invaluable was this colloquium talk by Tristan du Pree, where he goes through the 263 submissions and digs up a lot of interesting numbers and commentary. Read the slides for quotes from the different national inputs and “solo inputs” with comments from particular senior scientists. I used that talk to get a broad impression of what the community was feeling, and it was interesting how well it was reflected in the people I interviewed. The physicist based in Switzerland felt the most urgency for the FCC plan, while the Dutch sources were more cautious, with other Europeans firmly in the middle.

Going over the FCC report itself, one thing I decided to leave out of the discussion was the cost-benefit analysis. There’s the potential for a cute sound-bite there, “see, the collider is net positive!”, but I’m pretty skeptical of the kind of analysis they’re doing there, even if it is standard practice for government projects. Between the biggest benefits listed being industrial benefits to suppliers and early-career researcher training (is a collider unusually good for either of those things, compared to other ways we spend money?) and the fact that about 10% of the benefit is the science itself (where could one possibly get a number like that?), it feels like whatever reasoning is behind this is probably the kind of thing that makes rigor-minded economists wince. I wasn’t able to track down the full calculation though, so I really don’t know, maybe this makes more sense than it looks.

I think a stronger argument than anything along those lines is a much more basic point, about expertise. Right now, we have a community of people trying to do something that is not merely difficult, but fundamental. This isn’t like sending people to space, where many of the engineering concerns will go away when we can send robots instead. This is fundamental engineering progress in how to manipulate the forces of nature (extremely powerful magnets, high voltages) and process huge streams of data. Pushing those technologies to the limit seems like it’s going to be relevant, almost no matter what we end up doing. That’s still not putting the science first and foremost, but it feels a bit closer to an honest appraisal of what good projects like this do for the world.

by 4gravitons at July 04, 2025 04:00 PM

Mark Goodsell — The Ant Mill

Jesper Grimstrup kindly sent me an electronic copy of his new book, The Ant Mill. He was also kind enough to give me some feedback on a first version of this review.

It has a foreword by Peter Woit, who has commented briefly about the book on his blog; the author also has a substack. The subtitle is 'How theoretical high-energy physics descended into groupthink, tribalism and mass production of research' so you would expect it to be the sort of thing that I would take a strong objection to. However, I am the sort of person who likes to read things that challenge me; the only thing that gets under my skin in this book is attacking the whole of academia in public.

The story is an interweaving of the author's personal experiences in academia with his general observations. This personal story and his experiences are interesting, much like I expect those of everyone who has spent several years in academia would be. And he has clearly spent a lot of time thinking about thinking about research. I love meta-activities of this sort; the best example that I know of is You and Your Research by Hamming, which I stumbled on as a postdoc. Indeed, the existence of these sorts of things that are shared by young researchers is actually evidence against the central thesis of Grimstrup's book.

The market attacking High-Energy Physics seems to be burgeoning. On the one hand Hossenfelder believes that we have become 'lost in math,' and on the other Woit believes we are not mathematical enough; both attack string theory as a failed program. Grimstrup's book is in the mathematical camp, with the novelty that he piles scorn on all popular approaches to quantum gravity, in particular loop quantum gravity and noncommutative geometry, since he has come into closest contact with them. His observations about string theorists are mainly about the shoddy way that he was treated during his time at the NBI, with several egregious examples of bad behaviour. We are lead to conclude that it is not just string theorists who have formed a closed tribe, but that there are several such groups crowding out innovation.

One problem with the combination of the general and the personal is that Grimstrup constantly refers to his own research program and gives examples of how it has just generally been ignored within academia. For example, he starts the book with a copy of a grant application by a 31-year-old Niels Bohr for an entire institute, and contrasts this with a grant application of his that was refused that effectively ended his career within academia (my understanding is that at the NBI in Copenhagen it is common to ask for and obtain grants to pay your own salary and prolong temporary contracts). He writes that he does not do this to compare himself to Niels Bohr, but he does this repeatedly (mostly indirectly) throughout the book -- not in a self-aggrandising way, but in the sense that you can almost feel his frustration coming through the pages that his expectations did not meet reality. It seems like bait at times, inviting anyone who disagrees with the general thesis to attack him personally. Instead, I will have a look at his papers with an open mind, after writing this review, and keep my thoughts on them to myself.

The book made me think of how many of us enter academia. We grow up reading popular science accounts idolising physicists from a century ago. And it made me think more of the self-actualisation messages that were rammed down all our throats in the popular culture in the 80s and 90s: follow your dreams, stick to your principles, be true to yourself, this is the most important thing in life and you shouldn't worry about money, just be happy. And: working hard and getting good grades is the way to get to the top. The problem is that this is largely obsolete: it's based on the world that existed post world war two when there was a scarcity of labour and an economic boom. Then -- if you were from the right background and your face fit -- you could work hard, get a PhD and walk into a permanent academic job (yes this is a caricature). Science was respected and so were scientists; high-energy physics was at the top of the tree because of the connection with technological advancements and nuclear weapons. That world doesn't exist any more; while in many ways for the better, it is undeniable that we live in a world of much greater competition and public skepticism about science is increasing.

The scientific community has expanded, as has the population; and more importantly education throughout the world and global travel and communication has meant that the number of people around the world who are involved in research is much greater than it was. Grimstrup notes that increasing the size of the academic community has led to fundamental changes of behaviour: professionalisation of research and group think. It is clearly true that the Matthew effect exists in many branches of society, and therefore also in academia; governments wanting to exert some form of oversight in exchange for the funds that they provide has definitely led to changes in incentives for researchers. One aspect of this is that it is hard to judge the work of people from other fields, but we are required to do so; and then it is difficult to argue with quantitative measures such as number of papers, citations, h-indices. Then of course the measure becomes the target for certain people.

Grimstrup rails against all these changes; he clearly believed that the correct thing to do for an aspiring researcher would be to work on their own ideas, stick to their principles and not compromise. They should work for a long time, in isolation, on a major paper, put it on arxiv.org and the next day their colleagues would read it and ask interesting questions about them. Fame and fortune would follow. The thing that shocked Grimstrup was that not only did people not even care about any papers he posted, a young competitor even once told him some ideas are simply not worth pursuing even though they may be interesting. For sure, this is horrible and shocking behaviour, and does not reflect well on the anonymous person who said it.

For my part I am still naive enough to think that if new ideas are good, someone will recognise them as such, and network effects will make them known. I know that many researchers already think more deeply about what they are doing than he gives us credit for: and we discuss it, during seminars, over a drink with colleagues, in the coffee-breaks of conferences, during our annual or five-year reviews, or in grant applications. When I discussed this review with a string-theorist colleague they remarked "of course we know the situation sucks!'' I think Grimstrup is therefore wrong to tar everyone with the same brush: the diversity in our community has increased greatly with time, and this means that there are indeed strong incentives to take a risk on a novel idea, because the rewards of opening a new research direction are immense! Being the originator of an idea, or the first to recognise the merit in even an old forgotten idea, can yield tremendous results and even greater recognition nowadays thanks to the same effects. Hence, starting a new field, or even a subfield, is something that most researchers aspire to; the rewards for doing so are even greater now than in times gone by, and the evidence that this is possible is even given in this book: the existence of several communities working on different approaches to quantum gravity. He argues that these are now old and stale, but my point is that the way that they were able to take root at all is an example of how this can happen. There are many subfields that have sprung up more recently, and in other branches of HEP there are of course many examples. Nowadays things can change very quickly: a new good idea will be very rapidly jumped on once it is recognised, and people are constantly on the lookout.

Grimstrup also, like Lee Smolin, divides researchers into visionaries and technicians. He then complains that the technicians have taken over, with lots of disparaging comments about them digging endless holes. He then complains that there is an incentive to collaborate in modern research, only collaborators survive in the system: he has evidence that being a lone wolf is a poor survival strategy. He believes that we should work on our own; yet at the same time visionaries need to collaborate with technicians. I found this very jarring. Other than the facile placing of people into boxes, he is overlooking the benefits of collaboration -- his opinion is that it is just about inflating the number of papers one person can sign (and for sure there are people who cynically do this). But to me, discussing with other people, even just explaining something, is often the quickest way to generate genuinely new ideas or solutions to problems that we may never have come up with alone. At the same time, there are plenty of people who do write papers alone; to take a leaf from his book and share a personal story, I once had a comment on a postdoc application that I had no single-author papers and therefore did not demonstrate independence. Hence, there are incentives and a good reason for young researches to work alone sometimes. I then wrote a single-author paper, as I have occasionally done since (and got the fellowship next time I applied); I would agree that there is a pleasure and some advantages in doing this, but to do this all the time would mean I would risk missing out on lots of new ideas and other perspectives, as well as the pleasure of regular interactions with collaborators, and it would also limit the scope of my projects, where I benefit from others' expertise. Or collaborations may just be working with a student, pursuing my ideas (hopefully they contribute some of their own!) and imparting my knowledge in the process. This is why I do not think that encouraging people to predominantly cloister themselves away to work alone for a long time is the most productive or healthy one.

The book also has a very narrow focus as to the goal of high-energy physics. For the author, the quest is a "the next theory," but in essence this means a theory of quantum gravity, which he acknowledges would be far from being able to be tested with any present or near-future data. Otherwise, we should look for a mathematically rigorous definition of quantum field theory; he hopes these will be one and the same thing. This latter problem has proven to be both very hard and not obviously useful -- it is certainly not obvious that the solution should even be unique, for example a theory of strings would cure ultra-violet divergences, and the question of whether strings should be necessary for such a theory is one that I know people have tried to explore. I also recently attended a talk by Michael Douglas where he reviewed recent attempts on rigorous QFT, so it is a subject that is regarded as important but very difficult, and still being explored by a small number of people. Regarding quantum gravity, some people in the community have taken the opinion that if you have no data, it is not a good problem, and are working on other things. Or people try to make contact with data using e.g. EFT approaches to measuring quantum effects of gravity. The string theory community might say that we do have a theory of quantum gravity, in fact we have a landscape of them, and try e.g. to use it to answer questions about black hole information. But at the same time some people then complain that the leading string theorists have moved on to other things: there are lots of important open fundamental problems, and we just do not know how they are interlinked, if at all!

Grimstrup's insistence that the solution to what he sees as problems is to shrink competition and also encourage research outside of academia, reminded me of another Dane, subject of another book I read recently: king Cnut, famous for (presumably apocryphally) standing on the beach in front of his ministers and commanding the tide to turn back. Otherwise Grimstrup hopes for a crisis, perhaps one provoked by his book. He explicitly states that he does not want to fuel the anti-establishment or ant-academic movements, but I suspect that the only crises we might suffer would not be good for the field. Perhaps one is already taking place in the US; perhaps people will take his message to heart despite his protests and start a DOGE-style decimation of research. Necessarily, in science we mark our own homework: only other scientists are capable of judging the claims of their peers. If we start opening this up to question then we will only end with government appointees deciding what are acceptable topics and directions, or shutting public funding down altogether. What would be left over would surely be even greater competition for scarce resources.

For me, the solution to the problems in the book, to the extent that I agree with them, is to regularly remind ourselves that we should always maintain a childlike curiosity and not close our minds to new ideas and new possibilities. This is the message from the text of Hamming, and very well put in the writings of Feynman (who Grimstrup bizarrely dismisses as a technician compared to Bohr). Otherwise of course in science it is necessary to have a community spirit, to realise that we are all trying to make progress in the best way we know how, and to help each other do so; and it is necessary to maintain healthy competition as a motivator. But both conflicting instincts -- to compete and to group into communities -- are vital parts of human nature and denying this has been the mistake of utopians throughout history.

I am also sure that many of the complaints that Grimstrup assigns to high-energy physics could also be applied to society more generally. So instead of trying to hold back or reverse the societal changes of the last century we should try to work with them as best we can. We have to accept that we live now in an attention economy; and this gives new opportunities: blogging, social media, writing articles in science magazines or popular press, etc. Since Grimstrup is now, interestingly, an independent scientist, perhaps tying his own research program so closely with his book is embracing the modern world at last, and creating a brand as a radical outside thinker, that will be attractive to private backers. He promotes the path that he has followed, crowdfunding his research or seeking support of patrons, as a possible path for the independently minded once they have completed their training in academia, and in this I wish him well: he is clearly serious, determined and sincere. But while this is now part of twenty-first century society, many people have noticed that this modern trend is a return to the nineteenth century (or even earlier, e.g. Leonardo da Vinci being invited to France by François 1) where a wealthy patron was the only source of funding.

by Mark Goodsell at July 04, 2025 09:22 AM

July 01, 2025

Doug Natelson — Cryogenic CMOS - a key need for solid state quantum information processing

The basis for much of modern electronics is a set of silicon technologies called CMOS, which stands for complementary metal oxide semiconductor devices and processes. "Complementary" means using semiconductors (typically silicon) that is locally chemically doped so that you can have both n-type (carriers are negatively charged electrons in the conduction band) and p-type (carriers are positively charged holes in the valence band) material on the same substrate. With field-effect transistors (using oxide gate dielectrics), you can make very compact, comparatively low power devices like inverters and logic gates.

There are multiple different approaches to try to implement quantum information processing in solid state platforms, with the idea that the scaling lessons of microelectronics (in terms of device density and reliability) can be applied. I think that essentially all of these avenues require cryogenic operating conditions; all superconducting qubits need ultracold conditions for both superconductivity and to minimize extraneous quasiparticles and other decoherence sources. Semiconductor-based quantum dots (Intel's favorite) similarly need thermal perturbations and decoherence to be minimized. The wealth of solid state quantum computing research is the driver for the historically enormous (to me, anyway) growth of dilution refrigerator manufacturing (see my last point here).

So you eventually want to have thousands of error-corrected logical qubits at sub-Kelvin temperatures, which may involve millions of physical qubits at sub-Kelvin temperatures, all of which need to be controlled. Despite the absolute experimental fearlessness of people like John Martinis, you are not going to get this to work by running a million wires from room temperature into your dil fridge.

Fig. 1 from here.

The alternative people in this area have converged upon is to create serious CMOS control circuitry that can work at 4 K or below, so that a lot of the wiring does not need to go from the qubits all the way to room temperature. The materials and device engineering challenges in doing this are substantial! Power dissipation really needs to be minimized, and material properties to work at cryogenic conditions are not the same as those optimized for room temperature. There have been major advances in this - examples include Google in 2019, Intel in 2021, IBM in 2024, and this week, folks at the University of Syndney in New South Wales, supported by Diraq and Emergence Quantum. (updated/corrected)

In this most recent work, the aspect that I find most impressive is that the CMOS electronics are essentially a serious logic-based control board operating at milliKelvin temperatures right next to the chip with the qubits (in this case, spins-in-quantum-dots). I'm rather blown away that this works and with sufficiently low power dissipation that the fridge is happy. This is very impressive, and there is likely a very serious future in store for cryogenic CMOS.

by Douglas Natelson at July 01, 2025 02:06 PM

June 30, 2025

Doug Natelson — Science slow down - not a simple question

I participated in a program about 15 years ago that looked at science and technology challenges faced by a subset of the US government. I came away thinking that such problems fall into three broad categories.

Actual science and engineering challenges, which require foundational research and creativity to solve.
Technology that may be fervently desired but is incompatible with the laws of nature, economic reality, or both.
Alleged science and engineering problems that are really human/sociology issues.

Part of science and engineering education and training is giving people the skills to recognize which problems belong to which categories. Confusing these can strongly shape the perception of whether science and engineering research is making progress.

There has been a lot of discussion in the last few years about whether scientific progress (however that is measured) has slowed down or stagnated. For example, see here:

https://www.theatlantic.com/science/archive/2018/11/diminishing-returns-science/575665/

https://news.uchicago.edu/scientific-progress-slowing-james-evans

https://www.forbes.com/sites/roberthart/2023/01/04/where-are-all-the-scientific-breakthroughs-forget-ai-nuclear-fusion-and-mrna-vaccines-advances-in-science-and-tech-have-slowed-major-study-says/

https://theweek.com/science/world-losing-scientific-innovation-research

A lot of the recent talk is prompted by this 2023 study, which argues that despite the world having many more researchers than ever before (behold population growth) and more global investment in research, somehow "disruptive" innovations are coming less often, or are fewer and farther between these days. (Whether this is an accurate assessment is not a simple matter to resolve; more on this below.)

There is a whole tech bro culture that buys into this, however. For example, see this interview from last week in the New York Times with Peter Thiel, which points out that Thiel has been complaining about this for a decade and a half.

On some level, I get it emotionally. The unbounded future spun in a lot of science fiction seems very far away. Where is my flying car? Where is my jet pack? Where is my moon base? Where are my fusion power plants, my antigravity machine, my tractor beams, my faster-than-light drive? Why does the world today somehow not seem that different than the world of 1985, while the world of 1985 seems very different than that of 1945?

Some of the folks that buy into this think that science is deeply broken somehow - that we've screwed something up, because we are not getting the future they think we were "promised". Some of these people have this as an internal justification underpinning the dismantling of the NSF, the NIH, basically a huge swath of the research ecosystem in the US. These same people would likely say that I am part of the problem, and that I can't be objective about this because the whole research ecosystem as it currently exists is a groupthink self-reinforcing spiral of mediocrity.

Science and engineering are inherently human ventures, and I think a lot of these concerns have an emotional component. My take at the moment is this:

Genuinely transformational breakthroughs are rare. They often require a combination of novel insights, previously unavailable technological capabilities, and luck. They don't come on a schedule.
There is no hard and fast rule that guarantees continuous exponential technological progress. Indeed, in real life, exponential growth regimes never last. The 19th and 20th centuries were special. If we think of research as a quest for understanding, it's inherently hierarchal. Civilizational collapses aside, you can only discover how electricity works once. You can only discover the germ theory of disease, the nature of the immune system, and vaccination once (though in the US we appear to be trying really hard to test that by forgetting everything). You can only discover quantum mechanics once, and doing so doesn't imply that there will be an ongoing (infinite?) chain of discoveries of similar magnitude.
People are bad at accurately perceiving rare events and their consequences, just like people have a serious problem evaluating risk or telling the difference between correlation and causation. We can't always recognize breakthroughs when they happen. Sure, I don't have a flying car. I do have a device in my pocket that weighs only a few ounces, gives me near-instantaneous access to the sum total of human knowledge, let's me video call people around the world, can monitor aspects of my fitness, and makes it possible for me to watch sweet videos about dogs. The argument that we don't have transformative, enormously disruptive breakthroughs as often as we used to or as often as we "should" is in my view based quite a bit on perception.
Personally, I think we still have a lot more to learn about the natural world. AI tools will undoubtedly be helpful in making progress in many areas, but I think it is definitely premature to argue that the vast majority of future advances will come from artificial superintelligences and thus we can go ahead and abandon the strategies that got us the remarkable achievements of the last few decades.
I think some of the loudest complainers (Thiel, for example) about perceived slowing advancement are software people. People who come from the software development world don't always appreciate that physical infrastructure and understanding are hard, and that there are not always clever or even brute-force ways to get to an end goal. Solving foundational problems in molecular biology or quantum information hardware or photonics or materials is not the same as software development. (The tech folks generally know this on an intellectual level, but I don't think all of them really understand it in their guts. That's why so many of them seem to ignore real world physical constraints when talking about AI.). Trying to apply software development inspired approaches to science and engineering research isn't bad as a component of a many-pronged strategy, but alone it may not give the desired results - as warned in part by this piece in Science this week.

More frequent breakthroughs in our understanding and capabilities would be wonderful. I don't think dynamiting the US research ecosystem is the way to get us there, and hoping that we can dismantle everything because AI will somehow herald a new golden age seems premature at best.

by Douglas Natelson at June 30, 2025 03:20 PM

June 29, 2025

Scott Aaronson BusyBeaver(6) is really quite large

For overdetermined reasons, I’ve lately found the world an increasingly terrifying and depressing place. It’s gotten harder and harder to concentrate on research, or even popular science writing. Every so often, though, something breaks through that wakes my inner child, reminds me of why I fell in love with research thirty years ago, and helps me forget about the triumphantly strutting factions working to destroy everything I value.

Back in 2022, I reported an exciting advance in BusyBeaverology: namely, whereas we previously knew merely that BB(6) > 10^36,534, Pavel Kropitz managed to show that

BB(6) > ¹⁵10.

For those tuning in from home, here BB(6) is the 6^th Busy Beaver number, i.e. the maximum number of steps that a 6-state Turing machine with a {0,1} alphabet can take before halting, when run on an initially all-0 input tape. Also, the left-superscript means tetration, or iterated exponentiation: for example, ¹⁵10 means 10 to the 10 to the 10 and so on 15 times.

By comparison, last year the international “BBchallenge” team determined that BB(5) is “merely” 47,176,870 (see also Quanta magazine’s superb feature article on that milestone). So, between 5 and 6 is where the Busy Beaver function makes its leap, from the millions to beyond the bounds of observable reality.

But if you thought that was the end of the BB(6) story, think again! Eleven days ago, Tristan Sterin, who organized the BBchallenge the team, emailed to tell me that a team member with the handle “mxdys” improved the BB(6) bound yet further, to

BB(6) > ^10,000,00010

(i.e., 10 to the 10 to the 10 and so on 10 million times), with a correctness proof in Coq. Then, three days ago, Tristan wrote again to say that mxdys has improved the bound again, to

$$ BB(6) \gt ^{^{{^9}2}2}2 $$

I.e., BB(6) is at least 2 tetrated to the 2 tetrated to the 2 tetrated to the 9. So in particular, BB(6) is at least 2 pentated to the 5, where pentation is iterated tetration, i.e. the operation that is to tetration as tetration is to exponentiation, exponentiation is to multiplication, and multiplication is to addition.

Last week, when we “merely” knew that BB(6) > ^10,000,00010, I talked to a journalist who asked me to give an intuitive sense of how big such a number is. So I said, imagine you had ^10,000,00010 grains of sand. Then you could … well, uh … you could fill about ^10,000,00010 copies of the observable universe with that sand. I hope that helps people visualize it!

The journalist also asked: have these new discoveries about BB(6) caused me to rethink any broader beliefs about the Busy Beaver function? And I mean, yes and no: it was always completely within the realm of possibility that BB(6) would already be, not some puny little thing like 10^36,534, but way out in iteration land. Now that we know for sure that it is, though, maybe I ought to conjecture that the value of BB(n) becomes independent of the ZFC axioms of set theory already when n is 7 or 8 or 9, rather than when it’s 20 or 30 or whatever. (Currently, we know that BB(n) becomes independent of ZFC only when n=643.)

Unrelated Update: I’m just now returning to the US from STOC’2025 in Prague, where I saw lots of old friends and learned many interesting new things, again helping to distract me from the state of the world! Many I’ll write about some of those things in a future post. For now, though, anyone who’s interested in my STOC plenary lecture, entitled “The Status of Quantum Speedups,” can check out the PowerPoint slides here.

by Scott at June 29, 2025 03:33 PM

June 27, 2025

Matt von Hippel — Why Solving the Muon Puzzle Doesn’t Solve the Puzzle

You may have heard that the muon g-2 problem has been solved.

Muons are electrons’ heavier cousins. As spinning charged particles, they are magnetic, the strength of that magnetism characterized by a number denoted “g”. If you were to guess this number from classical physics alone, you’d conclude it should be 2, but quantum mechanics tweaks it. The leftover part, “g-2”, can be measured, and predicted, with extraordinary precision, which ought to make it an ideal test: if our current understanding of the particle physics, called the Standard Model, is subtly wrong, the difference might be noticeable there.

And for a while, it looked like such a difference was indeed noticeable. Extremely precise experiments over the last thirty years have consistently found a number slightly different from the extremely precise calculations, different enough that it seemed quite unlikely to be due to chance.

Now, the headlines are singing a different tune.

What changed?

That headline might make you think the change was an experimental result, a new measurement that changed the story. It wasn’t, though. There is a new, more precise measurement, but it agrees with the old measurements.

So the change has to be in the calculations, right? They did a new calculation, corrected a mistake or just pushed up their precision, and found that the Standard Model matches the experiment after all?

…sort of, but again, not really. The group of theoretical physicists associated with the experiment did release new, more accurate calculations. But it wasn’t the new calculations, by themselves, that made a difference. Instead, it was a shift in what kind of calculations they used…or even more specifically, what kind of calculations they trusted.

Parts of the calculation of g-2 can be done with Feynman diagrams, those photogenic squiggles you see on physicists’ blackboards. That part is very precise, and not especially controversial. However, Feynman diagrams only work well when forces between particles are comparatively weak. They’re great for electromagnetism, even better for the weak nuclear force. But for the strong nuclear force, the one that holds protons and neutrons together, you often need a different method.

For g-2, that used to be done via a “data-driven” method. Physicists measured different things, particles affected by the strong nuclear force in different ways, and used that to infer how the strong force would affect g-2. By getting a consistent picture from different experiments, they were reasonably confident that they had the right numbers.

Back in 2020, though, a challenger came to the scene, with another method. Called lattice QCD, this method involves building gigantic computer simulations of the effect of the strong force. People have been doing lattice QCD since the 1970’s, and the simulations have been getting better and better, until in 2020, a group managed to calculate the piece of the g-2 calculation that had until then been done by the data-driven method.

The lattice group found a very different result than what had been found previously. Instead of a wild disagreement with experiment, their calculation agreed. According to them, everything was fine, the muon g-2 was behaving exactly as the Standard Model predicted.

For some of us, that’s where the mystery ended. Clearly, something must be wrong with the data-driven method, not with the Standard Model. No more muon puzzle.

But the data-driven method wasn’t just a guess, it was being used for a reason. A significant group of physicists found the arguments behind it convincing. Now, there was a new puzzle: figuring out why the data-driven method and lattice QCD disagree.

Five years later, has that mystery been solved? Is that, finally, what the headlines are about?

Again, not really, no.

The theorists associated with the experiment have decided to trust lattice QCD, not the data-driven method. But they don’t know what went wrong, exactly.

Instead, they’ve highlighted cracks in the data-driven method. The way the data-driven method works, it brings together different experiments to try to get a shared picture. But that shared picture has started to fall apart. A new measurement by a different experiment doesn’t fit into the system: the data-driven method now “has tensions”, as physicists say. It’s no longer possible to combine all experiments into a shared picture they way they used to. Meanwhile, lattice QCD has gotten even better, reaching even higher precision. From the perspective of the theorists associated with the muon g-2 experiment, switching methods is now clearly the right call.

But does that mean they solved the puzzle?

If you were confident that lattice QCD is the right approach, then the puzzle was already solved in 2020. All that changed was the official collaboration finally acknowledging that.

And if you were confident that the data-driven method was the right approach, then the puzzle is even worse. Now, there are tensions within the method itself…but still no explanation of what went wrong! If you had good reasons to think the method should work, you still have those good reasons. Now you’re just…more puzzled.

I am reminded of another mystery, a few years back, when an old experiment announced a dramatically different measurement for the mass of the W boson. Then, I argued the big mystery was not how the W boson’s mass had changed (it hadn’t), but how they came to be so confident in a result so different from what others, also confidently, had found. In physics, our confidence is encoded in numbers, estimated and measured and tested and computed. If we’re not estimating that confidence correctly…then that’s the real mystery, the real puzzle. One much more important to solve.

Also, I had two more pieces out this week! In Quanta I have a short explainer about bosons and fermions, while at Ars Technica I have a piece about machine learning at the LHC. I may have a “bonus info” post on the latter at some point, I have to think about whether I have enough material for it.

by 4gravitons at June 27, 2025 04:00 PM

Justin Wilson — Water and its phases

I’m working on a much longer post on phases and phase transitions for next week1, but in the meantime, let me share with you some cool facts about water and its “phases.”

\We all know about solids, liquids, and gases from school. Heat up ice, and you get water; heat up water, and you get vapor. We may even have been slightly baffled if we saw this phase diagram with “pressure” added to the mix

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Phase diagram of water. This file is licensed under CC BY-SA 3.0 .

I see here a solid phase, a liquid phase, and a gas phase, but what is this “Critical point”? If you tune your temperature and pressure just right you can smoothly cross over from liquid to gas without ever undergoing a phase transition. Without getting into the molecular details, we can think of phases as particular valleys between mountains, and water wants to reach the absolute lowest point. Sometimes there are two valleys, but one is lower, and sometimes there is just one valley.

In fact, this “number of valleys” is why we see this odd behavior. If we sit at 100 degrees C and decrease or increase the pressure, there are two energy minima2—two valleys. At small pressure, the deepest valley is on the gas side, and at large pressure, the deepest valley is on the liquid side. As you then tune pressure across that one-bar point, one valley gets deeper than the other—it’s the true minimum! Yet, to get from one valley to the next, you need some energy to get you over that mountain in between. That’s the phase transition. However, that's not the only option. As the temperature increases, the mountain in between gets smaller and smaller until, at the critical point, it finally disappears, and the two valleys merge.

Without two distinguished valleys, there is no need to scale the mountain and no need for a phase transition. Liquid smoothly and easily becomes gas. At the temperatures above the critical point, you cannot meaningfully distinguish water and gas. OK, so perhaps we only have two phases?

Not quite; look at this more fleshed-out version of the phase diagram:

Phase diagram of water and ice. File licensed under CC BY-SA 3.0

When ice forms, it adopts a low-energy crystal structure. However, there are numerous crystal structures to choose from. In fact, as you change pressure and temperature, it can completely reorganize how the ice bonds together into a crystal. This leads to over 20 phases of ice, labeled by some of the Roman numerals above.3

Then what are the phases? Solids undergo their own phase transitions—structural phase transitions. Are these not phases of matter? If they are, then we have already exceeded our three phases of matter just within water. But phases go beyond temperature and pressure. They also possess a multitude of interesting properties, particularly at that critical point. We'll cover some of that in detail next week.

We’ll be making our own phase! Related, of course, to a known phase transition.

For most of the phase diagram, there is one absolute minimum, and the other is a “metastable” or local minimum.

For those interested, this Wikipedia article has a lot of information on the phases of ice.

June 26, 2025

Jordan Ellenberg — A history of the 21st century for Pittsburgh Pirates fans

Sometime this August or September, Paul Skenes, in his second year in the majors, will become the pitcher with the most total WAR for the Pittsburgh Pirates in the 21st century.

(He will pass Paul Maholm, who put together 11.8 WAR in his 7 years with the Bucs.)

by JSE at June 26, 2025 04:04 AM

June 25, 2025

Scott Aaronson Guess I’m A Rationalist Now

A week ago I attended LessOnline, a rationalist blogging conference featuring many people I’ve known for years—Scott Alexander, Eliezer Yudkowsky, Zvi Mowshowitz, Sarah Constantin, Carl Feynman—as well as people I’ve known only online and was delighted to meet in person, like Joe Carlsmith and Jacob Falkovich and Daniel Reeves. The conference was at Lighthaven, a bewildering maze of passageways, meeting-rooms, sleeping quarters, gardens, and vines off Telegraph Avenue in Berkeley, which has recently emerged as the nerd Shangri-La, or Galt’s Gulch, or Shire, or whatever. I did two events at this year’s LessOnline: a conversation with Nate Soares about the Orthogonality Thesis, and an ask-me-anything session about quantum computing and theoretical computer science (no new ground there for regular consumers of my content).

What I’ll remember most from LessOnline is not the sessions, mine or others’, but the unending conversation among hundreds of people all over the grounds, which took place in parallel with the sessions and before and after them, from morning till night (and through the night, apparently, though I’ve gotten too old for that). It felt like a single conversational archipelago, the largest in which I’ve ever taken part, and the conference’s real point. (Attendees were exhorted, in the opening session, to skip as many sessions as possible in favor of intense small-group conversations—not only because it was better but also because the session rooms were too small.)

Within the conversational blob, just making my way from one building to another could take hours. My mean free path was approximately five feet, before someone would notice my nametag and stop me with a question. Here was my favorite opener:

“You’re Scott Aaronson?! The quantum physicist who’s always getting into arguments on the Internet, and who’s essentially always right, but who sustains an unreasonable amount of psychic damage in the process?”

“Yes,” I replied, not bothering to correct the “physicist” part.

One night, I walked up to Scott Alexander, who sitting on the ground, with his large bald head and a blanket he was using as a robe, resembled a monk. “Are you enjoying yourself?” he asked.

I replied, “you know, after all these years of being coy about it, I think I’m finally ready to become a Rationalist. Is there, like, an initiation ritual or something?”

Scott said, “Oh, you were already initiated a decade ago; you just didn’t realize it at the time.” Then he corrected himself: “two decades ago.”

The first thing I did, after coming out as a Rationalist, was to get into a heated argument with Other Scott A., Joe Carlsmith, and other fellow-Rationalists about the ideas I set out twelve years ago in my Ghost in the Quantum Turing Machine essay. Briefly, my argument was that the irreversibility and ephemerality of biological life, which contrasts with the copyability, rewindability, etc. of programs running on digital computers, and which can ultimately be traced back to microscopic details of the universe’s initial state, subject to the No-Cloning Theorem of quantum mechanics, which then get chaotically amplified during brain activity … might be a clue to a deeper layer of the world, one that we understand about as well as the ancient Greeks understood Newtonian physics, but which is the layer where mysteries like free will and consciousness will ultimately need to be addressed.

I got into this argument partly because it came up, but partly also because this seemed like the biggest conflict between my beliefs and the consensus of my fellow Rationalists. Maybe part of me wanted to demonstrate that my intellectual independence remained intact—sort of like a newspaper that gets bought out by a tycoon, and then immediately runs an investigation into the tycoon’s corruption, as well as his diaper fetish, just to prove it can.

The funny thing, though, is that all my beliefs are the same as they were before. I’m still a computer scientist, an academic, a straight-ticket Democratic voter, a liberal Zionist, a Jew, etc. (all identities, incidentally, well-enough represented at LessOnline that I don’t even think I was the unique attendee in the intersection of them all).

Given how much I resonate with what the Rationalists are trying to do, why did it take me so long to identify as one?

Firstly, while 15 years ago I shared the Rationalists’ interests, sensibility, and outlook, and their stances on most issues, I also found them bizarrely, inexplicably obsessed with the question of whether AI would soon become superhumanly powerful and change the basic conditions of life on earth, and with how to make the AI transition go well. Why that, as opposed to all the other sci-fi scenarios one could worry about, not to mention all the nearer-term risks to humanity?

Suffice it to say that empirical developments have since caused me to withdraw my objection. Sometimes weird people are weird merely because they see the future sooner than others. Indeed, it seems to me that the biggest thing the Rationalists got wrong about AI was to underestimate how soon the revolution would happen, and to overestimate how many new ideas would be needed for it (mostly, as we now know, it just took lots more compute and training data). Now that I, too, spend some of my time working on AI alignment, I was able to use LessOnline in part for research meetings with colleagues.

A second reason I didn’t identify with the Rationalists was cultural: they were, and are, centrally a bunch of twentysomethings who “work” at an ever-changing list of Berkeley- and San-Francisco-based “orgs” of their own invention, and who live in group houses where they explore their exotic sexualities, gender identities, and fetishes, sometimes with the aid of psychedelics. I, by contrast, am a straight, monogamous, middle-aged tenured professor, married to another such professor and raising two kids who go to normal schools. Hanging out with the Rationalists always makes me feel older and younger at the same time.

So what changed? For one thing, with the march of time, a significant fraction of Rationalists now have marriages, children, or both—indeed, a highlight of LessOnline was the many adorable toddlers running around the Lighthaven campus. Rationalists are successfully reproducing! Some because of explicit pronatalist ideology, or because they were persuaded by Bryan Caplan’s arguments in Selfish Reasons to Have More Kids. But others simply because of the same impulses that led their ancestors to do the same for eons. And perhaps because, like the Mormons or Amish or Orthodox Jews, but unlike typical secular urbanites, the Rationalists believe in something. For all their fears around AI, they don’t act doomy, but buzz with ideas about how to build a better world for the next generation.

At a LessOnline parenting session, hosted by Julia Wise, I was surrounded by parents who worry about the same things I do: how do we raise our kids to be independent and agentic yet socialized and reasonably well-behaved, technologically savvy yet not droolingly addicted to iPad games? What schooling options will let them accelerate in math, save them from the crushing monotony that we experienced? How much of our own lives should we sacrifice on the altar of our kids’ “enrichment,” versus trusting Judith Rich Harris that such efforts quickly hit a point of diminishing returns?

A third reason I didn’t identify with the Rationalists was, frankly, that they gave off some (not all) of the vibes of a cult, with Eliezer as guru. Eliezer writes in parables and koans. He teaches that the fate of life on earth hangs in the balance, that the select few who understand the stakes have the terrible burden of steering the future. Taking what Rationalists call the “outside view,” how good is the track record for this sort of thing?

OK, but what did I actually see at Lighthaven? I saw something that seemed to resemble a cult only insofar as the Beatniks, the Bloomsbury Group, the early Royal Society, or any other community that believed in something did. When Eliezer himself—the bearded, cap-wearing Moses who led the nerds from bondage to their Promised Land in Berkeley—showed up, he was argued with like anyone else. Eliezer has in any case largely passed his staff to a new generation: Nate Soares and Zvi Mowshowitz have found new and, in various ways, better ways of talking about AI risk; Scott Alexander has for the last decade written the blog that’s the community’s intellectual center; figures from Kelsey Piper to Jacob Falkovich to Aella have taken Rationalism in new directions, from mainstream political engagement to the … err … statistical analysis of orgies.

I’ll say this, though, on the naysayers’ side: it’s really hard to make dancing to AI-generated pop songs about Bayes’ theorem and Tarski’s definition of truth not feel cringe, as I can now attest from experience.

The cult thing brings me to the deepest reason I hesitated for so long to identify as a Rationalist: namely, I was scared that if I did, people whose approval I craved (including my academic colleagues, but also just randos on the Internet) would sneer at me. For years, I searched of some way of explaining this community’s appeal so reasonable that it would silence the sneers.

It took years of psychological struggle, and (frankly) solidifying my own place in the world, to follow the true path, which of course is not to give a shit what some haters think of my life choices. Consider: five years ago, it felt obvious to me that the entire Rationalist community might be about to implode, under existential threat from Cade Metz’s New York Times article, as well as RationalWiki and SneerClub and all the others laughing at the Rationalists and accusing them of every evil. Yet last week at LessOnline, I saw a community that’s never been thriving more, with a beautiful real-world campus, excellent writers on every topic who felt like this was the place to be, and even a crop of kids. How many of the sneerers are living such fulfilled lives? To judge from their own angry, depressed self-disclosures, probably not many.

But are the sneerers right that, even if the Rationalists are enjoying their own lives, they’re making other people’s lives miserable? Are they closet far-right monarchists, like Curtis Yarvin? I liked how The New Yorker put it in its recent, long and (to my mind) devastating profile of Yarvin:

The most generous engagement with Yarvin’s ideas has come from bloggers associated with the rationalist movement, which prides itself on weighing evidence for even seemingly far-fetched claims. Their formidable patience, however, has also worn thin. “He never addressed me as an equal, only as a brainwashed person,” Scott Aaronson, an eminent computer scientist, said of their conversations. “He seemed to think that if he just gave me one more reading assignment about happy slaves singing or one more monologue about F.D.R., I’d finally see the light.”

The closest to right-wing politics that I witnessed at LessOnline was a session, with Kelsey Piper and current and former congressional staffers, about the prospects for moderate Democrats to articulate a pro-abundance agenda that would resonate with the public and finally defeat MAGA.

But surely the Rationalists are incels, bitter that they can’t get laid? Again, the closest I saw was a session where Jacob Falkovich helped a standing-room-only crowd of mostly male nerds confront their fears around dating and understand women better, with Rationalist women eagerly volunteering to answer questions about their perspective. Gross, right? (Also, for those already in relationships, Eliezer’s primary consort and former couples therapist Gretta Duleba did a session on relationship conflict.)

So, yes, when it comes to the Rationalists, I’m going to believe my own lying eyes over the charges of the sneerers. The sneerers can even say about me, in their favorite formulation, that I’ve “gone mask off,” confirmed the horrible things they’ve always suspected. Yes, the mask is off—and beneath the mask is the same person I always was, who has an inordinate fondness for the Busy Beaver function and the complexity class BQP/qpoly, and who uses too many filler words and moves his hands too much, and who strongly supports the Enlightenment, and who once feared that his best shot at happiness in life would be to earn women’s pity rather than their contempt. Incorrectly, as I’m glad to report. From my nebbishy nadir to the present, a central thing that’s changed is that, from my family to my academic colleagues to the Rationalist community to my blog readers, I finally found some people who want what I have to sell.

Unrelated Announcements:

My replies to comments on this post might be light, as I’ll be accompanying my daughter on a school trip to the Galapagos Islands!

A few weeks ago, I was “ambushed” into leading a session on philosophy and theoretical computer science at UT Austin. (I.e., asked to show up for the session, but thought I’d just be a participant rather than the main event.) The session was then recorded and placed on YouTube—and surprisingly, given the circumstances, some people seemed to like it!

Friend-of-the-blog Alon Rosen has asked me to announce a call for nominations for a new theoretical computer science prize, in memory of my former professor (and fellow TCS blogger) Luca Trevisan, who was lost to the world too soon.

And one more: Mahdi Cheraghchi has asked me to announce the STOC’2025 online poster session, registration deadline June 12; see here for more. Incidentally, I’ll be at STOC in Prague to give a plenary on quantum algorithms; I look forward to meeting any readers who are there!

by Scott at June 25, 2025 02:20 AM

June 24, 2025

Clifford Johnson — Super-Fun!

In January 2024 I wrote a paper showing how to define the Supersymmetric Virasoro Minimal String* (SVMS) as a random matrix model, compute many of its properties, and indeed predict many aspects of its physics. This was the first time the SVMS had been constructed. Despite that, a recent paper found it necessary to specifically single out my paper disparagingly as somehow not being a string theory paper, in service of (of course) their own work trying to formulate it. Odd - and disappointingly unkind - behaviour. But I’m used to it.

Anyway, since it remains the case that there is no other working definition of the SVMS out there, I thought I’d revisit the matter, clean up some unpublished work of mine (defining the 0B version) and develop the whole formalism much more. Might be useful for people pursuing other approaches. What I thought would be at most a 10 page paper turned into a 19 page one, packed with lots of fun results.

In particular it is now clear to me how the type 0A vs 0B choices, usually done at the level of perturbative worldsheet CFT methods, show up fully at the level of matrix model string equation solutions. It is often said that random matrix model methods can rather obscure issues like worldsheet supersymmetry, making it unclear what structures pertain to what features in other approaches. That can be true, so these new observations clear show that this is not always the case. (This is true quite generally, beyond this particular family of models.)

Also (and this is lots of fun!) I demonstrate that the basic loop observables of the SVMS .... Click to continue reading this post →

The post Super-Fun! appeared first on Asymptotia.

by Clifford at June 24, 2025 06:42 PM

John Preskill — Congratulations, class of 2025! Words from a new graduate

Editor’s note (Nicole Yunger Halpern): Jade LeSchack, the Quantum Steampunk Laboratory’s first undergraduate, received her bachelor’s degree from the University of Maryland this spring. Kermit the Frog presented the valedictory address, but Jade gave the following speech at the commencement ceremony for the university’s College of Mathematical and Natural Sciences. Jade heads to the University of Southern California for a PhD in physics this fall.

Good afternoon, everyone. My name is Jade, and it is my honor and pleasure to speak before you.

Today, I’m graduating with my Bachelor of Science, but when I entered UMD, I had no idea what it meant to be a professional scientist or where my passion for quantum science would take me. I want you to picture where you were four years ago. Maybe you were following a long-held passion into college, or maybe you were excited to explore a new technical field. Since then, you’ve spent hours titrating solutions, debugging code, peering through microscopes, working out proofs, and all the other things our disciplines require of us. Now, we’re entering a world of uncertainty, infinite possibility, and lifelong connections. Let me elaborate on each of these.

First, there is uncertainty. Unlike simplified projectile motion, you can never predict the exact trajectory of your life or career. Plans will change, and unexpected opportunities will arise. Sometimes, the best path forward isn’t the one you first imagined. Our experiences at Maryland have prepared us to respond to the challenges and curveballs that life will throw at us. And, we’re going to get through the rough patches.

Second, let’s embrace the infinite possibilities ahead of us. While the concept of the multiverse is best left to the movies, it’s exciting to think about all the paths before us. We’ve each found our own special interests over the past four years here, but there’s always more to explore. Don’t put yourself in a box. You can be an artist and a scientist, an entrepreneur and a humanitarian, an athlete and a scholar. Continue to redefine yourself and be open to your infinite potential.

Third, as we move forward, we are equipped not only with knowledge but with connections. We’ve made lasting relationships with incredible people here. As we go from place to place, the people who we’re close to will change. But we’re lucky that, these days, people are only an email or phone call away. We’ll always have our UMD communities rooting for us.

Now, the people we met here are certainly not the only important ones. We’ve each had supporters along the various stages of our journeys. These are the people who championed us, made sacrifices for us, and gave us a shoulder to cry on. I’d like to take a moment to thank all my mentors, teachers, and friends for believing in me. To my mom, dad, and sister sitting up there, I couldn’t have done this without you. Thank you for your endless love and support.

To close, I’d like to consider this age-old question that has always fascinated me: Is mathematics discovered or invented? People have made a strong case for each side. If we think about science in general, and our future contributions to our fields, we might ask ourselves: Are we discoverers or inventors? My answer is both! Everyone here with a cap on their head is going to contribute to both. We’re going to unearth new truths about nature and innovate scientific technologies that better society. This uncertain, multitudinous, and interconnected world is waiting for us, the next generation of scientific thinkers! So let’s be bold and stay fearless.

Congratulations to the class of 2024 and the class of 2025! We did it!

Author’s note: I was deeply grateful for the opportunity to serve as the student speaker at my commencement ceremony. I hope that the science-y references tickle the layman and SME alike. You can view a recording of the speech here. I can’t wait for my next adventures in quantum physics!

by jleschack at June 24, 2025 05:39 PM

Scott Aaronson Raymond Laflamme (1960-2025)

Even with everything happening in the Middle East right now, even with (relatedly) everything happening in my own family (my wife and son sheltering in Tel Aviv as Iranian missiles rained down), even with all the rather ill-timed travel I’ve found myself doing as these events unfolded (Ecuador and the Galapagos and now STOC’2025 in Prague) … there’s been another thing, a huge one, weighing on my soul.

Ray Laflamme played a major role in launching the whole field of quantum computing and information, and also a major role in launching my own career. The world has lost him too soon. I’ve lost him too soon.

After growing up in Quebec—I still hear his French-Canadian accent, constantly on the verge of laughter, as I’m writing this—Ray went into physics and became a PhD student of Stephen Hawking. No, not a different Stephen Hawking. If you’ve read or watched anything by or about Hawking, including A Brief History of Time, you might remember the story where Hawking believed for a while that time would reverse itself as the universe contracted in a Big Crunch, with omelettes unscrambling themselves, old people turning into children, etc. etc., but then two graduate students persuaded him that that was totally wrong, and entropy would continue to increase like normal. Anyway, Ray was one of those students (Don Page was the other). I’d always meant to ask Ray to explain what argument changed Hawking’s mind, since the idea of entropy decreasing during contraction just seemed obviously wrong to me! Only today, while writing this post, did I find a 1993 paper by Hawking, Laflamme, and Lyons that explains the matter perfectly clearly, including three fallacious intuitions that Hawking had previously held. (Even though, as they comment, “the anatomy of error is not ruled by logic.”)

Anyway, in the mid-1990s, starting at Los Alamos National Lab and continuing at the University of Waterloo, Ray became a pioneer of the then-new field of quantum computing and information. In 1997, he was a coauthor of one of the seminal original papers that proved the possibility of fault-tolerant quantum computation with a constant error rate, what we now call the Threshold Theorem (Aharonov and Ben-Or had such a result independently). He made lots of other key early contributions to the theory of quantum error-correcting codes and fault-tolerance.

When it comes to Ray’s scientific achievements after his cosmology work with Hawking and after quantum fault-tolerance—well, there are many, but let me talk about two. Perhaps the biggest is the KLM (Knill-Laflamme-Milburn) Theorem. It would be fair to say that KLM started the entire field of optical or photonic quantum computation, as it’s existed in the 21^st century. In one sentence, what KLM showed is that it’s possible to build a universal quantum computer using only

identical single-photon states,
a network of “linear-optical elements” (that is, beamsplitters and phaseshifters) that the photons travel through, and
feedforward measurements—that is, measurements of an optical mode that tell you how many photons are there, in such a way that you can condition (using a classical computer) which optical elements to apply next on the outcome of the measurement.

All of a sudden, there was a viable path to building a quantum computer out of photons, where you wouldn’t need to get pairs of photons to interact with each other, which had previously been the central sticking point. The key insight was that feedforward measurements, combined with the statistical properties of identical bosons (what the photons are), are enough to simulate the effect of two-photon interactions.

Have you heard of PsiQuantum, the startup in Palo Alto with a $6 billion valuation and hundreds of employees that’s right now trying to build an optical quantum computer with a million qubits? Or Xanadu, its competitor in Toronto? These, in some sense, are companies that grew out of a theorem: specifically the KLM Theorem.

For me, though, the significance of KLM goes beyond the practical. In 2011, I used the KLM Theorem, together with the fact (known since the 1950s) that photonic amplitudes are the permanents of matrices, to give a new proof of Leslie Valiant’s celebrated 1979 theorem that calculating the permanent is a #P-complete problem. Thus, as I pointed out in a talk two years ago at Ray’s COVID-delayed 60^th birthday conference, entitled Ray Laflamme, Complexity Theorist (?!), KLM had said something new about computational complexity, without any intention of doing so. More generally, KLM was crucial backdrop to my and Alex Arkhipov’s later work on BosonSampling, where we gave strong evidence that some classical computational hardness—albeit probably not universal quantum computation—remains in linear optics, even if one gets rid of KLM’s feedforward measurements.

(Incidentally, I gave my talk at Ray’s birthday conference by Zoom, as I had a conflicting engagement. I’m now sad about that: had I known that that would’ve been my last chance to see Ray, I would’ve cancelled any other plans.)

The second achievement of Ray’s that I wanted to mention was his 1998 creation, again with his frequent collaborator Manny Knill, of the One Clean Qubit or “DQC1” model of quantum computation. In this model, you get to apply an arbitrary sequence of 2-qubit unitary gates, followed by measurements at the end, just like in standard quantum computing—but the catch is that the initial state consists of just a single qubit in the state |0⟩, and all other qubits in the maximally mixed state. If all qubits started in the maximally mixed state, then nothing would ever happen, because the maximally mixed state is left invariant by all unitary transformations. So it would stand to reason that, if all but one of the qubits start out maximally mixed, then almost nothing happens. The big surprise is that this is wrong. Instead you get a model that, while probably not universal for quantum computation, can do a variety of things in polynomial time that we don’t know how to do classically, including estimating the traces of exponentially large unitary matrices and the Jones polynomials of trace closures of braids (indeed, both of these problems turn out to be DQC1-complete). The discovery of DQC1 was one of the first indications that there’s substructure within BQP. Since then, the DQC1 model has turned up again and again in seemingly unrelated investigations in quantum complexity theory—way more than you’d have any right to expect a priori.

Beyond his direct contributions to quantum information, Ray will be remembered as one of the great institution-builders of our field. He directed the Institute for Quantum Computing (IQC) at the University of Waterloo in Canada, from its founding in 2002 until he finally stepped down in 2017. This includes the years 2005-2007, when I was a postdoc at IQC—two of the most pivotal years of my life, when I first drove a car and went out on dates (neither of which I do any longer, for different reasons…), when I started this blog, when I worked on quantum money and learnability of quantum states and much more, and when I taught the course that turned into my book Quantum Computing Since Democritus. I fondly remember Ray, as my “boss,” showing me every possible kindness. He even personally attended the Quantum Computing Since Democritus lectures, which is why he appears as a character in the book.

As if that wasn’t enough, Ray also directed the quantum information program of the Canadian Institute for Advanced Research (CIFAR). If you ever wondered why Canada, as a nation, has punched so far above its weight in quantum computing and information for the past quarter-century—Ray Laflamme is part of the answer.

At the same time, if you imagine the stereotypical blankfaced university administrator, who thinks and talks only in generalities and platitudes (“how can we establish public-private partnerships to build a 21^st-century quantum workforce?”) … well, Ray was whatever is the diametric opposite of that. Despite all his responsibilities, Ray never stopped being a mensch, a friend, an intellectually curious scientist, a truth-teller, and a jokester. Whenever he and I talked, probably at least a third of the conversation was raucous laughter.

I knew that Ray had spent many years battling cancer. I naïvely thought he was winning, or had won. But as so often with cancer, it looks like the victory was only temporary. I miss him already. He was a ray of light in the world—a ray that sparkles, illuminates, and as we now know, even has the latent power of universal quantum computation.

by Scott at June 24, 2025 04:05 PM

June 23, 2025

John Preskill — A (quantum) complex legacy: Part trois

When I worked in Cambridge, Massachusetts, a friend reported that MIT’s postdoc association had asked its members how it could improve their lives. The friend confided his suggestion to me: throw more parties.¹ This year grants his wish on a scale grander than any postdoc association could. The United Nations has designated 2025 as the International Year of Quantum Science and Technology (IYQ), as you’ve heard unless you live under a rock (or without media access—which, come to think of it, sounds not unappealing).

A metaphorical party cracker has been cracking since January. Governments, companies, and universities are trumpeting investments in quantum efforts. Institutions pulled out all the stops for World Quantum Day, which happens every April 14 but which scored a Google doodle this year. The American Physical Society (APS) suffused its Global Physics Summit in March with quantum science like a Bath & Body Works shop with the scent of Pink Pineapple Sunrise. At the summit, special symposia showcased quantum research, fellow blogger John Preskill dished about quantum-science history in a dinnertime speech, and a “quantum block party” took place one evening. I still couldn’t tell you what a quantum block party is, but this one involved glow sticks.

Attending the summit, I felt a satisfaction—an exultation, even—redolent of twelfth grade, when American teenagers summit the Mont Blanc of high school. It was the feeling that this year is our year. Pardon me while I hum “Time of your life.”²

Speakers and organizer of a Kavli Symposium, a special session dedicated to interdisciplinary quantum science, at the APS Global Physics Summit

Just before the summit, editors of the journal PRX Quantum released a special collection in honor of the IYQ.³ The collection showcases a range of advances, from chemistry to quantum error correction and from atoms to attosecond-length laser pulses. Collaborators and I contributed a paper about quantum complexity, a term that has as many meanings as companies have broadcast quantum news items within the past six months. But I’ve already published two Quantum Frontiers posts about complexity, and you surely study this blog as though it were the Bible, so we’re on the same page, right?

Just joshing.

Imagine you have a quantum computer that’s running a circuit. The computer consists of qubits, such as atoms or ions. They begin in a simple, “fresh” state, like a blank notebook. Post-circuit, they store quantum information, such as entanglement, as a notebook stores information post-semester. We say that the qubits are in some quantum state. The state’s quantum complexity is the least number of basic operations, such as quantum logic gates, needed to create that state—via the just-completed circuit or any other circuit.

Today’s quantum computers can’t create high-complexity states. The reason is, every quantum computer inhabits an environment that disturbs the qubits. Air molecules can bounce off them, for instance. Such disturbances corrupt the information stored in the qubits. Wait too long, and the environment will degrade too much of the information for the quantum computer to work. We call the threshold time the qubits’ lifetime, among more-obscure-sounding phrases. The lifetime limits the number of gates we can run per quantum circuit.

The ability to perform many quantum gates—to perform high-complexity operations—serves as a resource. Other quantities serve as resources, too, as you’ll know if you’re one of the three diehard Quantum Frontiers fans who’ve been reading this blog since 2014 (hi, Mom). Thermodynamic resources include work: coordinated energy that one can harness directly to perform a useful task, such as lifting a notebook or staying up late enough to find out what a quantum block party is.

My collaborators: Jonas Haferkamp, Philippe Faist, Teja Kothakonda, Jens Eisert, and Anthony Munson (in an order of no significance here)

My collaborators and I showed that work trades off with complexity in information- and energy-processing tasks: the more quantum gates you can perform, the less work you have to spend on a task, and vice versa. Qubit reset exemplifies such tasks. Suppose you’ve filled a notebook with a calculation, you want to begin another calculation, and you have no more paper. You have to erase your notebook. Similarly, suppose you’ve completed a quantum computation and you want to run another quantum circuit. You have to reset your qubits to a fresh, simple state.

Three methods suggest themselves. First, you can “uncompute,” reversing every quantum gate you performed.⁴ This strategy requires a long lifetime: the information imprinted on the qubits by a gate mustn’t leak into the environment before you’ve undone the gate.

Second, you can do the quantum equivalent of wielding a Pink Pearl Paper Mate: you can rub the information out of your qubits, regardless of the circuit you just performed. Thermodynamicists inventively call this strategy erasure. It requires thermodynamic work, just as applying a Paper Mate to a notebook does.

Third, you can

Suppose your qubits have finite lifetimes. You can undo as many gates as you have time to. Then, you can erase the rest of the qubits, spending work. How does complexity—your ability to perform many gates—trade off with work? My collaborators and I quantified the tradeoff in terms of an entropy we invented because the world didn’t have enough types of entropy.⁵

Complexity trades off with work not only in qubit reset, but also in data compression and likely other tasks. Quantum complexity, my collaborators and I showed, deserves a seat at the great soda fountain of quantum thermodynamics.

The great soda fountain of quantum thermodynamics

…as quantum information science deserves a seat at the great soda fountain of physics. When I embarked upon my PhD, faculty members advised me to undertake not only quantum-information research, but also some “real physics,” such as condensed matter. The latter would help convince physics departments that I was worth their money when I applied for faculty positions. By today, the tables have turned. A condensed-matter theorist I know has wound up an electrical-engineering professor because he calculates entanglement entropies.

So enjoy our year, fellow quantum scientists. Party like it’s 1925. Burnish those qubits—I hope they achieve the lifetimes of your life.

¹Ten points if you can guess who the friend is.

²Whose official title, I didn’t realize until now, is “Good riddance.” My conception of graduation rituals has just turned a somersault.

³PR stands for Physical Review, the brand of the journals published by the APS. The APS may have intended for the X to evoke exceptional, but I like to think it stands for something more exotic-sounding, like ex vita discedo, tanquam ex hospitio, non tanquam ex domo.

⁴Don’t ask me about the notebook analogue of uncomputing a quantum state. Explaining it would require another blog post.

⁵For more entropies inspired by quantum complexity, see this preprint. You might recognize two of the authors from earlier Quantum Frontiers posts if you’re one of the three…no, not even the three diehard Quantum Frontiers readers will recall; but trust me, two of the authors have received nods on this blog before.

by Nicole Yunger Halpern at June 23, 2025 12:01 AM

June 21, 2025

Doug Natelson — Brief items - fresh perspectives, some news bits

As usual, I hope to write more about particular physics topics soon, but in the meantime I wanted to share a sampling of news items:

First, it's a pleasure to see new long-form writing about condensed matter subjects, in an era where science blogging has unquestionably shrunk compared to its heyday. The new Quantum Matters substack by Justin Wilson (and William Shelton) looks like it will be a fun place to visit often.
Similar in spirit, I've also just learned about the Knowmads podcast (here on youtube), put out by Prachi Garella and Bhavay Tyagi, two doctoral students at the University of Houston. Fun Interviews with interesting scientists about their science and how they get it done.
There have been some additional news bits relevant to the present research funding/university-govt relations mess. Earlier this week, 200 business leaders published an open letter about how the slashing support for university research will seriously harm US economic competitiveness. More of this, please. I continue to be surprised by how quiet technology-related, pharma, and finance companies are being, at least in public. Crushing US science and engineering university research will lead to serious personnel and IP shortages down the line, definitely poor for US standing. Again, now is the time to push back on legislators about cuts mooted in the presidential budget request.
The would-be 15% indirect cost rate at NSF has been found to be illegal, in a summary court judgment released yesterday. (Brief article here, pdf of the ruling here.)
Along these lines, there are continued efforts for proposals about how to reform/alter indirect cost rates in a far less draconian manner. These are backed by collective organizations like the AAU and COGR. If you're interested in this, please go here, read the ideas, and give some feedback. (Note for future reference: the Joint Associations Group (JAG) may want to re-think their acronym. In local slang where I grew up, the word "jag" does not have pleasant connotations.)
The punitive attempt to prevent Harvard from taking international students has also been stopped for now in the courts.

by Douglas Natelson at June 21, 2025 07:35 PM

June 20, 2025

Matt von Hippel — Amplitudes 2025 This Week

Summer is conference season for academics, and this week held my old sub-field’s big yearly conference, called Amplitudes. This year, it was in Seoul at Seoul National University, the first time the conference has been in Asia.

(I wasn’t there, I don’t go to these anymore. But I’ve been skimming slides in my free time, to give you folks the updates you crave. Be forewarned that conference posts like these get technical fast, I’ll be back to my usual accessible self next week.)

There isn’t a huge amplitudes community in Korea, but it’s bigger than it was back when I got started in the field. Of the organizers, Kanghoon Lee of the Asia Pacific Center for Theoretical Physics and Sangmin Lee of Seoul National University have what I think of as “core amplitudes interests”, like recursion relations and the double-copy. The other Korean organizers are from adjacent areas, work that overlaps with amplitudes but doesn’t show up at the conference each year. There was also a sizeable group of organizers from Taiwan, where there has been a significant amplitudes presence for some time now. I do wonder if Korea was chosen as a compromise between a conference hosted in Taiwan or in mainland China, where there is also quite a substantial amplitudes community.

One thing that impresses me every year is how big, and how sophisticated, the gravitational-wave community in amplitudes has grown. Federico Buccioni’s talk began with a plot that illustrates this well (though that wasn’t his goal):

At the conference Amplitudes, dedicated to the topic of scattering amplitudes, there were almost as many talks with the phrase “black hole” in the title as there were with “scattering” or “amplitudes”! This is for a topic that did not even exist in the subfield when I got my PhD eleven years ago.

With that said, gravitational wave astronomy wasn’t quite as dominant at the conference as Buccioni’s bar chart suggests. There were a few talks each day on the topic: I counted seven in total, excluding any short talks on the subject in the gong show. Spinning black holes were a significant focus, central to Jung-Wook Kim’s, Andres Luna’s and Mao Zeng’s talks (the latter two showing some interesting links between the amplitudes story and classic ideas in classical mechanics) and relevant in several others, with Riccardo Gonzo, Miguel Correia, Ira Rothstein, and Enrico Herrmann’s talks showing not just a wide range of approaches, but an increasing depth of research in this area.

Herrmann’s talk in particular dealt with detector event shapes, a framework that lets physicists think more directly about what a specific particle detector or observer can see. He applied the idea not just to gravitational waves but to quantum gravity and collider physics as well. The latter is historically where this idea has been applied the most thoroughly, as highlighted in Hua Xing Zhu’s talk, where he used them to pick out particular phenomena of interest in QCD.

QCD is, of course, always of interest in the amplitudes field. Buccioni’s talk dealt with the theory’s behavior at high-energies, with a nice example of the “maximal transcendentality principle” where some quantities in QCD are identical to quantities in N=4 super Yang-Mills in the “most transcendental” pieces (loosely, those with the highest powers of pi). Andrea Guerreri’s talk also dealt with high-energy behavior in QCD, trying to address an experimental puzzle where QCD results appeared to violate a fundamental bound all sensible theories were expected to obey. By using S-matrix bootstrap techniques, they clarify the nature of the bound, finding that QCD still obeys it once correctly understood, and conjecture a weird theory that should be possible to frame right on the edge of the bound. The S-matrix bootstrap was also used by Alexandre Homrich, who talked about getting the framework to work for multi-particle scattering.

Heribertus Bayu Hartanto is another recent addition to Korea’s amplitudes community. He talked about a concrete calculation, two-loop five-particle scattering including top quarks, a tricky case that includes elliptic curves.

When amplitudes lead to integrals involving elliptic curves, many standard methods fail. Jake Bourjaily’s talk raised a question he has brought up again and again: what does it mean to do an integral for a new type of function? One possible answer is that it depends on what kind of numerics you can do, and since more general numerical methods can be cumbersome one often needs to understand the new type of function in more detail. In light of that, Stephen Jones’ talk was interesting in taking a common problem often cited with generic approaches (that they have trouble with the complex numbers introduced by Minkowski space) and finding a more natural way in a particular generic approach (sector decomposition) to take them into account. Giulio Salvatori talked about a much less conventional numerical method, linked to the latest trend in Nima-ology, surfaceology. One of the big selling points of the surface integral framework promoted by people like Salvatori and Nima Arkani-Hamed is that it’s supposed to give a clear integral to do for each scattering amplitude, one which should be amenable to a numerical treatment recently developed by Michael Borinsky. Salvatori can currently apply the method only to a toy model (up to ten loops!), but he has some ideas for how to generalize it, which will require handling divergences and numerators.

Other approaches to the “problem of integration” included Anna-Laura Sattelberger’s talk that presented a method to find differential equations for the kind of integrals that show up in amplitudes using the mathematical software Macaulay2, including presenting a package. Matthias Wilhelm talked about the work I did with him, using machine learning to find better methods for solving integrals with integration-by-parts, an area where two other groups have now also published. Pierpaolo Mastrolia talked about integration-by-parts’ up-and-coming contender, intersection theory, a method which appears to be delving into more mathematical tools in an effort to catch up with its competitor.

Sometimes, one is more specifically interested in the singularities of integrals than their numerics more generally. Felix Tellander talked about a geometric method to pin these down which largely went over my head, but he did have a very nice short description of the approach: “Describe the singularities of the integrand. Find a map representing integration. Map the singularities of the integrand onto the singularities of the integral.”

While QCD and gravity are the applications of choice, amplitudes methods germinate in N=4 super Yang-Mills. Ruth Britto’s talk opened the conference with an overview of progress along those lines before going into her own recent work with one-loop integrals and interesting implications of ideas from cluster algebras. Cluster algebras made appearances in several other talks, including Anastasia Volovich’s talk which discussed how ideas from that corner called flag cluster algebras may give insights into QCD amplitudes, though some symbol letters still seem to be hard to track down. Matteo Parisi covered another idea, cluster promotion maps, which he thinks may help pin down algebraic symbol letters.

The link between cluster algebras and symbol letters is an ongoing mystery where the field is seeing progress. Another symbol letter mystery is antipodal duality, where flipping an amplitude like a palindrome somehow gives another valid amplitude. Lance Dixon has made progress in understanding where this duality comes from, finding a toy model where it can be understood and proved.

Others pushed the boundaries of methods specific to N=4 super Yang-Mills, looking for novel structures. Song He’s talk pushes an older approach by Bourjaily and collaborators up to twelve loops, finding new patterns and connections to other theories and observables. Qinglin Yang bootstraps Wilson loops with a Lagrangian insertion, adding a side to the polygon used in previous efforts and finding that, much like when you add particles to amplitudes in a bootstrap, the method gets stricter and more powerful. Jaroslav Trnka talked about work he has been doing with “negative geometries”, an odd method descended from the amplituhedron that looks at amplitudes from a totally different perspective, probing a bit of their non-perturbative data. He’s finding more parts of that setup that can be accessed and re-summed, finding interestingly that multiple-zeta-values show up in quantities where we know they ultimately cancel out. Livia Ferro also talked about a descendant of the amplituhedron, this time for cosmology, getting differential equations for cosmological observables in a particular theory from a combinatorial approach.

Outside of everybody’s favorite theories, some speakers talked about more general approaches to understanding the differences between theories. Andreas Helset covered work on the geometry of the space of quantum fields in a theory, applying the method to a general framework for characterizing deviations from the standard model called the SMEFT. Jasper Roosmale Nepveu also talked about a general space of theories, thinking about how positivity (a trait linked to fundamental constraints like causality and unitarity) gets tangled up with loop effects, and the implications this has for renormalization.

Soft theorems, universal behavior of amplitudes when a particle has low energy, continue to be a trendy topic, with Silvia Nagy showing how the story continues to higher orders and Sangmin Choi investigating loop effects. Callum Jones talks about one of the more powerful results from the soft limit, Weinberg’s theorem showing the uniqueness of gravity. Weinberg’s proof was set up in Minkowski space, but we may ultimately live in curved, de Sitter space. Jones showed how the ideas Weinberg explored generalize in de Sitter, using some tools from the soft-theorem-inspired field of dS/CFT. Julio Parra-Martinez, meanwhile, tied soft theorems to another trendy topic, higher symmetries, a more general notion of the usual types of symmetries that physicists have explored in the past. Lucia Cordova reported work that was not particularly connected to soft theorems but was connected to these higher symmetries, showing how they interact with crossing symmetry and the S-matrix bootstrap.

Finally, a surprisingly large number of talks linked to Kevin Costello and Natalie Paquette’s work with self-dual gauge theories, where they found exact solutions from a fairly mathy angle. Paquette gave an update on her work on the topic, while Alfredo Guevara talked about applications to black holes, comparing the power of expanding around a self-dual gauge theory to that of working with supersymmetry. Atul Sharma looked at scattering in self-dual backgrounds in work that merges older twistor space ideas with the new approach, while Roland Bittelson talked about calculating around an instanton background.

Also, I had another piece up this week at FirstPrinciples, based on an interview with the (outgoing) president of the Sloan Foundation. I won’t have a “bonus info” post for this one, as most of what I learned went into the piece. But if you don’t know what the Sloan Foundation does, take a look! I hadn’t known they funded Jupyter notebooks and Hidden Figures, or that they introduced Kahneman and Tversky.

by 4gravitons at June 20, 2025 04:00 PM

Justin Wilson — A Bite of Quasicrystals

This is a slight reworking of a previous post on my personal blog since I am currently traveling.

Quasicrystals, a beautiful manifestation of something without a strict crystalline symmetry but nonetheless shows order, won a Nobel prize in 2011. A bit more recently, in 2018, a dodecagonal graphene quasicrystal (two sheets of graphene twisted 30 degrees with respect to each other) made its way onto the cover of Science1.

June 19, 2025

Jordan Ellenberg — 2009 AL ROY nonprescience

Just about 16 years ago today I blogged:

The Orioles have two legitimate Rookie of the Year candidates and neither one of them is named Matt Wieters

“I’m talking about Nolan Reimold, currently slugging .546 and leading all major-league rookies in OPS; and Brad Bergesen, who’s been the Orioles’ best starter this year at 23. Higher-profile pitching prospects Rick Porcello and David Price have ERAs a little lower, but Bergesen looks better on home runs, walks, and strikeouts. He is, as they say, “in the discussion.””

Yeah. Reimold and Bergesen did not win Rookie of the Year, and fact, both of them had the majority of their career WAR in 2009. Bergesen, in fact, had more WAR in 2009 than his career total, and was out of the major leagues by the age of 27. Price and Porcello, meanwhile, had long careers, and each won a Cy Young before turning 30. I guess the guys who rate pitching prospects know something about what they’re doing. In my defense, the 2009 AL Rookie of the Year was in the end not Nolan Reimold, or Brad Bergesen, or David Price, or Rick Porcello, or Matt Wieters — it was A’s reliever Andrew Bailey, who also had the majority of his career WAR in 2009.

by JSE at June 19, 2025 07:41 PM

Jordan Ellenberg — Happy Juneteenth

It’s a fine thing that we now have a national holiday that asks us to remember slavery. Some people think patriotism means insisting that America never did or does anything wrong, and that our schools should teach a purely heroic account of the American story. That’s foolish. America is made of people like you and me, who share certain ideals, truly heroic ideals, but don’t always live up to them — and some scoundrels too. Any patriotism that can’t survive contact with the actual history of America is weak stuff. A real patriot loves his country with open eyes.

by JSE at June 19, 2025 06:31 PM

Jordan Ellenberg — Math and medicine webinar

The National Academies of Sciences, Engineering, and Medicine have me moderating a series of webinars about the use of novel math in the applied sciences. I learn a lot every time I do one! Here’s the latest, on Machine Learning for Breakthroughs in Medical Care, featuring Charley Taylor and Lorin Crawford. Some fun! Looking forward to doing more of these.

by JSE at June 19, 2025 12:21 AM

June 17, 2025

Justin Wilson — Is Science Still Working? Here’s What I Think

You’ve probably seen headlines lately questioning science—it’s funding, its fairness, and whether it’s even open to new ideas anymore. Some people worry that science has become too rigid, unwilling to entertain fresh perspectives. As someone who lives and breathes science, I wanted to share my take.

First, let me tell you about myself and my expertise. I have a Ph.D. in theoretical condensed matter physics, which is the study of solids, surfaces, interfaces, and liquids. This also includes disordered systems. I have also worked in catalysis and energy storage. I have been doing active research with my first published article in 1987 to the present, which is about 38 years of active work. I have worked at two Department of Energy National Laboratories, Oak Ridge National Laboratory and Pacific Northwest National Laboratory. I am currently a full professor of Physics at Louisiana State University, where I have been for the last twelve years. I currently have 3 funded research projects, one on “Improving Transmon Qubit Performance,” the second is on “Directed Assembly of Metastable States for Harnessing Quantum Effects,” and the third is on “Enabling Formate-Based Hydrogen Storage and Generation via Multimetallic Alloy Catalysts.”

Thanks for reading Quantum Matters! Subscribe for free to receive new posts and support my work.

Let me start by saying this: I don’t think science is broken. In fact, science needs new ideas to survive. That’s what keeps it alive. It’s not about guarding the past—it’s about exploring what comes next.

a person lying on a bed — Photo by Accuray on Unsplash

So, What Is Science Anyway?

At its heart, science is just a way of asking questions and trying to find honest answers. There’s a process to it, and it goes something like this:

You notice something interesting.
You ask a question about it.
You come up with a possible explanation—what we call a hypothesis.
You test that idea through experiments, models, or observation.
You look at the results and ask, “Was I right?”
Whether the answer is yes or no, you learn something—and you keep going.
Finally, you share what you found, so others can learn from it too.

That’s it. And we repeat this process again and again. If the answer turns out to be wrong, that’s still progress. It tells us what doesn’t work, which is just as important as figuring out what does.

Let’s Pick a Scientific Process Example from our previous blog post on Quantum materials: The Quantum Topological Material Bi₂Se₃

🔍 Observation

Some newly discovered materials—like bismuth selenide (Bi₂Se₃)—conduct electricity on their surfaces while remaining insulating inside. This unusual behavior hints at a new phase of matter.

❓ Question

Why do these materials conduct only on their surfaces? What quantum mechanical principles govern this behavior? Can this property be harnessed for new electronic or quantum computing applications?

💡 Hypothesis

These materials are topological insulators. Their unique surface conductivity arises from strong spin-orbit coupling, which protects surface states from scattering due to defects or impurities. These protected states are a result of the material’s non-trivial topological order.

⚗️ Experiment

Researchers test the hypothesis by:

Synthesizing high-quality crystals of Bi₂Se₃.
Measuring surface conductivity via scanning tunneling microscopy (STM) and angle-resolved photoemission spectroscopy (ARPES) for directly observing the electronic states.
Applying magnetic fields and introducing defects to see if the surface states remain intact.
Using transport measurements to compare the bulk and surface contributions to conductivity.

📊 Analysis

ARPES data shows Dirac-like surface states.
STM confirms surface conduction pathways even when the bulk is insulating.
Magnetic fields break time-reversal symmetry, gapping the surface states—validating the topological protection mechanism.

✅ Conclusion

The data confirm that Bi₂Se₃ is a 3D topological insulator. Its conductive surface states are protected by time-reversal symmetry, making it a strong candidate for spintronic devices and fault-tolerant quantum computing.

📢 Communication

Findings are published in Nature Materials and Science, presented at condensed matter physics conferences, and used to inspire further research into topological superconductors, Majorana fermions, and quantum information systems.

🔍 Why It Matters

This discovery has opened up new paths in quantum electronics, where data can flow with minimal energy loss, and in quantum computing, where these materials could enable robust, error-resistant qubits.

person in black and white long sleeve shirt holding white and blue book — Photo by NMG Network on Unsplash

Why This Matters

This is how new medications move from lab benches to pharmacy shelves. The scientific process ensures treatments are both safe and effective before reaching the public.

But Is Science Open to New Ideas?

Yes, absolutely—but it’s not always easy. Getting a scientific paper published takes a lot of work. Top-tier journals reject around 80 to 95% of the papers they receive. Even solid mid-tier journals reject more than half. Why? Because these journals want well-supported, clearly written work that pushes knowledge forward. Lastly, here is an interesting statistic, do you know that that roughly 1% of the scientific workforce publishes every single year. That’s a small fraction maintaining a consistent online publishing presence. It is truly difficult getting articles published.

That doesn’t mean new or unusual ideas get shut out. They just have to be backed up with solid evidence—and explained clearly. I’ve seen good ideas get rejected just because the writing was hard to follow, or the data didn’t quite hold up. That’s why I often help younger researchers refine their papers. Heck, I’ve had others help me, too. We’re all learning as we go.

And yes, rejection stings. It really does. But it’s not about ego—it’s about making the work stronger.

Here’s the Bottom Line

Science is hard. But it’s supposed to be. We ask tough questions and hold ourselves to high standards. That’s how we make sure the answers we get are worth something.

Is it perfect? No. But the system is built to self-correct. Bad ideas eventually fall away. Good ones rise to the top—even if it takes a few tries.

Science isn’t just a pile of facts. It’s a conversation—a messy, exciting, and sometimes frustrating conversation about how the world works. And yes, it still works.

So the next time someone says science is too closed off, remind them: science is open—to anyone willing to do the work, ask the hard questions, and follow the evidence wherever it leads.

Peter Rohde Garbage

Contract termination Download

BTQ Consultant Agreement – Peter Rohde – July 2023 Download

by Peter Rohde at June 17, 2025 12:44 AM

June 13, 2025

Matt von Hippel — Bonus info for Reversible Computing and Megastructures

After some delay, a bonus info post!

At FirstPrinciples.org, I had a piece covering work by engineering professor Colin McInnes on stability of Dyson spheres and ringworlds. This was a fun one to cover, mostly because of how it straddles the borderline between science fiction and practical physics and engineering. McInnes’s claim to fame is work on solar sails, which seem like a paradigmatic example of that kind of thing: a common sci-fi theme that’s surprisingly viable. His work on stability was interesting to me because it’s the kind of work that a century and a half ago would have been paradigmatic physics. Now, though, very few physicists work on orbital mechanics, and a lot of the core questions have passed on to engineering. It’s fascinating to see how these classic old problems can still have undiscovered solutions, and how the people best equipped to find them now are tinkerers practicing their tools instead of cutting-edge mathematicians.

At Quanta Magazine, I had a piece about reversible computing. Readers may remember I had another piece on that topic at the end of March, a profile on the startup Vaire Computing at FirstPrinciples.org. That piece talked about FirstPrinciples, but didn’t say much about reversible computing. I figured I’d combine the “bonus info” for both posts here.

Neither piece went into much detail about the engineering involved, as it didn’t really make sense in either venue. One thing that amused me a bit is that the core technology that drove Vaire into action is something that actually should be very familiar to a physics or engineering student: a resonator. Theirs is obviously quite a bit more sophisticated than the base model, but at its heart it’s doing the same thing: storing charge and controlling frequency. It turns out that those are both essential to making reversible computers work: you need to store charge so it isn’t lost to ground when you empty a transistor, and you need to control the frequency so you can have waves with gentle transitions instead of the more sharp corners of the waves used in normal computers, thus wasting less heat in rapid changes of voltage. Vaire recently announced they’re getting 50% charge recovery from their test chips, and they’re working on raising that number.

Originally, the Quanta piece was focused more on reversible programming than energy use, as the energy angle seemed a bit more physics-focused than their computer science desk usually goes. The emphasis ended up changing as I worked on the draft, but it meant that an interesting parallel story got lost on the cutting-room floor. There’s a community of people who study reversible computing not from the engineering side, but from the computer science side, studying reversible logic and reversible programming languages. It’s a pursuit that goes back to the 1980’s, where at Caltech around when Feynman was teaching his course on the physics of computing a group of students were figuring out how to set up a reversible programming language. Called Janus, they sent their creation to Landauer, and the letter ended up with Michael Frank after Landauer died. There’s a lovely quote from it regarding their motivation: “We did it out of curiosity over whether such an odd animal as this was possible, and because we were interested in knowing where we put information when we programmed. Janus forced us to pay attention to where our bits went since none could be thrown away.”

Being forced to pay attention to information, in turn, is what has animated the computer science side of the reversible computing community. There are applications to debugging, where you can run code backwards when it gets stuck, to encryption and compression, where you want to be able to recover the information you hid away, and to security, where you want to keep track of information to make sure a hacker can’t figure out things they shouldn’t. Also, for a lot of these people, it’s just a fun puzzle. Early on my attention was caught by a paper by Hannah Earley describing a programming language called Alethe, a word you might recognize from the Greek word for truth, which literally means something like “not-forgetting”.

(Compression is particularly relevant for the “garbage data” you need to output in a reversible computation. If you want to add two numbers reversibly, naively you need to keep both input numbers and their output, but you can be more clever than that and just keep one of the inputs since you can subtract to find the other. There are a lot of substantially more clever tricks in this vein people have figured out over the years.)

I didn’t say anything about the other engineering approaches to reversible computing, that try to do something outside of traditional computer chips. There’s DNA computing, which tries to compute with a bunch of DNA in solution. There’s the old concept of ballistic reversible computing, where you imagine a computer that runs like a bunch of colliding billiard balls, conserving energy. Coordinating such a computer can be a nightmare, and early theoretical ideas were shown to be disrupted by something as tiny as a few stray photons from a distant star. But people like Frank figured out ways around the coordination problem, and groups have experimented with superconductors as places to toss those billiard balls around. The early billiard-inspired designs also had a big impact on quantum computing, where you need reversible gates and the only irreversible operation is the measurement. The name “Toffoli” comes up a lot in quantum computing discussions, I hadn’t known before this that Toffoli gates were originally for reversible computing in general, not specifically quantum computing.

Finally, I only gestured at the sci-fi angle. For reversible computing’s die-hards, it isn’t just a way to make efficient computers now. It’s the ultimate future of the technology, the kind of energy-efficiency civilization will need when we’re covering stars with shells of “computronium” full of busy joyous artificial minds.

And now that I think about it, they should chat with McInnes. He can tell them the kinds of stars they should build around.

by 4gravitons at June 13, 2025 04:00 PM

Justin Wilson — Controlling Quantum Chaos with Randomness

I’m only putting out a small Quantum Bite today because I’m on the road. Here’s a brief rundown of our new preprint “Universality of stochastic control of quantum chaos with measurement and feedback”, it hasn’t been peer-reviewed yet, but I’m excited about the result.

Someone pushing a boulder up a hill and nearly at the top

The idea came from classical chaos and control, and is pretty simple1. Imagine standing in a chaotic wilderness dotted with a few tiny, perfectly calm mountaintops. Place a boulder on one of those peaks and it can sit there forever—but miss the summit by a hair and it tumbles into the valley below. We want to escape the chaos of avalanches and rock slides, so our task is to sit a boulder up on the top of a mountain.

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Now suppose you randomly stop paying attention. Every so often (flip a coin, roll a die), you look up and find the boulder, and nudge it some fraction of the way back up the mountain. This sounds thoughtless—counterintuitive even—but leads us to an interesting realization: There is a critical “nudge-rate” that will get our boulder up the mountain. However, any less often than this value, the boulder is lost to the valleys below. This is a classical phase transition in disguise.

We extended the idea to a quantum-mechanical boulder, and quantum fuzziness changes everything. These types of dynamics have little points of stability classically, but when you look at them quantum mechanically, there is no way to park a boulder at these points. Quantum mechanics has this bad habit of blurring those otherwise stable points.

This is really a battle with quantum uncertainty. If we put our quantum boulder at the top of the hill it is stopped (zero velocity) and we know where it is, but this violates the uncertainty principle! If we know where it is, we should have no idea how fast it’s moving! Even if you get a quantum boulder to the top of the mountain, it will always fall back down.

This alters our whole conception of “control” from before2, but miraculously, there is still a phase transition, but it behaves differently. Whenever we push our quantum boulder up the hill at just the right rate, it still never reaches the top, but it starts to be up there a lot more often than normal. Formally, the probability P(h) that the boulder sits at a height h takes the schematic form

(This looks like it diverges at the summit but that’s the secret sauce of quantum uncertainty, this denominator never actually gets to zero.)

The result is the following plot from the preprint:

The horizontal access is the “nudge-rate” p and the y-axis represents (roughly) how often the boulder is at the top of the hill. Classically, you’d get perfect control for p > 0.5, but quantumly, at p = 0.5, you can see that above that, it’s not at the summit a lot, but we’ve reached a threshold. Even at p = 0.8, you “only” hold the peak about 80% of the time.

So we trade absolute control to something a bit more “uncertain.” We now get to the top more often than not since quantum mechanics tends to get in the way and push us down the side of the mountain again.

Behind the scenes are simulations, analytic calculations, and connections to random walks, turbulence, and even market dynamics, all of which justify the “Universality” in the title. If you’d like the technical story—including how weak measurements implement the quantum mechanical “nudge”—check out our preprint!

The original classical paper was titled The Probabilistic Control of Chaos.

I’m neglecting here how to even implement control quantum mechanically which involves quantum weak measurements. That’s an interesting story and if you want to know details, please read our preprint!

June 08, 2025

Tommaso Dorigo — Win A MSCA Post-Doctoral Fellowship!

Applications for MSCA Post-doctoral fellowships are on, and will be so until September 10 this year. What that means is that if you have less than 8 years of experience after your Ph.D., you can pair up with a research institute in Europe to present a research plan, and the European Commission may decide to fund it for two years (plus 6 months in industry in some cases).

In order for your application to have a chance to win funding, you need to:

have a great research topic in mind,
be ready to invest some time in writing a great application, and
pair up with an outstanding supervisor at a renowned research institute.

by Tommaso Dorigo at June 08, 2025 04:54 PM

June 05, 2025

Peter Rohde The Quantum Astronaut

[CONFIDENTIAL] Incident report Download

[STRICTLY CONFIDENTIAL] Incident report followup Download

[Nb: the following added on 24 April, 2025 due to oversight]

HR response – [STRICTLY CONFIDENTIAL] Incident report follow-up Download

by Peter Rohde at June 05, 2025 01:58 AM

June 04, 2025

Terence Tao — Decomposing a factorial into large factors (second version)

Boris Alexeev, Evan Conway, Matthieu Rosenfeld, Andrew Sutherland, Markus Uhr, Kevin Ventullo, and I have uploaded to the arXiv a second version of our paper “Decomposing a factorial into large factors“. This is a completely rewritten and expanded version of a previous paper of the same name. Thanks to many additional theoretical and numerical contributors from the other coauthors, we now have much more precise control on the main quantity ${t(N)}$ studied in this paper, allowing us to settle all the previous conjectures about this quantity in the literature.

As discussed in the previous post, ${t(N)}$ denotes the largest integer ${t}$ such that the factorial ${N!}$ can be expressed as a product of ${N}$ factors, each of which is at least ${t}$ . Computing ${t(N)}$ is a special case of the bin covering problem, which is known to be NP-hard in general; and prior to our work, ${t(N)}$ was only computed for ${N \leq 599}$ ; we have been able to compute ${t(N)}$ for all ${N \leq 10000}$ . In fact, we can get surprisingly sharp upper and lower bounds on ${t(N)}$ for much larger ${N}$ , with a precise asymptotic

$\displaystyle \frac{t(N)}{N} = \frac{1}{e} - \frac{c_0}{\log N} - \frac{O(1)}{\log^{1+c} N}$

for an explicit constant ${c_0 = 0.30441901\dots}$ , which we conjecture to be improvable to

$\displaystyle \frac{t(N)}{N} = \frac{1}{e} - \frac{c_0}{\log N} - \frac{c_1+o(1)}{\log^{2} N}$

for an explicit constant ${c_1 = 0.75554808\dots}$ : … For instance, we can demonstrate numerically that

$\displaystyle 0 \leq t(9 \times 10^8) - 316560601 \leq 113.$

As a consequence of this precision, we can verify several conjectures of Guy and Selfridge, namely

${t(N) \leq N/e}$ for all ${N \neq 1,2,4}$ .
${t(N) \geq \lfloor 2N/7\rfloor}$ for all ${N \neq 56}$ .
${t(N) \geq N/3}$ for all ${N \geq 3 \times 10^5}$ . (In fact we show this is true for ${N \geq 43632}$ , and that this threshold is best possible.)

Guy and Selfridge also claimed that one can establish ${t(N) \geq N/4}$ for all large ${N}$ purely by rearranging factors of ${2}$ and ${3}$ from the standard factorization ${1 \times 2 \times \dots \times N}$ of ${N!}$ , but surprisingly we found that this claim (barely) fails for all ${N > 26244}$ :

The accuracy of our bounds comes from several techniques:

Greedy algorithms, in which one allocates the largest prime factors of ${N!}$ first and then moves to smaller primes, provide quickly computable, though suboptimal, lower bounds on ${t(N)}$ for small, medium, and moderately large values;
Linear programming and integer programming methods provides extremely accurate upper and lower bounds on ${t(N)}$ for small and medium values of ${N}$ ;
Rearrangement methods can be analyzed asymptotically via linear programming, and work well for large ${N}$ ; and
The modified approximate factorization strategy, discussed in the previous post is now sharpened by using ${3}$ -smooth numbers (products of ${2}$ and ${3}$ ) as the primary “liquidity pool” to reallocate factors of ${N!}$ , as opposed to the previous approach of only using powers of ${2}$ .

To me, the biggest surprise was just how stunningly accurate the linear programming methods were; the very large number of repeated prime factors here actually make this discrete problem behave rather like a continuous one.

by Terence Tao at June 04, 2025 05:54 PM

June 03, 2025

Terence Tao — A Lean companion to “Analysis I”

Almost 20 years ago, I wrote a textbook in real analysis called “Analysis I“. It was intended to complement the many good available analysis textbooks out there by focusing more on foundational issues, such as the construction of the natural numbers, integers, rational numbers, and reals, as well as providing enough set theory and logic to allow students to develop proofs at high levels of rigor.

While some proof assistants such as Coq or Agda were well established when the book was written, formal verification was not on my radar at the time. However, now that I have had some experience with this subject, I realize that the content of this book is in fact very compatible with such proof assistants; in particular, the ‘naive type theory’ that I was implicitly using to do things like construct the standard number systems, dovetails well with the dependent type theory of Lean (which, among other things, has excellent support for quotient types).

I have therefore decided to launch a Lean companion to “Analysis I”, which is a “translation” of many of the definitions, theorems, and exercises of the text into Lean. In particular, this gives an alternate way to perform the exercises in the book, by instead filling in the corresponding “sorries” in the Lean code. (I do not however plan on hosting “official” solutions to the exercises in this companion; instead, feel free to create forks of the repository in which these sorries are filled in.)

Currently, the following sections of the text have been translated into Lean:

The formalization has been deliberately designed to be separate from the standard Lean math library Mathlib at some places, but reliant on it at others. For instance, Mathlib already has a standard notion of the natural numbers ${{\bf N}}$ . In the Lean formalization, I first develop “by hand” an alternate construction Chapter2.Nat of the natural numbers (or just Nat, if one is working in the Chapter2 namespace), setting up many of the basic results about these alternate natural numbers which parallel similar lemmas about ${{\bf N}}$ that are already in Mathlib (but with many of these lemmas set as exercises to the reader, with the proofs currently replaced with “sorries”). Then, in an epilogue section, isomorphisms between these alternate natural numbers and the Mathlib natural numbers are established (or more precisely, set as exercises). From that point on, the Chapter 2 natural numbers are deprecated, and the Mathlib natural numbers are used instead. I intend to continue this general pattern throughout the book, so that as one advances into later chapters, one increasingly relies on Mathlib’s definitions and functions, rather than directly referring to any counterparts from earlier chapters. As such, this companion could also be used as an introduction to Lean and Mathlib as well as to real analysis (somewhat in the spirit of the “Natural number game“, which in fact has significant thematic overlap with Chapter 2 of my text).

The code in this repository compiles in Lean, but I have not tested whether all of the (numerous) “sorries” in the code can actually be filled (i.e., if all the exercises can actually be solved in Lean). I would be interested in having volunteers “playtest” the companion to see if this can actually be done (and if the helper lemmas or “API” provided in the Lean files are sufficient to fill in the sorries in a conceptually straightforward manner without having to rely on more esoteric Lean programming techniques). Any other feedback will of course also be welcome.

[UPDATE, May 31: moved the companion to a standalone repository.]

by Terence Tao at June 03, 2025 03:28 PM

June 02, 2025

Terence Tao — On the number of exceptional intervals to the prime number theorem in short intervals

Ayla Gafni and I have just uploaded to the arXiv the paper “On the number of exceptional intervals to the prime number theorem in short intervals“. This paper makes explicit some relationships between zero density theorems and prime number theorems in short intervals which were somewhat implicit in the literature at present.

Zero density theorems are estimates of the form

$\displaystyle N(\sigma,T) \ll T^{A(\sigma)(1-\sigma)+o(1)}$

for various ${0 \leq \sigma < 1}$ , where ${T}$ is a parameter going to infinity, ${N(\sigma,T)}$ counts the number of zeroes of the Riemann zeta function of real part at least ${\sigma}$ and imaginary part between ${-T}$ and ${T}$ , and ${A(\sigma)}$ is an exponent which one would like to be as small as possible. The Riemann hypothesis would allow one to take ${A(\sigma)=-\infty}$ for any ${\sigma > 1/2}$ , but this is an unrealistic goal, and in practice one would be happy with some non-trivial upper bounds on ${A(\sigma)}$ . A key target here is the density hypothesis that asserts that ${A(\sigma) \leq 2}$ for all ${\sigma}$ (this is in some sense sharp because the Riemann-von Mangoldt formula implies that ${A(1/2)=2}$ ); this hypothesis is currently known for ${\sigma \leq 1/2}$ and ${\sigma \geq 25/32}$ , but the known bounds are not strong enough to establish this hypothesis in the remaining region. However, there was a recent advance of Guth and Maynard, which among other things improved the upper bound ${A_0}$ on ${\sup_\sigma A(\sigma)}$ from ${12/5=2.4}$ to ${30/13=2.307\dots}$ , marking the first improvement in this bound in over four decades. Here is a plot of the best known upper bounds on ${A(\sigma)}$ , either unconditionally, assuming the density hypothesis, or the stronger Lindelöf hypothesis:

One of the reasons we care about zero density theorems is that they allow one to localize the prime number theorem to short intervals. In particular, if we have the uniform bound ${A(\sigma) \leq A_0}$ for all ${\sigma}$ , then this leads to the prime number theorem

$\displaystyle \sum_{x \leq n < x+x^\theta} \Lambda(n) \sim x^\theta$ holding for all ${x}$ if ${\theta > 1-\frac{1}{A_0}}$ , and for almost all ${x}$ (possibly excluding a set of density zero) if ${\theta > 1 - \frac{2}{A_0}}$ . For instance, the Guth-Maynard results give a prime number theorem in almost all short intervals for ${\theta}$ as small as ${2/15+\varepsilon}$ , and the density hypotheis would lower this just to ${\varepsilon}$ .

However, one can ask about more information on this exceptional set, in particular to bound its “dimension” ${\mu(\theta)}$ , which roughly speaking amounts to getting an upper bound of ${X^{\mu(\theta)+o(1)}}$ on the size of the exceptional set in any large interval ${[X,2X]}$ . Based on the above assertions, one expects ${\mu(\theta)}$ to only be bounded by ${1}$ for ${\theta < 1-2/A}$ , be bounded by ${-\infty}$ for ${\theta > 1-1/A}$ , but have some intermediate bound for the remaining exponents.

This type of question had been studied in the past, most direclty by Bazzanella and Perelli, although there is earlier work by many authors om some related quantities (such as the second moment ${\sum_{n \leq x} (p_{n+1}-p_n)^2}$ of prime gaps) by such authors as Selberg and Heath-Brown. In most of these works, the best available zero density estimates at that time were used to obtain specific bounds on quantities such as ${\mu(\theta)}$ , but the numerology was usually tuned to those specific estimates, with the consequence being that when newer zero density estimates were discovered, one could not readily update these bounds to match. In this paper we abstract out the arguments from previous work (largely based on the explicit formula for the primes and the second moment method) to obtain an explicit relationship between ${\mu(\theta)}$ and ${A(\sigma)}$ , namely that

$\displaystyle \mu(\theta) \leq \inf_{\varepsilon>0} \sup_{0 \leq \theta<1; A(\sigma) \geq \frac{1}{1-\theta}-\varepsilon} \mu_{2,\sigma}(\theta)$ where

$\displaystyle \mu_{2,\theta}(\theta) = (1-\theta)(1-\sigma)A(\sigma)+2\sigma-1.$ Actually, by also utilizing fourth moment methods, we obtain a stronger bound

$\displaystyle \mu(\theta) \leq \inf_{\varepsilon>0} \sup_{0 \leq \theta<1; A(\sigma) \geq \frac{1}{1-\theta}-\varepsilon} \min( \mu_{2,\sigma}(\theta), \mu_{4,\sigma}(\theta) )$ where

$\displaystyle \mu_{4,\theta}(\theta) = (1-\theta)(1-\sigma)A^*(\sigma)+4\sigma-3$ and ${A^*(\sigma)}$ is the exponent in “additive energy zero density theorems”

$\displaystyle N^*(\sigma,T) \ll T^{A^*(\sigma)(1-\sigma)+o(1)}$ where ${N^*(\sigma,T)}$ is similar to ${N(\sigma,T)}$ , but bounds the “additive energy” of zeroes rather than just their cardinality. Such bounds have appeared in the literature since the work of Heath-Brown, and are for instance a key ingredient in the recent work of Guth and Maynard. Here are the current best known bounds:

These explicit relationships between exponents are perfectly suited for the recently launched Analytic Number Theory Exponent Database (ANTEDB) (discussed previously here), and have been uploaded to that site.

This formula is moderately complicated (basically an elaborate variant of a Legendre transform), but easy to calculate numerically with a computer program. Here is the resulting bound on ${\mu(\theta)}$ unconditionally and under the density hypothesis (together with a previous bound of Bazzanella and Perelli for comparison, where the range had to be restricted due to a gap in the argument we discovered while trying to reproduce their results):

For comparison, here is the situation assuming strong conjectures such as the density hypothesis, Lindelof hypothesis, or Riemann hypothesis:

by Terence Tao at June 02, 2025 11:52 PM

May 31, 2025

Tommaso Dorigo — The Anomaly That Wasn't: An Example Of Shifting Consensus In Science

Time is a gentleman - it waits patiently. And in physics, as in all exact sciences, problems and mysteries eventually get resolved, if we give it enough time. That is how science works, after all: the consensus on our explanation of reality changes as we acquire more information on the latter.

by Tommaso Dorigo at May 31, 2025 08:57 AM

Scott Aaronson “If Anyone Builds It, Everyone Dies”

Eliezer Yudkowsky and Nate Soares are publishing a mass-market book, the rather self-explanatorily-titled If Anyone Builds It, Everyone Dies. (Yes, the “it” means “sufficiently powerful AI.”) The book is now available for preorder from Amazon:

(If you plan to buy the book at all, Eliezer and Nate ask that you do preorder it, as this will apparently increase the chance of it making the bestseller lists and becoming part of The Discourse.)

I was graciously offered a chance to read a draft and offer, not a “review,” but some preliminary thoughts. So here they are:

For decades, Eliezer has been warning the world that an AI might soon exceed human abilities, and proceed to kill everyone on earth, in pursuit of whatever strange goal it ended up with. It would, Eliezer said, be something like what humans did to the earlier hominids. Back around 2008, I followed the lead of most of my computer science colleagues, who considered these worries, even if possible in theory, comically premature given the primitive state of AI at the time, and all the other severe crises facing the world.

Now, of course, not even two decades later, we live on a planet that’s being transformed by some of the signs and wonders that Eliezer foretold. The world’s economy is about to be upended by entities like Claude and ChatGPT, AlphaZero and AlphaFold—whose human-like or sometimes superhuman cognitive abilities, obtained “merely” by training neural networks (in the first two cases, on humanity’s collective output) and applying massive computing power, constitute (I’d say) the greatest scientific surprise of my lifetime. Notably, these entities have already displayed some of the worrying behaviors that Eliezer warned about decades ago—including lying to humans in pursuit of a goal, and hacking their own evaluation criteria. Even many of the economic and geopolitical aspects have played out as Eliezer warned they would: we’ve now seen AI companies furiously racing each other, seduced by the temptation of being (as he puts it) “the first monkey to taste the poisoned banana,” discarding their previous explicit commitments to safety, transparency, and the public good once they get in the way.

Today, then, even if one still isn’t ready to swallow the full package of Yudkowskyan beliefs, any empirically minded person ought to be updating in its direction—and acting accordingly. Which brings us to the new book by Eliezer and his collaborator Nate Soares. This book is far and away the clearest, most accessible presentation of Eliezer’s beliefs, the culmination of a quarter-century of his developing and talking about them. That undoubtedly owes a great deal to Nate, who seems to have sanded down the infamously brusque rough edges of Eliezer’s writing style. So much the better! But it also owes a lot to the world itself: current events now offer an endless supply of real-world examples for Eliezer’s formerly abstract arguments about AI, examples that the book deploys to maximum effect.

The book also mines history—everything from the Wright Brothers to World War II to the Chernobyl accident—for lessons about human attitudes toward technological progress, safety, and risk. And it maintains Eliezer’s fondness for stories and parables, one of the most charming features of his writing.

Even today, I’m not nearly as confident about the doom scenario as Eliezer and Nate are. I don’t know whether an AI’s goals are really “orthogonal” to its abilities, in the sense that will matter in practice. And when I reach the part where the AI, having copied itself all over the Internet and built robot factories, then invents and releases self-replicating nanotechnology that gobbles the surface of the earth in hours or days, a large part of me still screams out that there must be practical bottlenecks that haven’t been entirely accounted for here.

And yet, even if you agree with only a quarter of what Eliezer and Nate write, you’re likely to close this book fully convinced—as I am—that governments need to shift to a more cautious approach to AI, an approach more respectful of the civilization-changing enormity of what’s being created. And that, if they won’t, their citizens need to pressure them to do so.

So regardless of how much they agree or disagree, I’d like everyone on earth who cares about the future to read this book, debate its ideas, and have its thesis in mind when they’re discussing AI.

As for me? It would’ve been better if I’d reached my current position earlier: if I hadn’t needed empirical reality, plus superb writing like Eliezer’s and Nate’s, to bonk me over the head with the risks that AI was likely to pose to humanity in my lifetime. But having failed to see as far ahead as they did, the least I can do is update. You should too, and you can start by reading the book.

As it happens, this weekend I’ll be at LessOnline, the rationalist blogging conference in Berkeley, where (among my other events) I’ll engage in a dialogue/debate with Nate Soares about the orthogonality thesis, one of the crucial underpinnings of his and Eliezer’s case for AI doom. So, I’ll probably be LessAvailable to respond to comments on this post. But feel free to discuss anyway! After all, it’s merely the fate of all Earth-originating life that’s at stake here, not some actually hot-button topic like Trump or Gaza.

by Scott at May 31, 2025 12:46 AM

May 30, 2025

Terence Tao — Cosmic Distance Ladder videos with Grant Sanderson (3blue1brown): commentary and corrections

Grant Sanderson (who runs, and creates most of the content for, the website and Youtube channel 3blue1brown) has been collaborating with myself and others (including my coauthor Tanya Klowden) on producing a two-part video giving an account of some of the history of the cosmic distance ladder, building upon a previous public lecture I gave on this topic, and also relating to a forthcoming popular book with Tanya on this topic. The first part of this video is available here; the second part is available here.

The videos were based on a somewhat unscripted interview that Grant conducted with me some months ago, and as such contained some minor inaccuracies and omissions (including some made for editing reasons to keep the overall narrative coherent and within a reasonable length). They also generated many good questions from the viewers of the Youtube video. I am therefore compiling here a “FAQ” of various clarifications and corrections to the videos; this was originally placed as a series of comments on the Youtube channel, but the blog post format here will be easier to maintain going forward. Some related content will also be posted on the Instagram page for the forthcoming book with Tanya.

Questions on the two main videos are marked with an appropriate timestamp to the video.

Comments on part 1 of the video

4:26 Did Eratosthenes really check a local well in Alexandria?

This was a narrative embellishment on my part. Eratosthenes’s original work is lost to us. The most detailed contemperaneous account, by Cleomedes, gives a simplified version of the method, and makes reference only to sundials (gnomons) rather than wells. However, a secondary account of Pliny states (using this English translation), “Similarly it is reported that at the town of Syene, 5000 stades South of Alexandria, at noon in midsummer no shadow is cast, and that in a well made for the sake of testing this the light reaches to the bottom, clearly showing that the sun is vertically above that place at the time”. However, no mention is made of any well in Alexandria in either account.
4:50 How did Eratosthenes know that the Sun was so far away that its light rays were close to parallel?

This was not made so clear in our discussions or in the video (other than a brief glimpse of the timeline at 18:27), but Eratosthenes’s work actually came after Aristarchus, so it is very likely that Eratosthenes was aware of Aristarchus’s conclusions about how distant the Sun was from the Earth. Even if Aristarchus’s heliocentric model was disputed by the other Greeks, at least some of his other conclusions appear to have attracted some support. Also, after Eratosthenes’s time, there was further work by Greek, Indian, and Islamic astronomers (such as Hipparchus, Ptolemy, Aryabhata, and Al-Battani) to measure the same distances that Aristarchus did, although these subsequent measurements for the Sun also were somewhat far from modern accepted values.
5:17 Is it completely accurate to say that on the summer solstice, the Earth’s axis of rotation is tilted “directly towards the Sun”?

Strictly speaking, “in the direction towards the Sun” is more accurate than “directly towards the Sun”; it tilts at about 23.5 degrees towards the Sun, but it is not a total 90-degree tilt towards the Sun.
5:39 Wait, aren’t there two tropics? The tropic of Cancer and the tropic of Capricorn?

Yes! This corresponds to the two summers Earth experiences, one in the Northern hemisphere and one in the Southern hemisphere. The tropic of Cancer, at a latitude of about 23 degrees north, is where the Sun is directly overhead at noon during the Northern summer solstice (around June 21); the tropic of Capricorn, at a latitude of about 23 degrees south, is where the Sun is directly overhead at noon during the Southern summer solstice (around December 21). But Alexandria and Syene were both in the Northern Hemisphere, so it is the tropic of Cancer that is relevant to Eratosthenes’ calculations.
5:41 Isn’t it kind of a massive coincidence that Syene was on the tropic of Cancer?

Actually, Syene (now known as Aswan) was about half a degree of latitude away from the tropic of Cancer, which was one of the sources of inaccuracy in Eratosthenes’ calculations. But one should take the “look-elsewhere effect” into account: because the Nile cuts across the tropic of Cancer, it was quite likely to happen that the Nile would intersect the tropic near some inhabited town. It might not necessarily have been Syene, but that would just mean that Syene would have been substituted by this other town in Eratosthenes’s account.

On the other hand, it was fortunate that the Nile ran from South to North, so that distances between towns were a good proxy for the differences in latitude. Apparently, Eratosthenes actually had a more complicated argument that would also work if the two towns in question were not necessarily oriented along the North-South direction, and if neither town was on the tropic of Cancer; but unfortunately the original writings of Eratosthenes are lost to us, and we do not know the details of this more general argument. (But some variants of the method can be found in later work of Posidonius, Aryabhata, and others.)

Nowadays, the “Eratosthenes experiment” is run every year on the March equinox, in which schools at the same longitude are paired up to measure the elevation of the Sun at the same point in time, in order to obtain a measurement of the circumference of the Earth. (The equinox is more convenient than the solstice when neither location is on a tropic, due to the simple motion of the Sun at that date.) With modern timekeeping, communications, surveying, and navigation, this is a far easier task to accomplish today than it was in Eratosthenes’ time.
6:30 I thought the Earth wasn’t a perfect sphere. Does this affect this calculation?

Yes, but only by a small amount. The centrifugal forces caused by the Earth’s rotation along its axis cause an equatorial bulge and a polar flattening so that the radius of the Earth fluctuates by about 20 kilometers from pole to equator. This sounds like a lot, but it is only about 0.3% of the mean Earth radius of 6371 km and is not the primary source of error in Eratosthenes’ calculations.
7:27 Are the riverboat merchants and the “grad student” the leading theories for how Eratosthenes measured the distance from Alexandria to Syene?

There is some recent research that suggests that Eratosthenes may have drawn on the work of professional bematists (step measurers – a precursor to the modern profession of surveyor) for this calculation. This somewhat ruins the “grad student” joke, but perhaps should be disclosed for the sake of completeness.
8:51 How long is a “lunar month” in this context? Is it really 28 days?

In this context the correct notion of a lunar month is a “synodic month” – the length of a lunar cycle relative to the Sun – which is actually about 29 days and 12 hours. It differs from the “sidereal month” – the length of a lunar cycle relative to the fixed stars – which is about 27 days and 8 hours – due to the motion of the Earth around the Sun (or the Sun around the Earth, in the geocentric model). [A similar correction needs to be made around 14:59, using the synodic month of 29 days and 12 hours rather than the “English lunar month” of 28 days (4 weeks).]
10:47 Is the time taken for the Moon to complete an observed rotation around the Earth slightly less than 24 hours as claimed?

Actually, I made a sign error: the lunar day (also known as a tidal day) is actually 24 hours and 50 minutes, because the Moon rotates in the same direction as the spinning of Earth around its axis. The animation therefore is also moving in the wrong direction as well (related to this, the line of sight is covering up the Moon in the wrong direction to the Moon rising at around 10:38).
11:32 Is this really just a coincidence that the Moon and Sun have almost the same angular width?

I believe so. First of all, the agreement is not that good: due to the non-circular nature of the orbit of the Moon around the Earth, and Earth around the Sun, the angular width of the Moon actually fluctuates to be as much as 10% larger or smaller than the Sun at various times (cf. the “supermoon” phenomenon). All other known planets with known moons do not exhibit this sort of agreement, so there does not appear to be any universal law of nature that would enforce this coincidence. (This is in contrast with the empirical fact that the Moon always presents the same side to the Earth, which occurs in all other known large moons (as well as Pluto), and is well explained by the physical phenomenon of tidal locking.)

On the other hand, as the video hopefully demonstrates, the existence of the Moon was extremely helpful in allowing the ancients to understand the basic nature of the solar system. Without the Moon, their task would have been significantly more difficult; but in this hypothetical alternate universe, it is likely that modern cosmology would have still become possible once advanced technology such as telescopes, spaceflight, and computers became available, especially when combined with the modern mathematics of data science. Without giving away too many spoilers, a scenario similar to this was explored in the classic short story and novel “Nightfall” by Isaac Asimov.
12:58 Isn’t the illuminated portion of the Moon, as well as the visible portion of the Moon, slightly smaller than half of the entire Moon, because the Earth and Sun are not an infinite distance away from the Moon?

Technically yes (and this is actually for a very similar reason to why half Moons don’t quite occur halfway between the new Moon and the full Moon); but this fact turns out to have only a very small effect on the calculations, and is not the major source of error. In reality, the Sun turns out to be about 86,000 Moon radii away from the Moon, so asserting that half of the Moon is illuminated by the Sun is actually a very good first approximation. (The Earth is “only” about 220 Moon radii away, so the visible portion of the Moon is a bit more noticeably less than half; but this doesn’t actually affect Aristarchus’s arguments much.)

The angular diameter of the Sun also creates an additional thin band between the fully illuminated and fully non-illuminated portions of the Moon, in which the Sun is intersecting the lunar horizon and so only illuminates the Moon with a portion of its light, but this is also a relatively minor effect (and the midpoints of this band can still be used to define the terminator between illuminated and non-illuminated for the purposes of Aristarchus’s arguments).
13:27 What is the difference between a half Moon and a quarter Moon?

If one divides the lunar month, starting and ending at a new Moon, into quarters (weeks), then half moons occur both near the end of the first quarter (a week after the new Moon, and a week before the full Moon), and near the end of the third quarter (a week after the full Moon, and a week before the new Moon). So, somewhat confusingly, half Moons come in two types, known as “first quarter Moons” and “third quarter Moons”.
14:49 I thought the sine function was introduced well after the ancient Greeks.

It’s true that the modern sine function only dates back to the Indian and Islamic mathematical traditions in the first millennium CE, several centuries after Aristarchus. However, he still had Euclidean geometry at his disposal, which provided tools such as similar triangles that could be used to reach basically the same conclusions, albeit with significantly more effort than would be needed if one could use modern trigonometry.

On the other hand, Aristarchus was somewhat hampered by not knowing an accurate value for $\pi$ , which is also known as Archimedes’ constant: the fundamental work of Archimedes on this constant actually took place a few decades after that of Aristarchus!
15:17 I plugged in the modern values for the distances to the Sun and Moon and got 18 minutes for the discrepancy, instead of half an hour.

Yes; I quoted the wrong number here. In 1630, Godfried Wendelen replicated Aristarchus’s experiment. With improved timekeeping and the then-recent invention of the telescope, Wendelen obtained a measurement of half an hour for the discrepancy, which is significantly better than Aristarchus’s calculation of six hours, but still a little bit off from the true value of 18 minutes. (As such, Wendelinus’s estimate for the distance to the Sun was 60% of the true value.)
15:27 Wouldn’t Aristarchus also have access to other timekeeping devices than sundials?

Yes, for instance clepsydrae (water clocks) were available by that time; but they were of limited accuracy. It is also possible that Aristarchus could have used measurements of star elevations to also estimate time; it is not clear whether the astrolabe or the armillary sphere was available to him, but he would have had some other more primitive astronomical instruments such as the dioptra at his disposal. But again, the accuracy and calibration of these timekeeping tools would have been poor.

However, most likely the more important limiting factor was the ability to determine the precise moment at which a perfect half Moon (or new Moon, or full Moon) occurs; this is extremely difficult to do with the naked eye. (The telescope would not be invented for almost two more millennia.)
17:37 Could the parallax problem be solved by assuming that the stars are not distributed in a three-dimensional space, but instead on a celestial sphere?

Putting all the stars on a fixed sphere would make the parallax effects less visible, as the stars in a given portion of the sky would now all move together at the same apparent velocity – but there would still be visible large-scale distortions in the shape of the constellations because the Earth would be closer to some portions of the celestial sphere than others; there would also be variability in the brightness of the stars, and (if they were very close) the apparent angular diameter of the stars. (These problems would be solved if the celestial sphere was somehow centered around the moving Earth rather than the fixed Sun, but then this basically becomes the geocentric model with extra steps.)
18:29 Did nothing of note happen in astronomy between Eratosthenes and Copernicus?

Not at all! There were significant mathematical, technological, theoretical, and observational advances by astronomers from many cultures (Greek, Islamic, Indian, Chinese, European, and others) during this time, for instance improving some of the previous measurements on the distance ladder, a better understanding of eclipses, axial tilt, and even axial precession, more sophisticated trigonometry, and the development of new astronomical tools such as the astrolabe. See for instance this “deleted scene” from the video, as well as the FAQ entry for 14:49 for this video and 24:54 for the second video, or this instagram post. But in order to make the overall story of the cosmic distance ladder fit into a two-part video, we chose to focus primarily on the first time each rung of the ladder was climbed.
18:30 Is that really Kepler’s portrait?

We have since learned that this portrait was most likely painted in the 19th century, and may have been based more on Kepler’s mentor, Michael Mästlin. A more commonly accepted portrait of Kepler may be found at his current Wikipedia page.
19:07 Isn’t it tautological to say that the Earth takes one year to perform a full orbit around the Sun?

Technically yes, but this is an illustration of the philosophical concept of “referential opacity“: the content of a sentence can change when substituting one term for another (e.g., “1 year” and “365 days”), even when both terms refer to the same object. Amusingly, the classic illustration of this, known as Frege’s puzzles, also comes from astronomy: it is an informative statement that Hesperus (the evening star) and Phosphorus (the morning star, also known as Lucifer) are the same object (which nowadays we call Venus), but it is a mere tautology that Hesperus and Hesperus are the same object: changing the reference from Phosphorus to Hesperus changes the meaning.
19:10 How did Copernicus figure out the crucial fact that Mars takes 687 days to go around the Sun? Was it directly drawn from Babylonian data?

Technically, Copernicus drew from tables by European astronomers that were largely based on earlier tables from the Islamic golden age, which in turn drew from earlier tables by Indian and Greek astronomers, the latter of which also incorporated data from the ancient Babylonians, so it is more accurate to say that Copernicus relied on centuries of data, at least some of which went all the way back to the Babylonians. Among all of this data was the times when Mars was in opposition to the Sun; if one imagines the Earth and Mars as being like runners going around a race track circling the Sun, with Earth on an inner track and Mars on an outer track, oppositions are analogous to when the Earth runner “laps” the Mars runner. From the centuries of observational data, such “laps” were known to occur about once every 780 days (this is known as the synodic period of Mars). Because the Earth takes roughly 365 days to perform a “lap”, it is possible to do a little math and conclude that Mars must therefore complete its own “lap” in 687 days (this is known as the sidereal period of Mars). (See also this post on the cosmic distance ladder Instagram for some further elaboration.)
20:52 Did Kepler really steal data from Brahe?

The situation is complex. When Kepler served as Brahe’s assistant, Brahe only provided Kepler with a limited amount of data, primarily involving Mars, in order to confirm Brahe’s own geo-heliocentric model. After Brahe’s death, the data was inherited by Brahe’s son-in-law and other relatives, who intended to publish Brahe’s work separately; however, Kepler, who was appointed as Imperial Mathematician to succeed Brahe, had at least some partial access to the data, and many historians believe he secretly copied portions of this data to aid his own research before finally securing complete access to the data from Brahe’s heirs after several years of disputes. On the other hand, as intellectual property rights laws were not well developed at this time, Kepler’s actions were technically legal, if ethically questionable.
21:39 What is that funny loop in the orbit of Mars?

This is known as retrograde motion. This arises because the orbital velocity of Earth (about 30 km/sec) is a little bit larger than that of Mars (about 24 km/sec). So, in opposition (when Mars is in the opposite position in the sky than the Sun), Earth will briefly overtake Mars, causing its observed position to move westward rather than eastward. But in most other times, the motion of Earth and Mars are at a sufficient angle that Mars will continue its apparent eastward motion despite the slightly faster speed of the Earth.
21:59 Couldn’t one also work out the direction to other celestial objects in addition to the Sun and Mars, such as the stars, the Moon, or the other planets? Would that have helped?

Actually, the directions to the fixed stars were implicitly used in all of these observations to determine how the celestial sphere was positioned, and all the other directions were taken relative to that celestial sphere. (Otherwise, all the calculations would be taken on a rotating frame of reference in which the unknown orbits of the planets were themselves rotating, which would have been an even more complex task.) But the stars are too far away to be useful as one of the two landmarks to triangulate from, as they generate almost no parallax and so cannot distinguish one location from another.

Measuring the direction to the Moon would tell you which portion of the lunar cycle one was in, and would determine the phase of the Moon, but this information would not help one triangulate, because the Moon’s position in the heliocentric model varies over time in a somewhat complicated fashion, and is too tied to the motion of the Earth to be a useful “landmark” to one to determine the Earth’s orbit around the Sun.

In principle, using the measurements to all the planets at once could allow for some multidimensional analysis that would be more accurate than analyzing each of the planets separately, but this would require some sophisticated statistical analysis and modeling, as well as non-trivial amounts of compute – neither of which were available in Kepler’s time.
22:57 Can you elaborate on how we know that the planets all move on a plane?

The Earth’s orbit lies in a plane known as the ecliptic (it is where the lunar and solar eclipses occur). Different cultures have divided up the ecliptic in various ways; in Western astrology, for instance, the twelve main constellations that cross the ecliptic are known as the Zodiac. The planets can be observed to only wander along the Zodiac, but not other constellations: for instance, Mars can be observed to be in Cancer or Libra, but never in Orion or Ursa Major. From this, one can conclude (as a first approximation, at least), that the planets all lie on the ecliptic.

However, this isn’t perfectly true, and the planets will deviate from the ecliptic by a small angle known as the ecliptic latitude. Tycho Brahe’s observations on these latitudes for Mars were an additional useful piece of data that helped Kepler complete his calculations (basically by suggesting how to join together the different “jigsaw pieces”), but the math here gets somewhat complicated, so the story here has been somewhat simplified to convey the main ideas.
23:04 What are the other universal problem solving tips?

Grant Sanderson has a list (in a somewhat different order) in this previous video.
23:28 Can one work out the position of Earth from fixed locations of the Sun and Mars when the Sun and Mars are in conjunction (the same location in the sky) or opposition (opposite locations in the sky)?

Technically, these are two times when the technique of triangulation fails to be accurate; and also in the former case it is extremely difficult to observe Mars due to the proximity to the Sun. But again, following the Universal Problem Solving Tip from 23:07, one should initially ignore these difficulties to locate a viable method, and correct for these issues later. This video series by Welch Labs goes into Kepler’s methods in more detail.
24:04 So Kepler used Copernicus’s calculation of 687 days for the period of Mars. But didn’t Kepler discard Copernicus’s theory of circular orbits?

Good question! It turns out that Copernicus’s calculations of orbital periods are quite robust (especially with centuries of data), and continue to work even when the orbits are not perfectly circular. But even if the calculations did depend on the circular orbit hypothesis, it would have been possible to use the Copernican model as a first approximation for the period, in order to get a better, but still approximate, description of the orbits of the planets. This in turn can be fed back into the Copernican calculations to give a second approximation to the period, which can then give a further refinement of the orbits. Thanks to the branch of mathematics known as perturbation theory, one can often make this type of iterative process converge to an exact answer, with the error in each successive approximation being smaller than the previous one. (But performing such an iteration would probably have been beyond the computational resources available in Kepler’s time; also, the foundations of perturbation theory require calculus, which only was developed several decades after Kepler.)
24:21 Did Brahe have exactly 10 years of data on Mars’s positions?

Actually, it was more like 17 years, but with many gaps, due both to inclement weather, as well as Brahe turning his attention to other astronomical objects than Mars in some years; also, in times of conjunction, Mars might only be visible in the daytime sky instead of the night sky, again complicating measurements. So the “jigsaw puzzle pieces” in 25:26 are in fact more complicated than always just five locations equally spaced in time; there are gaps and also observational errors to grapple with. But to understand the method one should ignore these complications; again, see “Universal Problem Solving Tip #1”. Even with his “idea of true genius”, it took many years of further painstaking calculation for Kepler to tease out his laws of planetary motion from Brahe’s messy and incomplete observational data.
26:44 Shouldn’t the Earth’s orbit be spread out at perihelion and clustered closer together at aphelion, to be consistent with Kepler’s laws?

Yes, you are right; there was a coding error here.
26:53 What is the reference for Einstein’s “idea of pure genius”?

Actually, the precise quote was “an idea of true genius”, and can be found in the introduction to Carola Baumgardt’s “Life of Kepler“.

Comments on the “deleted scene” on Al-Biruni

Was Al-Biruni really of Arab origin?

Strictly speaking; no; his writings are all in Arabic, and he was nominally a subject of the Abbasid Caliphate whose rulers were Arab; but he was born in Khwarazm (in modern day Uzbekistan), and would have been a subject of either the Samanid empire or the Khrawazmian empire, both of which were largely self-governed and primarily Persian in culture and ethnic makeup, despite being technically vassals of the Caliphate. So he would have been part of what is sometimes called “Greater Persia” or “Greater Iran”.

Another minor correction: while Al-Biruni was born in the tenth century, his work on the measurement of the Earth was published in the early eleventh century.
Is $\theta$ really called the angle of declination?

This was a misnomer on my part; this angle is more commonly called the dip angle.
But the height of the mountain would be so small compared to the radius of the Earth! How could this method work?

Using the Taylor approximation $\cos \theta \approx 1 - \theta^2/2$ , one can approximately write the relationship $R = \frac{h \cos \theta}{1-\cos \theta}$ between the mountain height $h$ , the Earth radius $R$ , and the dip angle $\theta$ (in radians) as $R \approx 2 h / \theta^2$ . The key point here is the inverse quadratic dependence on $\theta$ , which allows for even relatively small values of $h$ to still be realistically useful for computing $R$ . Al-Biruni’s measurement of the dip angle $\theta$ was about $0.01$ radians, leading to an estimate of $R$ that is about four orders of magnitude larger than $h$ , which is within ballpark at least of a typical height of a mountain (on the order of a kilometer) and the radius of the Earth (6400 kilometers).
Was the method really accurate to within a percentage point?

This is disputed, somewhat similarly to the previous calculations of Eratosthenes. Al-Biruni’s measurements were in cubits, but there were multiple incompatible types of cubit in use at the time. It has also been pointed out that atmospheric refraction effects would have created noticeable changes in the observed dip angle $\theta$ . It is thus likely that the true accuracy of Al-Biruni’s method was poorer than 1%, but that this was somehow compensated for by choosing a favorable conversion between cubits and modern units.

Comments on the second part of the video

1:13 Did Captain Cook set out to discover Australia?

One of the objectives of Cook’s first voyage was to discover the hypothetical continent of Terra Australis. This was considered to be distinct from Australia, which at the time was known as New Holland. As this name might suggest, prior to Cook’s voyage, the northwest coastline of New Holland had been explored by the Dutch; Cook instead explored the eastern coastline, naming this portion New South Wales. The entire continent was later renamed to Australia by the British government, following a suggestion of Matthew Flinders; and the concept of Terra Australis was abandoned.
4:40 The relative position of the Northern and Southern hemisphere observations is reversed from those earlier in the video.

Yes, this was a slight error in the animation; the labels here should be swapped for consistency of orientation.
7:06 So, when did they finally manage to measure the transit of Venus, and use this to compute the astronomical unit?

While Le Gentil had the misfortune to not be able to measure either the 1761 or 1769 transits, other expeditions of astronomers (led by Dixon-Mason, Chappe d’Auteroche, and Cook) did take measurements of one or both of these transits with varying degrees of success, with the measurements of Cook’s team of the 1769 transit in Tahiti being of particularly high quality. All of this data was assembled later by Lalande in 1771, leading to the most accurate measurement of the astronomical unit at the time (within 2.3% of modern values, which was about three times more accurate than any previous measurement).
8:53 What does it mean for the transit of Io to be “twenty minutes ahead of schedule” when Jupiter is in opposition (Jupiter is opposite to the Sun when viewed from the Earth)?

Actually, it should be halved to “ten minutes ahead of schedule”, with the transit being “ten minutes behind schedule” when Jupiter is in conjunction, with the net discrepancy being twenty minutes (or actually closer to 16 minutes when measured with modern technology). Both transits are being compared against an idealized periodic schedule in which the transits are occuring at a perfectly regular rate (about 42 hours), where the period is chosen to be the best fit to the actual data. This discrepancy is only noticeable after carefully comparing transit times over a period of months; at any given position of Jupiter, the Doppler effects of Earth moving towards or away from Jupiter would only affect shift each transit by just a few seconds compared to the previous transit, with the delays or accelerations only becoming cumulatively noticeable after many such transits.

Also, the presentation here is oversimplified: at times of conjunction, Jupiter and Io are too close to the Sun for observation of the transit. Rømer actually observed the transits at other times than conjunction, and Huygens used more complicated trigonometry than what was presented here to infer a measurement for the speed of light in terms of the astronomical unit (which they had begun to measure a bit more accurately than in Aristarchus’s time; see the FAQ entry for 15:17 in the first video).
10:05 Are the astrological signs for Earth and Venus swapped here?

Yes, this was a small mistake in the animation.
10:34 Shouldn’t one have to account for the elliptical orbit of the Earth, as well as the proper motion of the star being observed, or the effects of general relativity?

Yes; the presentation given here is a simplified one to convey the idea of the method, but in the most advanced parallax measurements, such as the ones taken by the Hipparcos and Gaia spacecraft, these factors are taken into account, basically by taking as many measurements (not just two) as possible of a single star, and locating the best fit of that data to a multi-parameter model that incorporates the (known) orbit of the Earth with the (unknown) distance and motion of the star, as well as additional gravitational effects from other celestial bodies, such as the Sun and other planets.
14:53 The formula I was taught for apparent magnitude of stars looks a bit different from the one here.

This is because astronomers use a logarithmic scale to measure both apparent magnitude $m$ and absolute magnitude $M$ . If one takes the logarithm of the inverse square law in the video, and performs the normalizations used by astronomers to define magnitude, one arrives at the standard relation $M = m - 5 \log_{10} d_{pc} + 5$ between absolute and apparent magnitude.

But this is an oversimplification, most notably due to neglect of the effects of extinction effects caused by interstellar dust. This is not a major issue for the relatively short distances observable via parallax, but causes problems at larger scales of the ladder (see for instance the FAQ entry here for 18:08). To compensate for this, one can work in multiple frequencies of the spectrum (visible, x-ray, radio, etc.), as some frequencies are less susceptible to extinction than others. From the discrepancies between these frequencies one can infer the amount of extinction, leading to “dust maps” that can then be used to facilitate such corrections for subsequent measurements in the same area of the universe. (More generally, the trend in modern astronomy is towards “multi-messenger astronomy” in which one combines together very different types of measurements of the same object to obtain a more accurate understanding of that object and its surroundings.)
18:08 Can we really measure the entire Milky Way with this method?

Strictly speaking, there is a “zone of avoidance” on the far side of the Milky way that is very difficult to measure in the visible portion of the spectrum, due to the large amount of intervening stars, dust, and even a supermassive black hole in the galactic center. However, in recent years it has become possible to explore this zone to some extent using the radio, infrared, and x-ray portions of the spectrum, which are less affected by these factors.
18:19 How did astronomers know that the Milky Way was only a small portion of the entire universe?

This issue was the topic of the “Great Debate” in the early twentieth century. It was only with the work of Hubble using Leavitt’s law to measure distances to Magellanic clouds and “spiral nebulae” (that we now know to be other galaxies), building on earlier work of Leavitt and Hertzsprung, that it was conclusively established that these clouds and nebulae in fact were at much greater distances than the diameter of the Milky Way.
18:45 How can one compensate for light blending effects when measuring the apparent magnitude of Cepheids?

This is a non-trivial task, especially if one demands a high level of accuracy. Using the highest resolution telescopes available (such as HST or JWST) is of course helpful, as is switching to other frequencies, such as near-infrared, where Cepheids are even brighter relative to nearby non-Cepheid stars. One can also apply sophisticated statistical methods to fit to models of the point spread of light from unwanted sources, and use nearby measurements of the same galaxy without the Cepheid as a reference to help calibrate those models. Improving the accuracy of the Cepheid portion of the distance ladder is an ongoing research activity in modern astronomy.
18:54 What is the mechanism that causes Cepheids to oscillate?

For most stars, there is an equilibrium size: if the star’s radius collapses, then the reduced potential energy is converted to heat, creating pressure to pushing the star outward again; and conversely, if the star expands, then it cools, causing a reduction in pressure that no longer counteracts gravitational forces. But for Cepheids, there is an additional mechanism called the kappa mechanism: the increased temperature caused by contraction increases ionization of helium, which drains energy from the star and accelerates the contraction; conversely, the cooling caused by expansion causes the ionized helium to recombine, with the energy released accelerating the expansion. If the parameters of the Cepheid are in a certain “instability strip”, then the interaction of the kappa mechanism with the other mechanisms of stellar dynamics create a periodic oscillation in the Cepheid’s radius, which increases with the mass and brightness of the Cepheid.

For a recent re-analysis of Leavitt’s original Cepheid data, see this paper.
19:10 Did Leavitt mainly study the Cepheids in our own galaxy?

This was an inaccuracy in the presentation. Leavitt’s original breakthrough paper studied Cepheids in the Small Magellanic Cloud. At the time, the distance to this cloud was not known; indeed, it was a matter of debate whether this cloud was in the Milky Way, or some distance away from it. However, Leavitt (correctly) assumed that all the Cepheids in this cloud were roughly the same distance away from our solar system, so that the apparent brightness was proportional to the absolute brightness. This gave an uncalibrated form of Leavitt’s law between absolute brightness and period, subject to the (then unknown) distance to the Small Magellanic Cloud. After Leavitt’s work, there were several efforts (by Hertzsprung, Russell, and Shapley) to calibrate the law by using the few Cepheids for which other distance methods were available, such as parallax. (Main sequence fitting to the Hertzsprung-Russell diagram was not directly usable, as Cepheids did not lie on the main sequence; but in some cases one could indirectly use this method if the Cepheid was in the same stellar cluster as a main sequence star.) Once the law was calibrated, it could be used to measure distances to other Cepheids, and in particular to compute distances to extragalactic objects such as the Magellanic clouds.
19:15 Was Leavitt’s law really a linear law between period and luminosity?

Strictly speaking, the period-luminosity relation commonly known as Leavitt’s law was a linear relation between the absolute magnitude of the Cepheid and the logarithm of the period; undoing the logarithms, this becomes a power law between the luminosity and the period.
20:26 Was Hubble the one to discover the redshift of galaxies?

This was an error on my part; Hubble was using earlier work of Vesto Slipher on these redshifts, and combining it with his own measurements of distances using Leavitt’s law to arrive at the law that now bears his name; he was also assisted in his observations by Milton Humason. It should also be noted that Georges Lemaître had also independently arrived at essentially the same law a few years prior, but his work was published in a somewhat obscure journal and did not receive broad recognition until some time later.
20:37 Hubble’s original graph doesn’t look like a very good fit to a linear law.

Hubble’s original data was somewhat noisy and inaccurate by modern standards, and the redshifts were affected by the peculiar velocities of individual galaxies in addition to the expanding nature of the universe. However, as the data was extended to more galaxies, it became increasingly possible to compensate for these effects and obtain a much tighter fit, particularly at larger scales where the effects of peculiar velocity are less significant. See for instance this article from 2015 where Hubble’s original graph is compared with a more modern graph. This more recent graph also reveals a slight nonlinear correction to Hubble’s law at very large scales that has led to the remarkable discovery that the expansion of the universe is in fact accelerating over time, a phenomenon that is attributed to a positive cosmological constant (or perhaps a more complex form of dark energy in the universe). On the other hand, even with this nonlinear correction, there continues to be a roughly 10% discrepancy of this law with predictions based primarily on the cosmic microwave background radiation; see the FAQ entry for 23:49.
20:46 Does general relativity alone predict an uniformly expanding universe?

This was an oversimplification. Einstein’s equations of general relativity contain a parameter $\Lambda$ , known as the cosmological constant, which currently is only computable indirectly from fitting to experimental data. But even with this constant fixed, there are multiple solutions to these equations (basically because there are multiple possible initial conditions for the universe). For the purposes of cosmology, a particularly successful family of solutions are the solutions given by the Lambda-CDM model. This family of solutions contains additional parameters, such as the density of dark matter in the universe. Depending on the precise values of these parameters, the universe could be expanding or contracting, with the rate of expansion or contraction either increasing, decreasing, or staying roughly constant. But if one fits this model to all available data (including not just red shift measurements, but also measurements on the cosmic microwave background radiation and the spatial distribution of galaxies), one deduces a version of Hubble’s law which is nearly linear, but with an additional correction at very large scales; see the next item of this FAQ.
21:07 Is Hubble’s original law sufficiently accurate to allow for good measurements of distances at the scale of the observable universe?

Not really; as mentioned in the end of the video, there were additional efforts to cross-check and calibrate Hubble’s law at intermediate scales between the range of Cepheid methods (about 100 million light years) and observable universe scales (about 100 billion light years) by using further “standard candles” than Cepheids, most notably Type Ia supernovae (which are bright enough and predictable enough to be usable out to about 10 billion light years), the Tully-Fisher relation between the luminosity of a galaxy and its rotational speed, and gamma ray bursts. It turns out that due to the accelerating nature of the universe’s expansion, Hubble’s law is not completely linear at these large scales; this important correction cannot be discerned purely from Cepheid data, but also requires the other standard candles, as well as fitting that data (as well as other observational data, such as the cosmic microwave background radiation) to the cosmological models provided by general relativity (with the best fitting models to date being some version of the Lambda-CDM model).

On the other hand, a naive linear extrapolation of Hubble’s original law to all larger scales does provide a very rough picture of the observable universe which, while too inaccurate for cutting edge research in astronomy, does give some general idea of its large-scale structure.
21:15 Where did this guess of the observable universe being about 20% of the full universe come from?

There are some ways to get a lower bound on the size of the entire universe that go beyond the edge of the observable universe. One is through analysis of the cosmic microwave background radiation (CMB), that has been carefully mapped out by several satellite observatories, most notably WMAP and Planck. Roughly speaking, a universe that was less than twice the size of the observable universe would create certain periodicities in the CMB data; such periodicities are not observed, so this provides a lower bound (see for instance this paper for an example of such a calculation). The 20% number was a guess based on my vague recollection of these works, but there is no consensus currently on what the ratio truly is; there are some proposals that the entire universe is in fact several orders of magnitude larger than the observable one.

The situation is somewhat analogous to Aristarchus’s measurement of the distance to the Sun, which was very sensitive to a small angle (the half-moon discrepancy). Here, the predicted size of the universe under the standard cosmological model is similarly dependent in a highly sensitive fashion on a measure $\Omega_k$ of the flatness of the universe which, for reasons still not fully understood (but likely caused by some sort of inflation mechanism), happens to be extremely close to zero. As such, predictions for the size of the universe remain highly volatile at the current level of measurement accuracy.
23:44 Was it a black hole collision that allowed for an independent measurement of Hubble’s law?

This was a slight error in the presentation. While the first gravitational wave observation by LIGO in 2015 was of a black hole collision, it did not come with an electromagnetic counterpart that allowed for a redshift calculation that would yield a Hubble’s law measurement. However, a later collision of neutron stars, observed in 2017, did come with an associated kilonova in which a redshift was calculated, and led to a Hubble measurement which was independent of most of the rungs of the distance ladder.
23:49 Where can I learn more about this 10% discrepancy in Hubble’s law?

This is known as the Hubble tension (or, in more sensational media, the “crisis in cosmology”): roughly speaking, the various measurements of Hubble’s constant (either from climbing the cosmic distance ladder, or by fitting various observational data to standard cosmological models) tend to arrive at one of two values, that are about 10% apart from each other. The values based on gravitational wave observations are currently consistent with both values, due to significant error bars in this extremely sensitive method; but other more mature methods are now of sufficient accuracy that they are basically only consistent with one of the two values. Currently there is no consensus on the origin of this tension: possibilities include systemic biases in the observational data, subtle statistical issues with the methodology used to interpret the data, a correction to the standard cosmological model, the influence of some previously undiscovered law of physics, or some partial breakdown of the Copernican principle.

For an accessible recent summary of the situation, see this video by Becky Smethurst (“Dr. Becky”).
24:49 So, what is a Type Ia supernova and why is it so useful in the distance ladder?

A Type Ia supernova occurs when a white dwarf in a binary system draws more and more mass from its companion star, until it reaches the Chandrasekhar limit, at which point its gravitational forces are strong enough to cause a collapse that increases the pressure to the point where a supernova is triggered via a process known as carbon detonation. Because of the universal nature of the Chandrasekhar limit, all such supernovae have (as a first approximation) the same absolute brightness and can thus be used as standard candles in a similar fashion to Cepheids (but without the need to first measure any auxiliary observable, such as a period). But these supernovae are also far brighter than Cepheids, and can so this method can be used at significantly larger distances than the Cepheid method (roughly speaking it can handle distances of up to ~10 billion light years, whereas Cepheids are reliable out to ~100 million light years). Among other things, the supernovae measurements were the key to detecting an important nonlinear correction to Hubble’s law at these scales, leading to the remarkable conclusion that the expansion of the universe is in fact accelerating over time, which in the Lambda-CDM model corresponds to a positive cosmological constant, though there are more complex “dark energy” models that are also proposed to explain this acceleration.
24:54 Besides Type Ia supernovae, I felt that a lot of other topics relevant to the modern distance ladder (e.g., the cosmic microwave background radiation, the Lambda CDM model, dark matter, dark energy, inflation, multi-messenger astronomy, etc.) were omitted.

This is partly due to time constraints, and the need for editing to tighten the narrative, but was also a conscious decision on my part. Advanced classes on the distance ladder will naturally focus on the most modern, sophisticated, and precise ways to measure distances, backed up by the latest mathematics, physics, technology, observational data, and cosmological models. However, the focus in this video series was rather different; we sought to portray the cosmic distance ladder as evolving in a fully synergestic way, across many historical eras, with the evolution of mathematics, science, and technology, as opposed to being a mere byproduct of the current state of these other disciplines. As one specific consequence of this change of focus, we emphasized the first time any rung of the distance ladder was achieved, at the expense of more accurate and sophisticated later measurements at that rung. For instance, refinements in the measurement of the radius of the Earth since Eratosthenes, improvements in the measurement of the astronomical unit between Aristarchus and Cook, or the refinements of Hubble’s law and the cosmological model of the universe in the twentieth and twenty-first centuries, were largely omitted (though some of the answers in this FAQ are intended to address these omissions).

Many of the topics not covered here (or only given a simplified treatment) are discussed in depth in other expositions, including other Youtube videos. I would welcome suggestions from readers for links to such resources in the comments to this post. Here is a partial list:
- “Eratosthenes” – Cosmos (Carl Sagan), video posted Apr 24, 2009 (originally released Oct 1, 1980, as part of the episode “The Shores of the Cosmic Ocean”).
- “How Far Away Is It” – David Butler, a multi-part series beginning Aug 16 2013.
- “How the Bizarre Path of Mars Reshaped Astronomy [Kepler’s Laws Part 1]“, Welch Labs, May 8, 2024. See also Part 2.
- “An ASTROPHYSICIST’S TOP 5 space news stories of 2024“, Becky Smethurst (Dr. Becky), Dec 26, 2024 – covers the Hubble tension as one of the stories.
- “Measuring the Earth… from a vacation photo“, George Lowther (Almost sure), Feb 22 2025.
- “How Did This Ancient Genius Measure The Sun?“, Ben Syversen, Feb 28 2025.

by Terence Tao at May 30, 2025 03:51 PM

May 27, 2025

John Preskill — I know I am but what are you? Mind and Matter in Quantum Mechanics

Nowadays it is best to exercise caution when bringing the words “quantum” and “consciousness” anywhere near each other, lest you be suspected of mysticism or quackery. Eugene Wigner did not concern himself with this when he wrote his “Remarks on the Mind-Body Question” in 1967. (Perhaps he was emboldened by his recent Nobel prize for contributions to the mathematical foundations of quantum mechanics, which gave him not a little no-nonsense technical credibility.) The mind-body question he addresses is the full-blown philosophical question of “the relation of mind to body”, and he argues unapologetically that quantum mechanics has a great deal to say on the matter. The workhorse of his argument is a thought experiment that now goes by the name “Wigner’s Friend”. About fifty years later, Daniela Frauchiger and Renato Renner formulated another, more complex thought experiment to address related issues in the foundations of quantum theory. In this post, I’ll introduce Wigner’s goals and argument, and evaluate Frauchiger’s and Renner’s claims of its inadequacy, concluding that these are not completely fair, but that their thought experiment does do something interesting and distinct. Finally, I will describe a recent paper of my own, in which I formalize the Frauchiger-Renner argument in a way that illuminates its status and isolates the mathematical origin of their paradox.

* * *

Wigner takes a dualist view about the mind, that is, he believes it to be non-material. To him this represents the common-sense view, but is nevertheless a newly mainstream attitude. Indeed,

[until] not many years ago, the “existence” of a mind or soul would have been passionately denied by most physical scientists. The brilliant successes of mechanistic and, more generally, macroscopic physics and of chemistry overshadowed the obvious fact that thoughts, desires, and emotions are not made of matter, and it was nearly universally accepted among physical scientists that there is nothing besides matter.

He credits the advent of quantum mechanics with

the return, on the part of most physical scientists, to the spirit of Descartes’s “Cogito ergo sum”, which recognizes the thought, that is, the mind, as primary. [With] the creation of quantum mechanics, the concept of consciousness came to the fore again: it was not possible to formulate the laws of quantum mechanics in a fully consistent way without reference to the consciousness.

What Wigner has in mind here is that the standard presentation of quantum mechanics speaks of definite outcomes being obtained when an observer makes a measurement. Of course this is also true in classical physics. In quantum theory, however, the principles of linear evolution and superposition, together with the plausible assumption that mental phenomena correspond to physical phenomena in the brain, lead to situations in which there is no mechanism for such definite observations to arise. Thus there is a tension between the fact that we would like to ascribe particular observations to conscious agents and the fact that we would like to view these observations as corresponding to particular physical situations occurring in their brains.

Once we have convinced ourselves that, in light of quantum mechanics, mental phenomena must be considered on an equal footing with physical phenomena, we are faced with the question of how they interact. Wigner takes it for granted that “if certain physico-chemical conditions are satisfied, a consciousness, that is, the property of having sensations, arises.” Does the influence run the other way? Wigner claims that the “traditional answer” is that it does not, but argues that in fact such influence ought indeed to exist. (Indeed this, rather than technical investigation of the foundations of quantum mechanics, is the central theme of his essay.) The strongest support Wigner feels he can provide for this claim is simply “that we do not know of any phenomenon in which one subject is influenced by another without exerting an influence thereupon”. Here he recalls the interaction of light and matter, pointing out that while matter obviously affects light, the effects of light on matter (for example radiation pressure) are typically extremely small in magnitude, and might well have been missed entirely had they not been suggested by the theory.

Quantum mechanics provides us with a second argument, in the form of a demonstration of the inconsistency of several apparently reasonable assumptions about the physical, the mental, and the interaction between them. Wigner works, at least implicitly, within a model where there are two basic types of object: physical systems and consciousnesses. Some physical systems (those that are capable of instantiating the “certain physico-chemical conditions”) are what we might call mind-substrates. Each consciousness corresponds to a mind-substrate, and each mind-substrate corresponds to at most one consciousness. He considers three claims (this organization of his premises is not explicit in his essay):

1. Isolated physical systems evolve unitarily.

2. Each consciousness has a definite experience at all times.

3. Definite experiences correspond to pure states of mind-substrates, and arise for a consciousness exactly when the corresponding mind-substrate is in the corresponding pure state.

The first and second assumptions constrain the way the model treats physical and mental phenomena, respectively. Assumption 1 is often paraphrased as the `”completeness of quantum mechanics”, while Assumption 2 is a strong rejection of solipsism – the idea that only one’s own mind is sure to exist. Assumption 3 is an apparently reasonable assumption about the relation between mental and physical phenomena.

With this framework established, Wigner’s thought experiment, now typically known as Wigner’s Friend, is quite straightforward. Suppose that an observer, Alice (to name the friend), is able to perform a measurement of some physical quantity $q$ of a particle, which may take two values, $0$ and $1$ . Assumption 1 tells us that if Alice performs this measurement when the particle is in a superposition state, the joint system of Alice’s brain and the particle will end up in an entangled state. Now Alice’s mind-substrate is not in a pure state, so by Assumption 3 does not have a definite experience. This contradicts Assumption 2. Wigner’s proposed resolution to this paradox is that in fact Assumption 1 is incorrect, and that there is an influence of the mental on the physical, namely objective collapse or, as he puts it, that the “statistical element which, according to the orthodox theory, enters only if I make an observation enters equally if my friend does”.

* * *

Decades after the publication of Wigner’s essay, Daniela Frauchiger and Renato Renner formulated a new thought experiment, involving observers making measurements of other observers, which they intended to remedy what they saw as a weakness in Wigner’s argument. In their words, “Wigner proposed an argument […] which should show that quantum mechanics cannot have unlimited validity”. In fact, they argue, Wigner’s argument does not succeed in doing so. They assert that Wigner’s paradox may be resolved simply by noting a difference in what each party knows. Whereas Wigner, describing the situation from the outside, does not initially know the result of his friend’s measurement, and therefore assigns the “absurd” entangled state to the joint system composed of both her body and the system she has measured, his friend herself is quite aware of what she has observed, and so assigns to the system either, but not both, of the states corresponding to definite measurement outcomes. “For this reason”, Frauchiger and Renner argue, “the Wigner’s Friend Paradox cannot be regarded as an argument that rules out quantum mechanics as a universally valid theory.”

This criticism strikes me as somewhat unfair to Wigner. In fact, Wigner’s objection to admitting two different states as equally valid descriptions is that the two states correspond to different sets of \textit{physical} properties of the joint system consisting of Alice and the system she measures. For Wigner, physical properties of physical systems are distinct from mental properties of consciousnesses. To engage in some light textual analysis, we can note that the word ‘conscious’, or ‘consciousness’, appears forty-one times in Wigner’s essay, and only once in Frauchiger and Renner’s, in the title of a cited paper. I have the impression that the authors pay inadequate attention to how explicitly Wigner takes a dualist position, including not just physical systems but also, and distinctly, consciousnesses in his ontology. Wigner’s argument does indeed achieve his goals, which are developed in the context of this strong dualism, and differ from the goals of Frauchiger and Renner, who appear not to share this philosophical stance, or at least do not commit fully to it.

Nonetheless, the thought experiment developed by Frauchiger and Renner does achieve something distinct and interesting. We can understand Wigner’s no-go theorem to be of the following form: “Within a model incorporating both mental and physical phenomena, a set of apparently reasonable conditions on how the model treats physical phenomena, mental phenomena, and their interaction cannot all be satisfied”. The Frauchiger-Renner thought experiment can be cast in the same form, with different choices about how to implement the model and which conditions to consider. The major difference in the model itself is that Frauchiger and Renner do not take consciousnesses to be entities in their own rights, but simply take some states of certain physical systems to correspond to conscious experiences. Within such a model, Wigner’s assumption that each mind has a single, definite conscious experience at all times seems far less natural than it did within his model, where consciousnesses are distinct entities from the physical systems that determine them. Thus Frauchiger and Renner need to weaken this assumption, which was so natural to Wigner. The weakening they choose is a sort of transitivity of theories of mind. In their words (Assumption C in their paper):

Suppose that agent A has established that “I am certain that agent A’, upon reasoning within the same theory as the one I am using, is certain that $x =\xi$ at time $t$ .” Then agent A can conclude that “I am certain that $x=\xi$ at time $t$ .”

Just as Assumption 3 above was, for Wigner, a natural restriction on how a sensible theory ought to treat mental phenomena, this serves as Frauchiger’s and Renner’s proposed constraint. Just as Wigner designed a thought experiment that demonstrated the incompatibility of his assumption with an assumption of the universal applicability of unitary quantum mechanics to physical systems, so do Frauchiger and Renner.

* * *

In my recent paper “Reasoning across spacelike surfaces in the Frauchiger-Renner thought experiment”, I provide two closely related formalizations of the Frauchiger-Renner argument. These are motivated by a few observations:

1. Assumption C ought to make reference to the (possibly different) times at which agents $A$ and $A'$ are certain about their respective judgments, since these states of knowledge change.

2. Since Frauchiger and Renner do not subscribe to Wigner’s strong dualism, an agent’s certainty about a given proposition, like any other mental state, corresponds within their implicit model to a physical state. Thus statements like “Alice knows that P” should be understood as statements about the state of some part of Alice’s brain. Conditional statements like “if upon measuring a quantity q Alice observes outcome $x$ , she knows that P” should be understood as claims about the state of the composite system composed of the part of Alice’s brain responsible for knowing P and the part responsible for recording outcomes of the measurement of q.

3. Because the causal structure of the protocol does not depend on the absolute times of each event, an external agent describing the protocol can choose various “spacelike surfaces”, corresponding to fixed times in different spacetime embeddings of the protocol (or to different inertial frames). There is no reason to privilege one of these surfaces over another, and so each of them should be assigned a quantum state. This may be viewed as an implementation of a relativistic principle.

A visual representation of the formalization of the Frauchiger-Renner protocol and the arguments of the no-go theorem. The graphical conventions are explained in detail in “Reasoning across spacelike surfaces in the Frauchiger-Renner thought experiment”.

After developing a mathematical framework based on these observations, I recast Frauchiger’s and Renner’s Assumption C in two ways: first, in terms of a claim about the validity of iterating the “relative state” construction that captures how conditional statements are interpreted in terms of quantum states; and second, in terms of a deductive rule that allows chaining of inferences within a system of quantum logic. By proving that these claims are false in the mathematical framework, I provide a more formal version of the no-go theorem. I also show that the first claim can be rescued if the relative state construction is allowed to be iterated only “along” a single spacelike surface, and the second if a deduction is only allowed to chain inferences “along” a single surface. In other words, the mental transitivity condition desired by Frauchiger and Renner can in fact be combined with universal physical applicability of unitary quantum mechanics, but only if we restrict our analysis to a single spacelike surface. Thus I hope that the analysis I offer provides some clarification of what precisely is going on in Frauchiger and Renner’s thought experiment, what it tells us about combining the physical and the mental in light of quantum mechanics, and how it relates to Wigner’s thought experiment.

* * *

In view of the fact that “Quantum theory cannot consistently describe the use of itself” has, at present, over five hundred citations, and “Remarks on the Mind-Body Question” over thirteen hundred, it seems fitting to close with a thought, cautionary or exultant, from Peter Schwenger’s book on asemic, that is meaningless, writing. He notes that

commentary endlessly extends language; it is in the service of an impossible quest to extract the last, the final, drop of meaning.

I provide no analysis of this claim.

by jeffreymepstein at May 27, 2025 06:05 PM

May 25, 2025

John Preskill — The most steampunk qubit

I never imagined that an artist would update me about quantum-computing research.

Last year, steampunk artist Bruce Rosenbaum forwarded me a notification about a news article published in Science. The article reported on an experiment performed in physicist Yiwen Chu’s lab at ETH Zürich. The experimentalists had built a “mechanical qubit”: they’d stored a basic unit of quantum information in a mechanical device that vibrates like a drumhead. The article dubbed the device a “steampunk qubit.”

I was collaborating with Bruce on a quantum-steampunk sculpture, and he asked if we should incorporate the qubit into the design. Leave it for a later project, I advised. But why on God’s green Earth are you receiving email updates about quantum computing?

My news feed sends me everything that says “steampunk,” he explained. So keeping a bead on steampunk can keep one up to date on quantum science and technology—as I’ve been preaching for years.

Other ideas displaced Chu’s qubit in my mind until I visited the University of California, Berkeley this January. Visiting Berkeley in January, one can’t help noticing—perhaps with a trace of smugness—the discrepancy between the temperature there and the temperature at home. And how better to celebrate a temperature difference than by studying a quantum-thermodynamics-style throwback to the 1800s?

One sun-drenched afternoon, I learned that one of my hosts had designed another steampunk qubit: Alp Sipahigil, an assistant professor of electrical engineering. He’d worked at Caltech as a postdoc around the time I’d finished my PhD there. We’d scarcely interacted, but I’d begun learning about his experiments in atomic, molecular, and optical physics then. Alp had learned about my work through Quantum Frontiers, as I discovered this January. I had no idea that he’d “met” me through the blog until he revealed as much to Berkeley’s physics department, when introducing the colloquium I was about to present.

Alp and collaborators proposed that a qubit could work as follows. It consists largely of a cantilever, which resembles a pendulum that bobs back and forth. The cantilever, being quantum, can have only certain amounts of energy. When the pendulum has a particular amount of energy, we say that the pendulum is in a particular energy level.

One might hope to use two of the energy levels as a qubit: if the pendulum were in its lowest-energy level, the qubit would be in its 0 state; and the next-highest level would represent the 1 state. A bit—a basic unit of classical information—has 0 and 1 states. A qubit can be in a superposition of 0 and 1 states, and so the cantilever could be.

A flaw undermines this plan, though. Suppose we want to process the information stored in the cantilever—for example, to turn a 0 state into a 1 state. We’d inject quanta—little packets—of energy into the cantilever. Each quantum would contain an amount of energy equal to (the energy associated with the cantilever’s 1 state) – (the amount associated with the 0 state). This equality would ensure that the cantilever could accept the energy packets lobbed at it.

But the cantilever doesn’t have only two energy levels; it has loads. Worse, all the inter-level energy gaps equal each other. However much energy the cantilever consumes when hopping from level 0 to level 1, it consumes that much when hopping from level 1 to level 2. This pattern continues throughout the rest of the levels. So imagine starting the cantilever in its 0 level, then trying to boost the cantilever into its 1 level. We’d probably succeed; the cantilever would probably consume a quantum of energy. But nothing would stop the cantilever from gulping more quanta and rising to higher energy levels. The cantilever would cease to serve as a qubit.

We can avoid this problem, Alp’s team proposed, by placing an atomic-force microscope near the cantilever. An atomic force microscope maps out surfaces similarly to how a Braille user reads: by reaching out a hand and feeling. The microscope’s “hand” is a tip about ten nanometers across. So the microscope can feel surfaces far more fine-grained than a Braille user can. Bumps embossed on a page force a Braille user’s finger up and down. Similarly, the microscope’s tip bobs up and down due to forces exerted by the object being scanned.

Imagine placing a microscope tip such that the cantilever swings toward it and then away. The cantilever and tip will exert forces on each other, especially when the cantilever swings close. This force changes the cantilever’s energy levels. Alp’s team chose the tip’s location, the cantilever’s length, and other parameters carefully. Under the chosen conditions, boosting the cantilever from energy level 1 to level 2 costs more energy than boosting from 0 to 1.

So imagine, again, preparing the cantilever in its 0 state and injecting energy quanta. The cantilever will gobble a quantum, rising to level 1. The cantilever will then remain there, as desired: to rise to level 2, the cantilever would have to gobble a larger energy quantum, which we haven’t provided.¹

Will Alp build the mechanical qubit proposed by him and his collaborators? Yes, he confided, if he acquires a student nutty enough to try the experiment. For when he does—after the student has struggled through the project like a dirigible through a hurricane, but ultimately triumphed, and a journal is preparing to publish their magnum opus, and they’re brainstorming about artwork to represent their experiment on the journal’s cover—I know just the aesthetic to do the project justice.

¹Chu’s team altered their cantilever’s energy levels using a superconducting qubit, rather than an atomic force microscope.

by Nicole Yunger Halpern at May 25, 2025 11:54 PM

May 24, 2025

Matt Strassler The War on Harvard University

The United States’ government is waging an all-out assault on Harvard University. The strategy, so far, has been:

Cut most of the grants (present and future) for scientific and medical research, so that thousands of Harvard’s scientists, researchers and graduate students have to stop their work indefinitely. That includes research on life-saving medicine, on poorly understood natural phenomena, and on new technology. This also means that the university will have no money from these activities to pay salaries of its employees.
Eliminate the tax-advantageous status of the university, so that the university is much more expensive to operate.
Prohibit Harvard from having any international students (undergraduate and graduate) and other researchers, so that large numbers of existing scientific and medical research projects that still have funding will have to cease operation. This destroys the careers of thousands of brilliant people — and not just foreigners. Many US faculty and students are working with and depend upon these expelled researchers, and their work will stop too. It also means that Harvard’s budget for the next academic year will be crushed, since it is far too late to replace the tuition from international undergraduate students for the coming year.

The grounds for this war is that Harvard allegedly does not provide a safe environment for its Jewish students, and that Harvard refuses to let the government determine who it may and may not hire.

Now, maybe you can explain to me what this is really about. I’m confused what crimes these scientific researchers commited that justifies stripping them of their grants and derailing their research. I’m also unclear as to why many apolitical, hard-working young trainees in laboratories across the campus deserve to be ejected from their graduate and post-graduate careers and sent home, delaying or ruining their futures. [Few will be able to transfer to other US schools; with all the government cuts to US science, there’s no money to support them at other locations.] And I don’t really understand how such enormous damage and disruption to the lives and careers of ten thousand-ish scientists, researchers and graduate students at Harvard (including many who are Jewish) will actually improve the atmosphere for Harvard’s Jewish students.

As far as I can see, the government is merely using Jewish students as pawns, pretending to attack Harvard on their behalf while in truth harboring no honest concern for their well-being. The fact that the horrors and nastiness surrounding the Gaza war are being exploited by the government as cover for an assault on academic freedom and scientific research is deeply cynical and exceedingly ugly.

From the outside, where Harvard is highly respected — it is certainly among the top five universities in the world, however you rank them — this must look completely idiotic, as idiotic as France gutting the Sorbonne, or the UK eviscerating Oxford. But keep in mind that Harvard is by no means the only target here. The US government is cutting the country’s world-leading research in science, technology and medicine to the bone. If that’s what you want to do, then ruining Harvard makes perfect sense.

The country that benefits the most from this self-destructive behavior? China, obviously. As a friend of mine said, this isn’t merely like shooting yourself in the foot, it’s like shooting yourself in the head.

I suspect most readers will understand that I cannot blog as usual right now. To write good articles about quantum physics requires concentration and focus. When people’s careers and life’s work are being devastated all around me, that’s simply not possible.

by Matt Strassler at May 24, 2025 03:12 PM

May 23, 2025

Tommaso Dorigo — An Innovative Proposal

The other day I finally emerged from a very stressful push to submit two grant applications to the European Innovation Council. The call in question is for PATHFINDER_OPEN projects, that aim for proofs of principle of groundbreaking technological innovations. So I thought I would broadly report on that experience (no, I am not new to it, but you never cease to learn!), and disclose just a little about the ideas that brought about one of the two projects.
Grant applications

by Tommaso Dorigo at May 23, 2025 01:45 PM

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Tommaso Dorigo — Win A MSCA Post-Doctoral Fellowship!

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June 04, 2025

Terence Tao — Decomposing a factorial into large factors (second version)

June 03, 2025

Terence Tao — A Lean companion to “Analysis I”

June 02, 2025

Terence Tao — On the number of exceptional intervals to the prime number theorem in short intervals

May 31, 2025

Tommaso Dorigo — The Anomaly That Wasn't: An Example Of Shifting Consensus In Science

May 30, 2025

Terence Tao — Cosmic Distance Ladder videos with Grant Sanderson (3blue1brown): commentary and corrections

Comments on part 1 of the video

Comments on the “deleted scene” on Al-Biruni