Planet Musings

A common task in analysis is to obtain bounds on sums

Here are some typical examples of such estimation problems, drawn from recent questions on MathOverflow:

• (i) (From this question) If and , is the expression

  

  finite?
• (ii) (From this question) If , how can one show that

  

• (iii) (From this question) Can one show that

  

  as for an explicit constant , and what is this constant?

Compared to other estimation tasks, such as that of controlling oscillatory integrals, exponential sums, singular integrals, or expressions involving one or more unknown functions (that are only known to lie in some function spaces, such as an space), high-dimensional geometry (or alternatively, large numbers of random variables), or number-theoretic structures (such as the primes), estimation of sums or integrals of non-negative elementary expressions is a relatively straightforward task, and can be accomplished by a variety of methods. The art of obtaining such estimates is typically not explicitly taught in textbooks, other than through some examples and exercises; it is typically picked up by analysts (or those working in adjacent areas, such as PDE, combinatorics, or theoretical computer science) as graduate students, while they work through their thesis or their first few papers in the subject.

Somewhat in the spirit of this previous post on analysis problem solving strategies, I am going to try here to collect some general principles and techniques that I have found useful for these sorts of problems. As with the previous post, I hope this will be something of a living document, and encourage others to add their own tips or suggestions in the comments.

— 1. Asymptotic arithmetic —

Asymptotic notation is designed so that many of the usual rules of algebra and inequality manipulation continue to hold, with the caveat that one has to be careful if subtraction or division is involved. For instance, if one knows that and , then one can immediately conclude that and , even if are negative (note that the notation or automatically forces to be non-negative). Equivalently, we have the rules

Another rule of inequalities that is inherited by asymptotic notation is that if one has two bounds

For instance, suppose one wanted to show that the sum

One of the key advantages of dealing with order of magnitude estimates, as opposed to sharp inequalities, is that the arithmetic becomes tropical. More explicitly, we have the important rule

The above divide and conquer strategy does not directly apply when one is decomposing into an unbounded number of pieces , . In such cases, one needs an additional gain in the index that is summable in in order to conclude. For instance, if one wants to establish a bound of the form , and one has located a decomposition or upper bound

Sometimes, when trying to prove an estimate such as , one has identified a promising decomposition with an unbounded number of terms

If one is faced with the task of estimating a doubly infinite sum , it can often be useful to first think about how one would proceed in estimating when is very large and positive, and how one would proceed when is very large and negative. In many cases, one can simply decompose the sum into two pieces such as and and use whatever methods you came up with to handle the two extreme cases; in some cases one also needs a third argument to handle the case when is of bounded (or somewhat bounded) size, in which case one may need to divide into three pieces such as , , and . Sometimes there will be a natural candidate for the places where one is cutting the sum, but in other situations it may be best to just leave these cut points as unspecified parameters initially, obtain bounds that depend on these parameters, and optimize at the end. (Typically, the optimization proceeds by trying to balance the magnitude of a term that is increasing with respect to a parameter, with one that is decreasing. For instance, if one ends up with a bound such as for some parameter and quantities , it makes sense to select to balance the two terms. Or, if faced with something like for some , then something like would be close to the optimal choice of parameter. And so forth.)

— 1.1. Psychological distinctions between exact and asymptotic arithmetic —

The adoption of the “divide and conquer” strategy requires a certain mental shift from the “simplify, simplify” strategy that one is taught in high school algebra. In the latter strategy, one tries to collect terms in an expression make them as short as possible, for instance by working with a common denominator, with the idea that unified and elegant-looking expressions are “simpler” than sprawling expressions with many terms. In contrast, the divide and conquer strategy is intentionally extremely willing to greatly increase the total length of the expressions to be estimated, so long as each individual component of the expressions appears easier to estimate than the original one. Both strategies are still trying to reduce the original problem to a simpler problem (or collection of simpler sub-problems), but the metric by which one judges whether the problem has become simpler is rather different.

A related mental shift that one needs to adopt in analysis is to move away from the exact identities that are so prized in algebra (and in undergraduate calculus), as the precision they offer is often unnecessary and distracting for the task at hand, and often fail to generalize to more complicated contexts in which exact identities are no longer available. As a simple example, consider the task of estimating the expression

As a general rule, instead of relying exclusively on exact formulae, one should seek approximations that are valid up to the degree of precision that one seeks in the final estimate. For instance, suppose one one wishes to establish the bound

The following table lists some common approximations that can be used to simplify expressions when one is only interested in order of magnitude bounds (with an arbitrary small constant):

The quantity…	has magnitude comparable to …	provided that…
		or
, ,
		real
		real
, ,
,
		real
		,
		,
		,
		,

On the other hand, some exact formulae are still very useful, particularly if the end result of that formula is clean and tractable to work with (as opposed to involving somewhat exotic functions such as the arctangent). The geometric series formula, for instance, is an extremely handy exact formula, so much so that it is often desirable to control summands by a geometric series purely to use this formula (we already saw an example of this in (7)). Exact integral identities, such as

Exercise 1 Suppose obeys the quasi-monotonicity property whenever . Show that for any integers .

Exercise 2 Use (11) to obtain the “cheap Stirling approximation”

for any natural number . (Hint: take logarithms to convert the product into a sum.)

With practice, you will be able to identify any term in a computation which is already “negligible” or “acceptable” in the sense that its contribution is always going to lead to an error that is smaller than the desired accuracy of the final estimate. One can then work “modulo” these negligible terms and discard them as soon as they appear. This can help remove a lot of clutter in one’s arguments. For instance, if one wishes to establish an asymptotic of the form

Another psychological shift when switching from algebraic simplification problems to estimation problems is that one has to be prepared to let go of constraints in an expression that complicate the analysis. Suppose for instance we now wish to estimate the variant

— 2. More on decomposition —

The way in which one decomposes a sum or integral such as or is often guided by the “geometry” of , and in particular where is large or small (or whether various component terms in are large or small relative to each other). For instance, if comes close to a maximum at some point , then it may make sense to decompose based on the distance to , or perhaps to treat the cases and separately. (Note that does not literally have to be the maximum in order for this to be a reasonable decomposition; if it is in “within reasonable distance” of the maximum, this could still be a good move. As such, it is often not worthwhile to try to compute the maximum of exactly, especially if this exact formula ends up being too complicated to be useful.)

If an expression involves a distance between two quantities , it is sometimes useful to split into the case where is much smaller than (so that ), the case where is much smaller than (so that ), or the case when neither of the two previous cases apply (so that ). The factors of here are not of critical importance; the point is that in each of these three cases, one has some hope of simplifying the expression into something more tractable. For instance, suppose one wants to estimate the expression

• (i) The region where .
• (ii) The region where .
• (iii) The region where .

(This is not the only way to cut up the integral, but it will suffice. Often there is no “canonical” or “elegant” way to perform the decomposition; one should just try to find a decomposition that is convenient for the problem at hand.)

The reason why we want to perform such a decomposition is that in each of the three cases, one can simplify how the integrand depends on . For instance, in region (i), we see from the triangle inequality that is now comparable to , so that this contribution to (13) is comparable to

A powerful and common type of decomposition is dyadic decomposition. If the summand or integrand involves some quantity in a key way, it is often useful to break up into dyadic regions such as , so that , and then sum over . (One can tweak the dyadic range here with minor variants such as , or replace the base by some other base, but these modifications mostly have a minor aesthetic impact on the arguments at best.) For instance, one could break up a sum

Exercise 3 Let be a smooth compact submanifold of . Establish the bound

for all , where the implied constants are allowed to depend on . (This can be accomplished either by a vertical dyadic decomposition, or a dyadic decomposition of the quantity .)

Exercise 4 Solve problem (ii) from the introduction to this post by dyadically decomposing in the variable.

Remark 5 By such tools as (10), (11), or Exercise 1, one could convert the dyadic sums one obtains from dyadic decomposition into integral variants. However, if one wished, one could “cut out the middle-man” and work with continuous dyadic decompositions rather than discrete ones. Indeed, from the integral identity

for any , together with the Fubini–Tonelli theorem, we obtain the continuous dyadic decomposition

for any quantity that is positive whenever is positive. Similarly if we work with integrals rather than sums. This version of dyadic decomposition is occasionally a little more convenient to work with, particularly if one then wants to perform various changes of variables in the parameter which would be tricky to execute if this were a discrete variable.

— 3. Exponential weights —

Many sums involve expressions that are “exponentially large” or “exponentially small” in some parameter. A basic rule of thumb is that any quantity that is “exponentially small” will likely give a negligible contribution when compared against quantities that are not exponentially small. For instance, if an expression involves a term of the form for some non-negative quantity , which can be bounded on at least one portion of the domain of summation or integration, then one expects the region where is bounded to provide the dominant contribution. For instance, if one wishes to estimate the integral

To make such a heuristic precise, one can perform a dyadic decomposition in the exponential weight , or equivalently perform an additive decomposition in the exponent , for instance writing

Exercise 6 Use this decomposition to rigorously establish the bound

for any .

Exercise 7 Solve problem (i) from the introduction to this post.

More generally, if one is working with a sum or integral such as

Exercise 8 In the converse direction, establish the upper bound

for some absolute constant and all .

Exercise 9 If for some , show that

(Hint: estimate the ratio between consecutive binomial coefficients and then control the sum by a geometric series).

When the maximum of the exponent occurs in the interior of the region of summation or integration, then one can get good results by some version of Laplace’s method. For simplicity we will discuss this method in the context of one-dimensional integrals

Let us illustrate how this argument can be made rigorous by considering the task of estimating the factorial of a large number. In contrast to what we did in Exercise 2, we will proceed using a version of Laplace’s method, relying on the integral representation

The main term is expected to be the middle term, so we shall use crude methods to bound the other two terms. For the first part where , is increasing so we can crudely bound and thus

so if we select for instance, we obtain the Stirling approximation

Remark 10 It can be convenient to do some initial rescalings to this analysis to achieve a nice normalization; see this previous blog post for details.

Exercise 11 Solve problem (iii) from the introduction. (Hint: extract out the term to write as the exponential factor , placing all the other terms (which are of polynomial size) in the amplitude function . The function will then attain a maximum at ; perform a Taylor expansion and mimic the arguments above.)

Today was not the first-ever Blanton–Hogg group meeting. But it was the first ever for me, since I missed the first two for health and travel reasons. It was great! Tardugno (NYU) showed simulations of gas disks with embedded planets. The planets affect the disk, and the disk causes the planets to interact. Daunt (NYU) showed that his method for inferring (simultaneously) the spectrum, tellurics, and radial velocities in stellar spectra all works. I am stoked! Novara (NYU) showed that he has a bug in his code! But we had a good discussion inspired by that bug about surfaces of section in a real dynamics problem. Gandhi (NYU) showed us a paper with questionable claims about the CMB light passing through galaxy halos?

Today I got invited to be on a panel discussion (hosted by Soledad Villar of JHU) with Alberto Bietti (Flatiron) about the theory and practice of machine learning. It was great! We talked about why ML works for scientific applications, and Bietti said something (obvious maybe) that I loved: Maybe ML only works because of properties of the data. That is, maybe when we are analyzing ML methods we are looking in the wrong place, and we should be analyizing the data to which they are successfully applied? I made fun of interpretation in ML, and that led to interesting comments from both Bietti and the audience. Several audience members suggested taking something more like a causal approach to interpretation: How does the method work under interventions or in conditional situations? That's interesting; it isn't what a physicist would consider interpetation, but it might be sufficient in many cases.

These are notes for the talk I’m giving at the Edinburgh Category Theory Seminar this Wednesday.

(No, the talk will not be recorded.)

Schur Functors

The representation theory of the symmetric groups is clarified by thinking of all representations of all these groups as objects of a single category: the category of Schur functors. These play a universal role in representation theory, since Schur functors act on the category of representations of any group. We can understand this as an example of categorification. A ‘rig’ is a ‘ring without negatives’, and the free rig on one generator is $\mathbb{N}[x]$, the rig of polynomials with natural number coefficients. Categorifying the concept of commutative rig we obtain the concept of ‘symmetric 2-rig’, and it turns out that the category of Schur functors is the free symmetric 2-rig on one generator. Thus, in a certain sense, Schur functors are the next step after polynomials.

Many important categories act like categorified rings:

• $\mathsf{Set}$, the category of sets, with coproduct and product giving $+$ and $\times$

• $\mathsf{Vect}_k$, the category of vector spaces over a field $k$, with the usual $\oplus$ and $\otimes$

and many more. Since they tend to lack subtraction they are really categorified ‘rigs’. Sometimes the multiplication is the categorical product but often not. Today I’m only interested in examples where the addition is coproduct, though there are examples where it’s not, like:

• $core(\mathsf{Set})$, the groupoid of sets with the $+$ and $\times$ coming from coproduct and product in $\mathsf{Set}$.

I definitely want multiplication to distribute over addition (up to natural isomorphism), but often it distributes over more general colimits. Some categorified rigs have all colimits and multiplication distributes over all of them: for example, $(\mathsf{Set}, +, \times)$ and $(\mathsf{Vect}_k, \oplus, \otimes)$ work this way. But I’m also interested in examples that don’t have all colimits, like

• $\mathsf{FinVect}_k$, the category of finite-dimensional vector spaces over a field $k$, with the usual $\oplus$ and $\otimes$

And I’m also interested in examples that don’t even have all finite colimits, like

• the category of complex vector bundles over a space with the usual $\oplus$ and $\otimes$

I’ve been working with Joe Moeller and Todd Trimble on a class of categorified rigs with a strong linear algebra flavor, including these examples:

• $\mathsf{Vect}_k$, the category of vector spaces over a field $k$, with the usual $\oplus$ and $\otimes$

• $\mathsf{FinVect}_k$, the category of finite-dimensional vector spaces over a field $k$, with the usual $\oplus$ and $\otimes$

• the category of complex vector bundles over a space with the usual $\oplus$ and $\otimes$

They still have absolute colimits, and one great thing is that multiplication automatically distribute over absolute colimits. But what are absolute colimits? For that, let me back up and review a few things about enriched category theory.

Let $\mathsf{V}$ be any cosmos: a complete and cocomplete symmetric monoidal closed category. I only care about the case where $\mathsf{V}$ is $\mathsf{Vect}_k$ made monoidal using its usual tensor product, but it’s fun to talk more generally.

We can define $\mathsf{V}$-categories, which are like ordinary categories but with $hom(x,y) \in V$, and with composition

$$\circ : hom(y,z) \otimes hom(x,y) \to hom(x,z)$$

being a morphism in $\mathf{V}$. We can define $\mathsf{V}$-functors and $\mathsf{V}$-natural transformations by following our nose. We can also define a tensor product $\mathsf{C} \boxtimes \mathsf{D}$ of $\mathsf{V}$-categories, generalizing the usual product of categories. This lets us define monoidal $\mathsf{V}$-categories, which have a tensor product

$$\phantom{|} \otimes \colon \mathsf{C} \boxtimes \mathsf{C} \to \mathsf{C}$$

We can also define symmetric monoidal $\mathsf{V}$-categories, etc.

In the world of $\mathsf{V}$-categories, ‘absolute’ colimits are those automatically preserved by all $\mathsf{V}$-functors. More precisely: in enriched category theory we use ‘weighted’ colimits, which are defined using not only a diagram $\mathsf{D}$ but also a ‘weight’ $\Phi: \mathsf{D}^{op} \to \mathsf{V}$. Then, absolute colimits are those whose weights give weighted colimits that are preserved by all $\mathsf{V}$-functors between all $\mathsf{V}$-categories.

For example, if $\mathsf{V} = \mathsf{Set}$ we’re back to ordinary colimits and essentially the only absolute colimits are split idempotents. (You can get all the rest from these.) It’s a good exercise to check that split idempotents are preserved by all functors.

But if $\mathsf{V} = \mathsf{Vect}_k$ there are more absolute colimits:

• the initial object (often called $0$)
• binary coproducts (often called direct sums)
• splittings of idempotents (explained below)

You can get all the rest from these.

Definition. A $\mathsf{V}$-category is Cauchy complete if it has all absolute colimits.

For example, the category of complex vector bundles over a space is a $\mathsf{Vect}_k$-enriched category that has all absolute colimits! It doesn’t have coequalizers of all parallel pairs of morphisms. But given $p \colon x \to x$ that’s idempotent ($p^2 = p$) you can form the coequalizer of $p$ and $0 \colon x \to x$ — or in other words, the cokernel $coker p$. This is called splitting the idempotent $p$ because $1-p$ is also idempotent, so you can also form $coker (1-p)$ and show

$$x \cong coker p \oplus coker (1-p)$$

Definition. An absolute 2-rig is a monoidal $\mathsf{V}$-category $\mathsf{R}$ that is Cauchy complete.

Note that for any object $x \in \mathsf{R}$, the functors

$$x \otimes - : \mathsf{R} \to \mathsf{R}$$

and

$$- \otimes x: \mathsf{R} \to \mathsf{R}$$

automatically preserve all absolute colimits, so we say tensor products distribute over absolute colimits. For example, we have a natural isomorphism

$$x \otimes (y \oplus z) \cong (x \otimes y) \, \oplus \, (x \otimes z)$$

as you’d hope for in a categorified ring.

From now on I’m going to say ‘2-rig’ when I mean symmetric 2-rig: that is, one where the tensor product is symmetric monoidal. It’s just like how algebraic geometers say ‘ring’ when they mean commutative ring. I used to be annoyed by how algebraic geometers do that, but now I see why: I’m interested in the symmetric case, and it gets really boring saying ‘symmetric’ all the time.

And from now on let’s take $\mathsf{V} = \mathsf{Vect}_k$. In this case the free 2-rig on one object turns out to be an important structure in mathematics, often called the category of Schur functors! But we’ll just work it out.

There is a 2-functor

$$F : \mathbf{Cat} \to \mathbf{2Rig}$$

that forms the free (symmetric) 2-rig on any category. We can get it by composing three other 2-functors

$$\mathbf{Cat} \xrightarrow{S} \mathbf{SymMonCat} \xrightarrow{k[\cdot]} \mathbf{SymMon}\mathsf{V}\mathbf{Cat} \xrightarrow{Cauchy \; completion} \mathbf{2Rig}$$

In fact all of these are left adjoints, or technically left pseudoadjoints since we’re working with 2-categories.

Here:

1) $S$ gives the free symmetric monoidal category on a category.

2) For starters, $k[\cdot]$ gives the free $\mathsf{Vect}_k$-category on a category, by replacing each homset $hom(x,y)$ by the free vector space on that set, which we call $k[hom(x,y)]$. Since $\mathsf{V} = \mathsf{Vect}_k$, this gives a 2-functor

$$k[\cdot] \colon \mathbf{Cat} \to \mathsf{V}\mathbf{Cat}$$

which we can then use to get

$$k[\cdot] \colon \mathbf{SymMonCat} \to \mathbf{SymMon}\mathsf{V}\mathbf{Cat}$$

3) If $\mathsf{C}$ is some $\mathsf{V}$-category, $\overline{\mathsf{C}}$ is a Cauchy complete category called its Cauchy completion. Cauchy completing a symmetric monoidal $\mathsf{V}$-category we get a 2-rig.

Let’s look at two examples!

As a warmup, let’s start with the empty category $\emptyset$. The free symmetric monoidal category on the empty category, $S\emptyset$, is just the terminal category $1$. The free $Vect_k$-enriched category on this, $k[1]$, still has one object $\ast$ but now it has a one-dimensional space of endomorphisms. So

$$hom(\ast, \ast) \cong k$$

and composition of morphisms is multiplication in the field $k$. This is actually a symmetric monoidal $\mathsf{V}$-category. What happens when we Cauchy complete it? All idempotents already split in $k[1]$, but it doesn’t have an initial object or binary coproducts. When we throw those in we get the category of finite-dimensional vector spaces! So

$$F(\emptyset) \simeq \mathsf{FinVect}_k$$

Yes: the free 2-rig on the empty category is $\mathsf{FinVect}_k$. And since left pseudoadjonts preserve initial objects, this means the initial 2-rig is $\mathsf{FinVect}_k$.

That was fun. But now let’s figure out the free 2-rig on one object. More precisely, let’s work out $F 1$ where $1$ is the terminal category.

$S 1$, the free symmetric monoidal category on one object, is equivalent to the groupoid of finite sets! If we use a skeleton it has objects $n \in \mathbb{N}$, and

$$hom(m,n) \cong \left\{\begin{array}{cc} S_n & if m = n \\ \emptyset & if m \ne n \end{array} \right.$$

So, $S 1$ is a groupoid combining all the symmetric groups.

Next we get $k[S 1]$, the free symmetric monoidal $Vect_k$-category on one object, by linearizing the homsets of $S 1$. So, it has the same objects but now

$$hom(m,n) \cong \left\{\begin{array}{cc} k[S_n] & if m = n \\ 0 & if m \ne n \end{array} \right.$$

Here $k[S_n]$ is the free vector space on the symmetric group $S_n$. Its usually called the ‘group algebra’ of $S_n$ because it gets a multiplication from multiplication in the group — and this multiplication is how we compose morphisms in $k[S 1]$. At least that’s the interesting part: there are also ‘zero morphisms’ from $m$ to $n$ when $m \ne n$, and composing with these is like multiplying by zero. I’ll summarize all this by writing

$$k[S 1] \simeq \bigoplus_{n \ge 0} k[S_n]$$

where the direct sum is a way of glomming together $\mathsf{Vect}_k$-categories (their coproduct).

Next we get $\overline{k[S 1]}$, the free 2-rig on one object, by taking the Cauchy completion of $k[S 1]$. To understand this, I’ll now assume $k$ has characteristic zero. Then for any finite group $G$ we have

$$\overline{k[G]} \simeq FinRep_k(G)$$

where $FinRep_k(G)$ is the category of representations of $G$ on finite-dimensional vector spaces over $k$. So, we get

$$F 1 \simeq \overline{k[S 1]} \simeq \bigoplus_{n \ge 0} FinRep_k(S_n)$$

Nice! We’re seeing the free 2-rig on one object contains the representation categories of all the symmetric groups, put together in one neat package!

Now I want to prove a fun theorem about the free 2-rig on one object which explains why it’s called the category of ‘Schur functors’. This theorem actually describes, not the free 2-rig $F 1$, but its underlying category, which I’ll call $U(F 1)$.

Theorem. Let $U \colon \mathbf{2Rig} \to \mathbf{Cat}$ be the forgetful 2-functor from 2-rigs to categories, and let $[U,U]$ be the category with

• pseudonatural transformations $\alpha \colon U \Rightarrow U$ as objects
• modifications between these as morphisms.

Then

$$[U, U] \simeq U(F 1)$$

Proof sketch. I’ll prove a simpler theorem, but the proof of the full-fledged one works just the same way. Let’s decategorify and look at the forgetful functor from rings to sets, $U \colon \mathsf{Ring} \to \mathsf{Set}$. In this case $[U,U]$ is just the set of natural transformations $\alpha \colon U \Rightarrow U$. I’ll show you that

$$[U, U] \simeq U(F 1)$$

where $F \colon \mathsf{Set} \to \mathsf{Ring}$ is the ‘free ring on a set’ functor. In this decategorified case $F 1$ is just $\mathbb{Z}[x]$, the algebra of polynomials in one variable.

The proof is quick. We use a little formula for the functor $U$:

$$U \cong \mathsf{Ring}(F 1, -) \qquad \qquad (\star)$$

In other words, for any ring $R$, the underlying set $U R$ is naturally isomorphic to $\mathsf{Ring}(F 1, R)$, which is the set of ring homomorphisms from $\mathbb{Z}[x]$ to $R$.

This is obvious, because such a ring homomorphism can send $x$ to any element of $R$, and that determines it. But let’s prove this fact in a way that generalizes! Note that since $F$ is left adjoint to $U$ we have

$$\mathsf{Ring}(F 1, R) \cong \mathsf{Set}(1, U R)$$

but for any set $X$ we have $\mathsf{Set}(1,X) \cong X$, so

$$\mathsf{Set}(1, U R) \cong U R$$

So, we’ve shown $(\star)$.

Next we calculate:

$$\begin{array}{ccll} [U,U] &\cong& \left[\mathsf{Ring}(F 1, -), \mathsf{Ring}(F 1, -) \right] & by \; (\star) \\ & \cong & \mathsf{Ring}(F 1, F 1) & by \; the \; Yoneda \; Lemma \\ & \cong & U(F 1) & by \; (\star) \; again \end{array}$$

We’re done!    █

Now, when you carefully look at this proof you’ll see it has nothing to do with rings, or 2-rigs. It’s extremely general! In the decategorified case all we needed was a right adjoint functor from any category to $\mathsf{Set}$. Similarly, in the full-fledged case all we need is a right pseudoadjoint 2-functor $U$ from any 2-category to $\mathbf{Cat}$. Then we recover $U(F 1)$ from the category of pseudonatural transformations of $U$. Experts would call this result a kind of ‘Tannaka reconstruction’ theorem.

But what does this result actually mean in the cases I’m talking about?

In the case of rings, $\mathbb{Z}[x]$ acts naturally on the underlying set of any ring. Say we have a polynomial

$$P = \sum_{n \ge 0} a_n x^n \in \mathbb{Z}[x]$$

Then for any ring $R$, we get a map

$$r \mapsto \sum_{n \ge 0} a_n r^n$$

It’s not a ring homomorphism, just a map from the underlying set $U R$ to itself. And its natural in $R$.

Simple enough. But the theorem says something deeper: every natural map $U(R) \to U(R)$ comes from a polynomial in $\mathbb{Z}[x]$.

In the case of 2-rigs the story is similar. $\overline{k[S 1]}$ acts pseudonaturally on the underlying category of any 2-rig. Say we have an object

$$P = \bigoplus_{n \ge 0} \rho_n \in \overline{k[S 1]}$$

Remember that now $\rho_n$ is a finite-dimensional representation of $S_n$. Then for any 2-rig $\mathsf{R}$, we get a map

$$x \mapsto \bigoplus_{n \ge 0} \rho_n \otimes_{k[S_n]} x^{\otimes n}$$

It’s not a 2-rig map, just a functor from the underlying category $U \mathsf{R}$ to itself. It’s called a Schur functor. And it’s pseudonatural in $\mathsf{R}$.

Postlude

Everything I explained is in here:

• John Baez, Joe Moeller and Todd Trimble, Schur functors and categorified plethysm, to appear in Higher Structures.

so you can look there for more details and references to previous work. But for the $n$-Category Café crowd, I’ll add that I guessed a primitive version of this theorem here on the n-Café back in 2007, in response to this question of Allen Knutson:

For as long as I’ve understood Schur functors, I’ve thought about them as functors $Vect_{\mathbb{C}} \to Vect_{\mathbb{C}}$. But now that we’re going through them in a reading course on Fulton’s Young Tableaux, I discover that the input isn’t really a complex vector space, but an arbitrary module over a commutative ring. (And maybe, just maybe, a bimodule over a noncommutative one, but I doubt it.)

In particular, the Schur functor commutes with base change, aka extension of scalars.

What is the right way to describe this object, categorically? (Or should I say, 2- or 3-categorically?)

You can see me guessing the theorem back then… but it took Todd Trimble to prove it. And in the process, he whittled down my assumptions and came up with the definition of 2-rig I’m using here. It doesn’t need to have lots of colimits, just absolute colimits.

When Joe, Todd and I wrote our paper, which includes a lot more material than this, we acknowledged Allen as “prime instigator”, and blogged about it here. But I never wrote a simple explanation of how we proved the theorem above! So here it is.

I’m teaching the undergraduate Quantum II course (“Atoms and Molecules”) this semester. We’ve come to the point where it’s time to discuss the fine structure of hydrogen. I had previously found this somewhat unsatisfactory. If one wanted to do a proper treatment, one would start with a relativistic theory and take the non-relativistic limit. But we’re not going to introduce the Dirac equation (much less QED). And, in any case, introducing the Dirac equation would get you the leading corrections but fail miserably to get various non-leading corrections (the Lamb shift, the anomalous magnetic moment, …).

Instead, various hand-waving arguments are invoked (“The electron has an intrinsic magnetic moment and since it’s moving in the electrostatic field of the proton, it sees a magnetic field …”) which give you the wrong answer for the spin-orbit coupling (off by a factor of two), which you then have to further correct (“Thomas precession”) and then there’s the Darwin term, with an even more hand-wavy explanation …

So I set about trying to find a better way. I want use as minimal as possible input from the relativistic theory and get the leading relativistic correction(s).

1. For a spinless particle, the correction amounts to replacing the nonrelativistic kinetic energy by the relativistic expression $$\frac{p^2}{2m} \to \sqrt{p^2 c^2 +m^2 c^4} - m c^2 = \frac{p^2}{2m} - \frac{(p^2)^2}{8m^3 c^2}+\dots$$
2. For a spin-1/2 particle, “$\vec{p}$” only appears dotted into the Pauli matrices, $\vec{\sigma}\cdot\vec{p}$.
   • In particular, this tells us how the spin couples to external magnetic fields $\vec{\sigma}\cdot\vec{p} \to \vec{\sigma}\cdot(\vec{p}-q \vec{A}/c)$.
   • What we previously wrote as “$p^2$” could just as well have been written as $(\vec{\sigma}\cdot\vec{p})^2$.
3. Parity and time-reversal invariance¹ imply only even powers of $\vec{\sigma}\cdot\vec{p}$ appear in the low-velocity expansion.
4. Shifting the potential energy, $V(\vec{r})\to V(\vec{r})+\text{const}$, should shift $H\to H+\text{const}$.

With those ingredients, at $O(\vec{v}^2/c^2)$ there is a unique term² (in addition to the correction to the kinetic energy that we found for a spinless particle) that can be written down for spin-1/2 particle. $$H = \frac{p^2}{2m} +V(\vec{r}) - \frac{(p^2)^2}{8m^3 c^2} - \frac{c_1}{m^2 c^2} [\vec{\sigma}\cdot\vec{p},[\vec{\sigma}\cdot\vec{p},V(\vec{r})]]$$ Expanding this out a bit, $$[\vec{\sigma}\cdot\vec{p},[\vec{\sigma}\cdot\vec{p},V]] = (p^2 V + V p^2) - 2 \vec{\sigma}\cdot\vec{p} V \vec{\sigma}\cdot\vec{p}$$ Both terms are separately Hermitian, but condition (4) fixes their relative coefficient.

Expanding this out, yet further (and letting $\vec{S}=\tfrac{\hbar}{2}\vec{\sigma}$) $$-\frac{c_1}{m^2 c^2} [\vec{\sigma}\cdot\vec{p},[\vec{\sigma}\cdot\vec{p},V]]= \frac{4c_1}{m^2 c^2} (\vec{\nabla}(V)\times \vec{p})\cdot\vec{S} + \frac{c_1\hbar^2}{m^2 c^2} \nabla^2(V)$$

For a central force potential, $\vec{\nabla}(V)= \vec{r}\frac{1}{r}\frac{d V}{d r}$ and the first term is the spin-orbit coupling, $\frac{4c_1}{m^2 c^2} \frac{1}{r}\frac{d V}{d r}\vec{L}\cdot\vec{S}$. The second term is the Darwin term. While I haven’t fixed the overall coefficient ($c_1=1/8$), I got the form of the spin-orbit coupling and of the Darwin term correct and I fixed their relative coefficient (correctly!).

No hand-wavy hocus-pocus was required.

And I did learn something that I never knew before, namely that this correction can be succinctly written as a double-commutator $[\vec{\sigma}\cdot\vec{p},[\vec{\sigma}\cdot\vec{p},V]]$. I don’t think I’ve ever seen that before …

Trying to spread the word:

The Department of Physics and Astronomy at Rice University in Houston, Texas invites applications for two tenure-track faculty positions, one experimental and one theoretical, in the area of quantum science using atomic, molecular, or optical methods. This encompasses quantum information processing, quantum sensing, quantum networks, quantum transduction, quantum many-body physics, and quantum simulation conducted on a variety of platforms. The ideal candidates will intellectually connect AMO physics to topics in condensed matter and quantum information theory. In both searches, we seek outstanding scientists whose research will complement and extend existing quantum activities within the Department and across the University. In addition to developing an independent and vigorous research program, the successful applicants will be expected to teach, on average, one undergraduate or graduate course each semester, and contribute to the service missions of the Department and University. The Department anticipates making the appointments at the assistant professor level. A Ph.D. in physics or related field is required by June 30, 2024.

Applications for these positions must be submitted electronically at apply.interfolio.com/131378 (experimental) and apply.interfolio.com/131379 (theoretical). Applicants will be required to submit the following: (1) cover letter; (2) curriculum vitae; (3) statement of research; (4) statement on teaching; (5) statement on diversity, mentoring, and outreach; (6) PDF copies of up to three publications; and (7) the names, affiliations, and email addresses of three professional references. Rice University, and the Department of Physics and Astronomy, are strongly committed to a culturally diverse intellectual community. In this spirit, we particularly welcome applications from all genders and members of historically underrepresented groups who exemplify diverse cultural experiences and who are especially qualified to mentor and advise all members of our diverse student population. We will begin reviewing applications by November 15, 2023. To receive full consideration, all application materials must be received by December 15, 2023. The expected appointment date is July 2024.

____

In addition, the Nobels will be announced this week.  For the nth year in a row, I will put forward my usual thought that it could be Aharonov and Berry for geometric phases in physics (though I know that Pancharatnam is intellectually in there and died in 1969).  Speculate away below in the comments.  I'm traveling, but I will try to follow the discussion.

Science and scholarship are global. If you want to find out the truth about the universe, you’ll have to employ the people best at figuring out that truth, regardless of where they come from. Research shuffles people around, driving them together to collaborate and apart to share their expertise.

(If you don’t care about figuring out the truth, and just want to make money? You still may want international employees. For plenty of jobs, the difference between the best person in the world and the best person in your country can be quite substantial.)

How do you get these international employees? You could pay them a lot, I guess, but that’s by definition expensive, and probably will annoy the locals. Instead, most of what you need to do to attract international employees isn’t to give them extra rewards: instead, it’s more important to level the playing field, and cover for the extra disadvantages an international employee will have.

You might be surprised when I mention disadvantages, but while international employees may be talented people, that doesn’t make moving to another country easy. If you stay in the same country you were born, you get involved in that country’s institutions in a regular way. Your rights and responsibilities, everything from driving to healthcare to taxes, are set up gradually over the course of your life. For someone moving to a new country, that means all of this has to be set up all at once.

This means that countries that can process these things quickly are much better for international employees. If your country takes six months to register someone for national healthcare, then new employees are at risk during that time or will have to pay extra for private insurance. If a national ID number is required to get a bank account, then whatever processing time that ID number takes must pass before the new employee can get paid. It also matters if the rules are clearly and consistently communicated, as new international employees can waste a lot of time and money if they’re given incorrect advice, or if different bureaucrats enforce different rules at their own discretion.

It also means that employers have an advantage if they can smooth entry into these institutions. In some countries it can be quite hard to find a primary care physician, as most people have the same doctor as their parents, switching only when a doctor retires. When I worked with the Perimeter Institute, they had a relationship with a local clinic that would accept their new employees as clients. In a city where it was otherwise quite hard to find a doctor, that was a real boon. Employers can also offer consistent advice even when their government doesn’t. They can keep track of their employees experiences and make reliable guides for how to navigate the system. If they can afford it, they can even keep an immigration lawyer on staff to advise about these questions.

An extremely important institution is the language itself. Moving internationally will often involve moving somewhere where you don’t speak the language, or don’t speak it very well. This gives countries an advantage if their immigrant-facing institutions are proficient in a language that’s common internationally, which at the moment largely means English. It also means countries have a big advantage if their immigrant-facing institutions are digital. If you communicate with immigrants with text, they can find online translations and at least try to figure things out. If you communicate in person, or worse through a staticky phone line, then you will try the patience even of people who do passably speak the language.

In the long term, of course, one cannot get by in one’s native language alone. As such, it is also important for countries to have good ways for people to learn the language. While I lived there, Denmark went back and forth on providing free language lessons for recent immigrants, sometimes providing them and sometimes not.

All of these things become twice as important in the case of spouses. You might think the idea that a country or employer should help out a new employee’s spouse is archaic, a product of an era of housewives discouraged from supporting themselves. But it is precisely because we don’t live in such an era that countries and employers need to take spouses into account. For an employer, hiring someone from another country is already an unusual event. Two partners getting hired to move to the same country by different employers at the same time is, barring special arrangements, extremely unlikely. That means that spouses of international employees should not have to wait for an employer to give them the same rights as their spouse: they need the same right to healthcare and employment and the like as their spouse, on arrival, so that they can find jobs and integrate without an unfair disadvantage. An employer can level the playing field further. The University of Copenhagen’s support for international spouses included social events (important because it’s hard to make new friends in a new country without the benefit of work friends), resume help (because each country has different conventions and expectations for job seekers), and even legal advice. At minimum, every resource you provide your employees that could in principle also be of use to their spouses (language classes, help with bureaucracy) should be considered.

In all your planning, as a country or an employer, keep in mind that not everyone has the same advantages. You can’t assume that someone moving to a new country will be able to integrate on their own. You have to help them, if not for fairness’ sake, then because if you don’t you won’t keep getting international employees to come at all.

Want to learn applied category theory? You can now read my lectures here:

• Lectures on applied category theory

There are a lot, but each one is bite-sized and basically covers just one idea. They’re self-contained, but you can also read them along with Fong and Spivak’s free book to get two outlooks on the same material:

• Brendan Fong and David Spivak, Seven Sketches in Compositionality: An Invitation to Applied Category Theory.

Huge thanks go to Simon Burton for making my lectures into nice web pages! But they still need work. If you see problems, please let me know.

Here’s one problem: I need to include more of my ‘Puzzles’ in these lectures. None of the links to puzzles work. Students in the original course also wrote up answers to all of these puzzles, and to many of Fong and Spivak’s exercises. But it would take quite a bit of work to put all those into webpage form, so I can’t promise to do that. 😢

Here are the lectures:

Chapter 1: Ordered Sets

• Lecture 1 - Introduction

• Lecture 2 - What is Applied Category Theory?

• Lecture 3 - Preorders

• Lecture 4 - Galois Connections

• Lecture 5 - Galois Connections

• Lecture 6 - Computing Adjoints

• Lecture 7 - Logic

• Lecture 8 - The Logic of Subsets

• Lecture 9 - Adjoints and the Logic of Subsets

• Lecture 10 - The Logic of Partitions

• Lecture 11 - The Poset of Partitions

• Lecture 12 - Generative Effects

• Lecture 13 - Pulling Back Partitions

• Lecture 14 - Adjoints, Joins and Meets

• Lecture 15 - Preserving Joins and Meets

• Lecture 16 - The Adjoint Functor Theorem for Posets

• Lecture 17 - The Grand Synthesis

Chapter 2: Resource Theories

• Lecture 18 - Resource Theories

• Lecture 19 - Chemistry and Scheduling

• Lecture 20 - Manufacturing

• Lecture 21 - Monoidal Preorders

• Lecture 22 - Symmetric Monoidal Preorders

• Lecture 23 - Commutative Monoidal Posets

• Lecture 24 - Pricing Resources

• Lecture 25 - Reaction Networks

• Lecture 26 - Monoidal Monotones

• Lecture 27 - Adjoints of Monoidal Monotones

• Lecture 28 - Ignoring Externalities

• Lecture 29 - Enriched Categories

• Lecture 30 - Preorders as Enriched Categories

• Lecture 31 - Lawvere Metric Spaces

• Lecture 32 - Enriched Functors

• Lecture 33 - Tying Up Loose Ends

Chapter 3: Databases

• Lecture 34 - Categories

• Lecture 35 - Categories versus Preorders

• Lecture 36 - Categories from Graphs

• Lecture 37 - Presentations of Categories

• Lecture 38 - Functors

• Lecture 39 - Databases

• Lecture 40 - Relations

• Lecture 41 - Composing Functors

• Lecture 42 - Transforming Databases

• Lecture 43 - Natural Transformations

• Lecture 44 - Categories, Functors and Natural Transformations

• Lecture 45 - Composing Natural Transformations

• Lecture 46 - Isomorphisms

• Lecture 47 - Adjoint Functors

• Lecture 48 - Adjoint Functors

• Lecture 49 - Kan Extensions

• Lecture 50 - Left Kan Extensions

• Lecture 51 - Right Kan Extensions

• Lecture 52 - The Hom-Functor

• Lecture 53 - Free and Forgetful Functors

• Lecture 54 - Tying Up Loose Ends

Chapter 4: Collaborative Design

• Lecture 55 - Enriched Profunctors and Collaborative Design

• Lecture 56 - Feasibility Relations

• Lecture 57 - Feasibility Relations

• Lecture 58 - Composing Feasibility Relations

• Lecture 59 - Cost-Enriched Profunctors

• Lecture 60 - Closed Monoidal Preorders

• Lecture 61 - Closed Monoidal Preorders

• Lecture 62 - Enriched Profunctors

• Lecture 63 - Composing Enriched Profunctors

• Lecture 64 - The Category of Enriched Profunctors

• Lecture 65 - Collaborative Design

• Lecture 66 - Collaborative Design

• Lecture 67 - Feedback in Collaborative Design

• Lecture 68 - Feedback in Collaborative Design

• Lecture 69 - Feedback in Collaborative Design

• Lecture 70 - Tensoring Enriched Profunctors

• Lecture 71 - Caps and Cups for Enriched Profunctors

• Lecture 72 - Monoidal Categories

• Lecture 73 - String Diagrams and Strictification

• Lecture 74 - Compact Closed Categories

• Lecture 75 - The Grand Synthesis

• Lecture 76 - The Grand Synthesis

• Lecture 77 - The End? No, the Beginning!

Want to learn applied category theory? You can now read my lectures here:

• Lectures on applied category theory

There are a lot of them, but each one is bite-sized and basically covers just one idea. They’re self-contained, but you can also read them along with Fong and Spivak’s free book to get two outlooks on the same material:

• Brendan Fong and David Spivak, Seven Sketches in Compositionality: An Invitation to Applied Category Theory.

Huge thanks go to Simon Burton for making my lectures into nice web pages! But they still need work. If you see problems, please let me know.

Here’s a problem: I need to include more of my ‘Puzzles’ in these lectures. None of the links to puzzles work. Students in the original course also wrote up answers to all of these puzzles, and to many of Fong and Spivak’s exercises. But it would take quite a bit of work to put all those into webpage form, so I can’t promise to do that.

Here are the lectures:

Chapter 1: Ordered Sets

• Lecture 1 – Introduction

• Lecture 2 – What is Applied Category Theory?

• Lecture 3 – Preorders

• Lecture 4 – Galois Connections

• Lecture 5 – Galois Connections

• Lecture 6 – Computing Adjoints

• Lecture 7 – Logic

• Lecture 8 – The Logic of Subsets

• Lecture 9 – Adjoints and the Logic of Subsets

• Lecture 10 – The Logic of Partitions

• Lecture 11 – The Poset of Partitions

• Lecture 12 – Generative Effects

• Lecture 13 – Pulling Back Partitions

• Lecture 14 – Adjoints, Joins and Meets

• Lecture 15 – Preserving Joins and Meets

• Lecture 16 – The Adjoint Functor Theorem for Posets

• Lecture 17 – The Grand Synthesis

Chapter 2: Resource Theories

• Lecture 18 – Resource Theories

• Lecture 19 – Chemistry and Scheduling

• Lecture 20 – Manufacturing

• Lecture 21 – Monoidal Preorders

• Lecture 22 – Symmetric Monoidal Preorders

• Lecture 23 – Commutative Monoidal Posets

• Lecture 24 – Pricing Resources

• Lecture 25 – Reaction Networks

• Lecture 26 – Monoidal Monotones

• Lecture 27 – Adjoints of Monoidal Monotones

• Lecture 28 – Ignoring Externalities

• Lecture 29 – Enriched Categories

• Lecture 30 – Preorders as Enriched Categories

• Lecture 31 – Lawvere Metric Spaces

• Lecture 32 – Enriched Functors

• Lecture 33 – Tying Up Loose Ends

Chapter 3: Databases

• Lecture 34 – Categories

• Lecture 35 – Categories versus Preorders

• Lecture 36 – Categories from Graphs

• Lecture 37 – Presentations of Categories

• Lecture 38 – Functors

• Lecture 39 – Databases

• Lecture 40 – Relations

• Lecture 41 – Composing Functors

• Lecture 42 – Transforming Databases

• Lecture 43 – Natural Transformations

• Lecture 44 – Categories, Functors and Natural Transformations

• Lecture 45 – Composing Natural Transformations

• Lecture 46 – Isomorphisms

• Lecture 47 – Adjoint Functors

• Lecture 48 – Adjoint Functors

• Lecture 49 – Kan Extensions

• Lecture 50 – Left Kan Extensions

• Lecture 51 – Right Kan Extensions

• Lecture 52 – The Hom-Functor

• Lecture 53 – Free and Forgetful Functors

• Lecture 54 – Tying Up Loose Ends

Chapter 4: Collaborative Design

• Lecture 55 – Enriched Profunctors and Collaborative Design

• Lecture 56 – Feasibility Relations

• Lecture 57 – Feasibility Relations

• Lecture 58 – Composing Feasibility Relations

• Lecture 59 – Cost-Enriched Profunctors

• Lecture 60 – Closed Monoidal Preorders

• Lecture 61 – Closed Monoidal Preorders

• Lecture 62 – Enriched Profunctors

• Lecture 63 – Composing Enriched Profunctors

• Lecture 64 – The Category of Enriched Profunctors

• Lecture 65 – Collaborative Design

• Lecture 66 – Collaborative Design

• Lecture 67 – Feedback in Collaborative Design

• Lecture 68 – Feedback in Collaborative Design

• Lecture 69 – Feedback in Collaborative Design

• Lecture 70 – Tensoring Enriched Profunctors

• Lecture 71 – Caps and Cups for Enriched Profunctors

• Lecture 72 – Monoidal Categories

• Lecture 73 – String Diagrams and Strictification

• Lecture 74 – Compact Closed Categories

• Lecture 75 – The Grand Synthesis

• Lecture 76 – The Grand Synthesis

• Lecture 77 – The End? No, the Beginning!

My day started with a long and very fun conversation with Monica Pate (NYU) about conservation laws in classical physics. As we all know, conservation laws are related to symmetries; each symmetry of the laws of physics creates a conservation law. Or does it? Well, it's a theorem! But it's a theorem when the laws of physics are lagrangian (or hamiltonian). That is, every symmetry of a hamiltonian system is associated with a conservation law in that system. So I asked: How do we know if or whether the world is lagrangian or hamiltonian? How could we know that? My best guess is that we know it because of these very conservation laws! The situation is complex.

At the end of last week I had a great conversation with Valentina Tardugno (NYU) and Phil Armitage (Flatiron) about how planets form. I spent the whole weekend thinking about it: If a few planets are forming in a proto-planetary disk, there are all sorts of interactions between the planets and the disk, and the planets and each other, and the disk with itself. You can think of this (at least) two different ways:

You can think of planets which are interacting not just directly with one another, but also with a disk, and with each other as mediated by that disk. This is the planet-centric view. In this view, the planets are what you are tracking, and the disk is a latent object that makes the planets interact and evolve.

Alternatively, you can think of the disk, with planets in it. In this disk-centric view, the planets are latent objects that modify the disk, creating gaps and spiral waves.

Both views are legitimate, and both have interesting science questions. We will explore and see where to work. I am partial to the planet-centric view: I want to know where planetary systems come from!

The press is full of excitement today at the news that anti-matter — hydrogen anti-atoms, specifically, made from positrons and anti-protons instead of electrons and protons — falls down rather than rising up. This has been shown in the ALPHA experiment at CERN. But no theoretical physicist is surprised. Today I’ll explain one of many reasons that no one in the halls of theoretical physics even blinked.

One basic point is that “anti-particle” is not a category but a relationship. We do not say “electrons are particles” and “positrons are anti-particles”, nor do we say “quarks are particles” and “anti-quarks are anti-particles.” Such statements would be inconsistent. Instead we say “quarks and anti-quarks are each others’ anti-particles.” (It’s like the term “opponent” in a football match.) That’s because some particles are their own anti-particles, including photons. If we tried to divide all the types of particles into two categories, it wouldn’t be clear where photons would go; we couldn’t say either that they are particles or that they are anti-particles.

Because of this, we can’t hope that all the types of particles separate into two groups, particles which fall and anti-particle which rise. At best, we would have to guess what would happen to particles that are their own anti-particles. Fortunately, this is easy. There is such a thing as a positronium atom, an exotic atom made from an electron and positron co-orbiting one other. A positronium atom is its own anti-particle; if we flip every particle for its anti-particle, the positronium atom’s electron and positron simply switch places. If indeed gravity pulls down on electrons and pushes up by the same amount on positrons, then gravity’s pull on a positronium atom would cancel. This atom would neither rise nor fall; it would float, feeling no net gravity.

This example would then lead us to expect that particles that are their own anti-particles will float. The logic would apply to photons; they would feel no gravity.

This would be a consistent picture. But experimentally, it is false: photons do feel gravity. The Sun bends the path of light, a fact that made Einstein famous in 1919, and an object’s strong gravity can create a gravitational lens that completely distorts the appearances of objects behind it. Photons can even orbit black holes. So experiment would force us to accept the picture shown below, if we want positrons to rise. Unfortunately, it is logically inconsistent.

The only consistent picture, then, is that everything falls. Are there any loopholes in this argument? Sure; perhaps the gravity of the Earth, made of atoms, causes electrons and quarks to fall rapidly and causes positrons and anti-quarks to rise slowly, so that positronium still falls, just more gently than ordinary atoms do. (The reverse would be true around a planet made of anti-atoms.) This gives us many more possibilities to consider, and we have to get into more complex questions of what experiment has and hasn’t excluded.

But we would still face another serious problem, because there are anti-quarks inside of protons. (One line of evidence for this is shown here.) If quarks and anti-quarks, which have the same inertial mass (the type of mass that determines how they change speed when pushed) had different gravitational mass (which determines how gravity affects them), then protons and neutrons, too, would not have equal inertial and gravitational mass. [The many gluons inside the protons and neutrons could make this even worse.] Since electrons do have equal inertial and gravitational mass, protons and neutrons would then fall at a different rate than electrons do. The consequence would be that different atoms would fall at slightly different rates. High precision experiments clearly say otherwise. This poses additional obstructions to the idea that anti-quarks (and the anti-protons and anti-neutrons that contain them) could rise in Earth’s gravity.

At best, when it comes to mass and gravity, existing experiments only allow for minor differences between atoms and anti-atoms. To look for subtle effects of any such differences is one of the real, long-term goals of the ALPHA experiment. What they’ve done so far is a neat experimental coup, but despite the headlines, it does not change our basic knowledge of anti-particles in general or of anti-atoms in particular. For that, we have a few years to wait.

Personally, I think that popular science books ought to devote more pages to the issue of how language is used in science. The words scientists choose are central to communication and miscommunication both among researchers and between scientists and non-scientists. The problem is that all language is full of misnomers and contradictory definitions, and scientific language is no exception.

One especially problematic scientific word is “matter.” It has multiple and partly contradictory meanings within particle physics, astronomy and cosmology. For instance,

• (Quote) It’s not even clear that “dark matter,” a term used widely by astronomers and particle physicists alike, is actually matter.
• (Endnote) Among possible dark matter particles are axions and dark photons, neither of which would obviously qualify as “matter.”*

Why might one not view them as matter?

This July, I came upon a museum called the Haus der Musik in one of Vienna’s former palaces. The museum contains a room dedicated to Johann Strauss II, king of the waltz. The room, dimly lit, resembles a twilit gazebo. I could almost believe that a hidden orchestra was playing the rendition of “The Blue Danube” that filled the room. Glass cases displayed dance cards and accessories that dancers would bring to a nineteenth-century ball.

A ball. Who hasn’t read about one in a novel or seen one in a film? A throng of youngsters and their chaperones, rustling in silk. The glint of candles, the vigor of movement, the thrill of interaction, the anxiety of establishing one’s place in society.

Another throng gathered a short walk from the Haus der Musik this summer. The Vienna University of Technology hosted the conference Quantum Thermodynamics (QTD) in the heart of the city. Don’t tell the other annual conferences, but QTD is my favorite. It spotlights the breed of quantum thermodynamics that’s surged throughout the past decade—the breed saturated with quantum information theory. I began attending QTD as a PhD student, and the conference shifts from city to city from year to year. I reveled in returning in person for the first time since the pandemic began.

Yet this QTD felt different. First, instead of being a PhD student, I brought a PhD student of my own. Second, granted, I enjoyed catching up with colleagues-cum-friends as much as ever. I especially relished seeing the “classmates” who belonged to my academic generation. Yet we were now congratulating each other on having founded research groups, and we were commiserating about the workload of primary investigators. 

Third, I found myself a panelist in the annual discussion traditionally called “Quo vadis, quantum thermodynamics?” The panel presented bird’s-eye views on quantum thermodynamics, analyzing trends and opining on the direction our field was taking (or should take).¹ Fourth, at the end of the conference, almost the last sentence spoken into any microphone was “See you in Maryland next year.” Colleagues and I will host QTD 2024.

The day after QTD ended, I boarded an Austrian Airlines flight. Waltzes composed by Strauss played over the loudspeakers. They flipped a switch in my mind: I’d come of age, I thought. I’d attended QTD 2017 as a debutante, presenting my first invited talk at the conference series. I’d danced through QTD 2018 in Santa Barbara, as well as the online iterations held during the pandemic. I’d reveled in the vigor of scientific argumentation, the thrill of learning, the glint of slides shining on projector screens (not really). Now, I was beginning to shoulder responsibilities like a ballgown-wearing chaperone.

As I came of age, so did QTD. The conference series budded around the time I started grad school and embarked upon quantum-thermodynamics research. In 2017, approximately 80 participants attended QTD. This year, 250 people registered to attend in person, and others attended online. Two hundred fifty! Quantum thermodynamics scarcely existed as a field of research fifteen years ago.

I’ve heard that organizers of another annual conference, Quantum Information Processing (QIP), reacted similarly to a 250-person registration list some years ago. Aram Harrow, a professor and quantum information theorist at MIT, has shared stories about co-organizing the first QIPs. As a PhD student, he’d sat in his advisor’s office, taking notes, while the local quantum-information theorists chose submissions to highlight. Nowadays, a small army of reviewers and subreviewers processes the hordes of submissions. And, from what I heard about this year’s attendance, you almost might as well navigate a Disney theme park on a holiday as the QIP crowd. 

Will QTD continue to grow like QIP? Would such growth strengthen or fracture the community? Perhaps we’ll discuss those questions at a “Quo vadis?” session in Maryland next year. But I, at least, hope to continue always to grow—and to dance.²

¹My opinion: Now that quantum thermodynamics has showered us with fundamental insights, we should apply it in practical applications. How? Collaborators and I suggest one path here.

²I confess to having danced the waltz step (gleaned during my 14 years of ballet training) around that Strauss room in the Haus der Musik. I didn’t waltz around the conference auditorium, though.

 It's been a very busy time, hence my lower posting frequency.  It was rather intense trying to attend both the KITP conference and the morning sessions of the DOE experimental condensed matter PI meeting (pdf of agenda here).  A few quick highlights that I thought were interesting:

• Kagome metals of the form AV3Sb5 are very complicated.  In these materials, in the a-b plane the V atoms form a Kagome lattice (before that one reader corrects me, I know that this is not formally a lattice from the crystallographic point of view, just using the term colloquially).  Band structure calculations show that there are rather flat bands (for an explanation, see here) near the Fermi level, and there are Dirac cones, van Hove singularities, Fermi surface nesting, etc.  These materials have nontrivial electronic topology, and CsV3Sb5 and KV3Sb5 both have charge density wave transitions and low-temperature superconductivity.  Here is a nice study of the CDW in CsV3Sb5, and here is a study that shows that there is no spontaneous breaking of time-reversal symmetry below that transition.  This paper shows that there is funky nonlinear electronic transport (apply a current at frequency \(\omega\), measure a voltage at frequency \(2 \omega\)) in CsV3Sb5 that is switchable in sign with an out-of-plane magnetic field.  Weirdly, that is not seen in KV3Sb5 even though the basic noninteracting band structures of the two materials are almost identical, implying that it has something to do with electronic correlation effects.
• Related to that last paper, here is a review article about using focused ion beams for sample preparation and material engineering.  It's pretty amazing what can be done with these tools, including carving out micro/nanostructured devices from originally bulk crystals of interesting materials.  
• The temperature-dependent part of the electrical resistivity of Fermi liquids is expected to scale like \(T^{2}\) as \(T \rightarrow 0\).  One can make a very general argument (that ignores actual kinematic restrictions on scattering) based on the Pauli exclusion principle that the inelastic e-e scattering rate should go like \(T^{2}\) (number of electron quasiparticles excited goes like \(T\), number of empty states available to scatter into also goes like \(T\)).  However, actually keeping track of momentum conservation, it turns out that one usually needs Umklapp scattering processes to get this.  That isn't necessary all the time, however.  In very low density metals, the Fermi wavevector is far from the Brillouin zone boundary and so Umklapp should not be important, but it is still possible to get \(T^{2}\) resistivity (see here as well).  Similarly, in 3He, a true Fermi liquid, there is no lattice, so there is no such thing as Umklapp, but at the lowest temperatures the \(T^{2}\) thermal conduction is still seen (though some weird things happen at higher temperatures). 

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I had two interactions today that made me think seriously about big-picture and design things. I like design language: How do you design your whole research program, and how do you design individual projects so they fit into it. One interaction was in the Astronomical Data Meeting at Flatiron, where Vivi Acquaviva (CUNY) talked about the intersection between what you are good at, what is important, and what brings you joy. That's a hard intersection to find. Or way too easy; I am not sure. The other interaction was a conversation with Jiayin Dong (Flatiron), who is thinking about faculty job applications and the like. How to talk about your research in terms of the next decade instead of the next year?

One comment that is frequently made by Hans-Walter Rix (MPIA) is that he feels like most early-career (and even mid-career) people spend too much time doing their science and not enough time planning and justifying their science. It is important to be able to answer “why” questions about your research, and in the medium term it helps all your projects.

I wrote a little article explaining the concept of ‘moduli space’ through an example. It’s due October 1st so I’d really appreciate it if you folks could take a look and see if it’s clear enough. It’s really short, and it’s written for people who know some math, but not necessarily anything about moduli spaces.

The cool part is the connection between the moduli space of acute triangles — that is, the space of all shapes an acute triangle can have — and the more famous moduli space of elliptic curves.

There’s a lot more one could do with this, e.g. describing the modular lambda function as the cross ratio of the 4 points on a sphere obtained by taking an acute triangle, dividing it into 4 similar acute triangles, and folding it up to form a tetrahedron, which is conformally equivalent to a sphere. But I didn’t have space for that here!

Okay, here goes.

The moduli space of acute triangles

In mathematics we often like to classify objects up to isomorphism. Sometimes the classification is discrete, but sometimes we have a notion of when two objects are ‘close’. Then we can make the set of isomorphism classes into a topological space called a ‘moduli space’. A simple example is the moduli space of acute triangles. In simple terms, this is the space of all possible shapes that an acute triangle can have, where we count two triangles as having the same shape if they are similar.

As a first step, consider triangles with labeled vertices in the complex plane. Every triangle is similar to one with its first vertex at $0$, the second at $1$, and the third at some point in the upper half-plane. This triangle is acute precisely when its third vertex lies in this set:

$$T = \left\{z \in \mathbb{C} \; \left\vert \; \mathrm{Im}(z) \gt 0, \; 0 \lt \mathrm{Re}(z) \lt 1, \; |z - \tfrac{1}{2}| \gt \tfrac{1}{2} \right. \right\}$$

So, we say $T$ is the moduli space of acute triangles with labeled vertices. This set is colored yellow and purple above; the yellow and purple regions on top extend infinitely upward.

To get the moduli space of acute triangles with unlabeled vertices, we must mod out $T$ by the action of $S_3$ that permutes the three vertices. The 6 yellow and purple regions in $T$ are ‘fundamental domains’ for this $S_3$ action: that is, they each contain exactly one point from each orbit. If we reflect a labeled triangle corresponding to a point in a yellow region we get a triangle corresponding to a point in a purple region, and vice versa. Points on the boundary between two regions correspond to isosceles triangles. All 6 regions meet at the point that corresponds to an equilateral triangle.

The moduli space of acute triangles is closely related to a more famous moduli space: that of elliptic curves. The group $\mathrm{GL}(2,\mathbb{Z})$, consisting of invertible $2 \times 2$ integer matrices, acts on the upper half-plane

$$\mathcal{H} = \left\{z \in \mathbb{C} \; \left\vert \; \mathrm{Im}(z) \gt 0 \right. \right\}$$

as follows:

$$\left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \colon z \mapsto \frac{a z + b}{c z + d} .$$

The light and dark regions shown above are fundamental domains for this group action. Elements of $\mathrm{GL}(2,\mathbb{Z})$ with determinant $-1$ map light regions to dark ones and vice versa. Elements with determinant $1$ map light regions to light ones and dark ones to dark ones. People more often study the action of the subgroup $\mathrm{SL}(2,\mathbb{Z})$ consisting of $g \in \mathrm{GL}(2,\mathbb{Z})$ with determinant 1. The union of a light region and a dark one, touching each other, forms a fundamental domain of $\mathrm{SL}(2,\mathbb{Z})$.

For any point $z \in \mathcal{H}$ we can form a parallelogram with vertices $0, 1, z$ and $z+1$. If we identify the opposite edges of this parallelogram we get an elliptic curve: a torus equipped with the structure of a complex manifold. We can get every elliptic curve this way, at least up to isomorphism. Moreover, two points $z,z' \in \mathcal{H}$ in the upper half-plane give isomorphic elliptic curves iff $z' = g z$ for some $g \in \mathrm{SL}(2,\mathbb{Z})$. Thus the quotient space $\mathcal{H}/\mathrm{SL}(2,\mathbb{Z})$ is the moduli space of elliptic curves: points in this space correspond to isomorphism classes of elliptic curves.

Since $T$ is the union of three fundamental domains for $\mathrm{SL}(2,\mathbb{Z})$, there is a map

$$p \colon T \to \mathcal{H}/\mathrm{SL}(2,\mathbb{Z})$$

from the moduli space of acute triangles to the moduli space of elliptic curves, and generically this map is three-to-one. This map is not onto, but if we take the closure of $T$ inside $\mathcal{H}$ we get a larger set

$$\overline{T} = \left\{z \in \mathbb{C} \; \vert \; \mathrm{Im}(z) \gt 0, \; 0 \le \mathrm{Re}(z) \le 1, \; |z - \tfrac{1}{2}| \ge \tfrac{1}{2} \right\}$$

whose boundary consists of points corresponding to right triangles. Then $p$ extends to an onto map

$$p \colon \overline{T} \to \mathcal{H}/\mathrm{SL}(2,\mathbb{Z}) .$$

The existence of this map suggests that from any acute or right triangle in the plane we can construct an elliptic curve, in such a way that similar triangles give isomorphic elliptic curves. This is in fact true! How can we understand this more directly?

Take any acute or right triangle with labelled vertices in the complex plane. Rotating it 180° around the midpoint of any edge we get another triangle. The union of these two triangles is a parallelogram. Identifying opposite edges of this parallelogram we get a torus with a complex structure — and this is an elliptic curve! There are three choices of how to build this parallelogram, one for each edge of the original triangle, but they give isomorphic elliptic curves. Also, similar triangles give isomorphic elliptic curves. Even better, every elliptic curve is isomorphic to one arising from this construction. So this construction gives a map from $\overline{T}$ onto $\mathcal{H}/\mathrm{SL}(2,\mathbb{Z})$, and with a little thought one can see that this map is $p$.

I learned about the moduli space of acute triangles from James Dolan. There has also been interesting work on the moduli space of all triangles in the plane. Gaspar and Neto [2] noticed that this space is a triangle, and Ian Stewart later gave a more geometrical explanation [3]. In fact all the moduli spaces mentioned here are better thought of as moduli ‘stacks’: stacks give a way to understand the special role of more symmetrical objects, like isosceles and equilateral triangles. Kai Behrend [1] has written an introduction to stacks using various moduli stacks of triangles and the moduli space of elliptic curves as examples. Though he does not describe the map $p$ (or its stacky analogue), his work is a nice way to dig deeper into some of the material discussed here.

References

[1] K. Behrend, An introduction to algebraic stacks, in Moduli Spaces, eds. L. Brambila-Paz, P. Newstead, R. P. Thomas and O. García-Prada, Cambridge U. Press, Cambridge 2014, pp. 1–131. Also available at https://secure.math.ubc.ca/~behrend/math615A/stacksintro.pdf.

[2] J. Gaspar and O. Neto, All triangles at once, Amer. Math. Monthly 122 (2015), 982–982.

[3] I. Stewart, Why do all triangles form a triangle?, Amer. Math. Monthly 124 (2017), 70–73.

Recently Quanta magazine came out with an article explaining modular forms:

• Jordana Cepelewicz, Behold modular forms, the ‘fifth fundamental operation’ of math, Quanta, 23 September 2023.

It does a heroically good job. One big thing it doesn’t do is explain these funny looking ‘fundamental domains’ in the upper half-plane:

That is: where does this picture come from and why is it important?

By sheer coincidence, I just wrote a little article explaining the concept of ‘moduli space’ through an example, which does touch on these fundamental domains. It’s due October 1st so I’d really appreciate it if you folks could take a look and see if it’s clear enough. It’s really short, and it’s written for people who know some math—more than your typical Quanta reader—but not necessarily anything about moduli spaces.

The cool part is the connection between the moduli space of acute triangles—that is, the space of all shapes an acute triangle can have—and the more famous moduli space of elliptic curves.

Okay, here goes.

The moduli space of acute triangles

In mathematics we often like to classify objects up to isomorphism. Sometimes the classification is discrete, but sometimes we have a notion of when two objects are ‘close’. Then we can make the set of isomorphism classes into a topological space called a ‘moduli space’. A simple example is the moduli space of acute triangles. In simple terms, this is the space of all possible shapes that an acute triangle can have, where we count two triangles as having the same shape if they are similar.

As a first step, consider triangles with labeled vertices in the complex plane. Every triangle is similar to one with its first vertex at the second at and the third at some point in the upper half-plane. This triangle is acute precisely when its third vertex lies in this set:

which is colored yellow and purple above. So, we say is the moduli space of acute triangles with labeled vertices.

To get the moduli space of acute triangles with unlabeled vertices, we must mod out by the action of that permutes the three vertices. The 6 yellow and purple regions in are fundamental domains for this action. If we reflect a labeled triangle corresponding to a point in a yellow region we get a triangle corresponding to a point in a purple region, and vice versa. Points on the boundary between two regions correspond to isosceles triangles. All 6 regions meet at the point that corresponds to an equilateral triangle.

The moduli space of acute triangles is closely related to a more famous moduli space: that of elliptic curves. The group consisting of invertible integer matrices, acts on the upper half-plane

as follows:

The light and dark regions shown above are fundamental domains for this group action. Elements of with determinant map light regions to dark ones and vice versa. Elements with determinant map light regions to light ones and dark ones to dark ones. People more often study the action of the subgroup consisting of with determinant 1. The union of a light region and a dark one, touching each other, forms a fundamental domain of

For any point we can form a parallelogram with vertices and If we identify the opposite edges of this parallelogram we get an elliptic curve: a torus equipped with the structure of a complex manifold. We can get every elliptic curve this way, at least up to isomorphism. Moreover, two points in the upper half-plane give isomorphic elliptic curves iff for some Thus the quotient space is the moduli space of elliptic curves: points in this space correspond to isomorphism classes of elliptic curves.

Since is the union of three fundamental domains for there is a map

from the moduli space of acute triangles to the moduli space of elliptic curves, and generically this map is three-to-one. This map is not onto, but if we take the closure of inside we get a larger set

whose boundary consists of points corresponding to right triangles. Then extends to an onto map

The existence of this map suggests that from any acute or right triangle in the plane we can construct an elliptic curve, in such a way that similar triangles give isomorphic elliptic curves. This is in fact true! How can we understand this more directly?

Take any acute or right triangle with labelled vertices in the complex plane. Rotating it 180° around the midpoint of any edge we get another triangle. The union of these two triangles is a parallelogram. Identifying opposite edges of this parallelogram we get a torus with a complex structure—and this is an elliptic curve! There are three choices of how to build this parallelogram, one for each edge of the original triangle, but they give isomorphic elliptic curves. Also, similar triangles give isomorphic elliptic curves. Even better, every elliptic curve is isomorphic to one arising from this construction. So this construction gives a map from onto and with a little thought one can see that this map is

I learned about the moduli space of acute triangles from James Dolan. There has also been interesting work on the moduli space of all triangles in the plane. Gaspar and Neto [2] noticed that this space is a triangle, and Ian Stewart later gave a more geometrical explanation [3]. In fact all the moduli spaces mentioned here are better thought of as moduli ‘stacks’: stacks give a way to understand the special role of more symmetrical objects, like isosceles and equilateral triangles. Kai Behrend [1] has written an introduction to stacks using various moduli stacks of triangles and the moduli space of elliptic curves as examples. Though he does not describe the map (or its stacky analogue), his work is a nice way to dig deeper into some of the material discussed here.

References

[1] K. Behrend, An introduction to algebraic stacks, in Moduli Spaces, eds. L. Brambila-Paz, P. Newstead, R. P. Thomas and O. García-Prada, Cambridge U. Press, Cambridge 2014, pp. 1–131. Also available at https://secure.math.ubc.ca/~behrend/math615A/stacksintro.pdf.

[2] J. Gaspar and O. Neto, All triangles at once, Amer.
Math. Monthly 122 (2015), 982–982.

[3] I. Stewart, Why do all triangles form a triangle?, Amer. Math. Monthly 124 (2017), 70–73.

Today a reader asked me “Out of the quantum fields which have mass, do any of them also have weight?” I thought other readers would be interested in my answer, so I’m putting it here. (Some of what is discussed below is covered in greater detail in my upcoming book.)

Before we start, we need to rephrase the question, because fields do not have mass.

I’m in Regensburg this week attending a workshop on Interactions of Proof Assistants and Mathematics. One of the lecture series is being given by John Harrison, a Senior Principal Applied Scientist in the Automated Reasoning Group at Amazon Web Services, and a lead developer of the HOL Light interactive theorem prover. He just told us about a very cool construction of the non-negative real numbers as sequences of natural numbers satisfying a property he calls “near multiplicativity”. In particular, the integers and the rational numbers aren’t needed at all! This is how the reals are constructed in HOL Light and is described in more detail in a book he wrote entitled Theorem Proving with the Real Numbers.

Edit: as the commenters note, these are also known as the Eudoxus reals and were apparently discovered by our very own Stephen Schanuel and disseminated by Ross Street. Thanks for pointing me to the history of this construction!

The idea

One of the standard constructions of the real numbers is as equivalence classes of Cauchy sequences of rationals. Let us consider a non-negative real number $a$. One way to say that a sequence $q \colon \mathbb{N} \to \mathbb{Q}$ of rational numbers converges to $a$ is to ask there to exist a constant $A \in \mathbb{N}$ so that for all $n \in \mathbb{N}$,

$$|q_n - a | \le \frac{A}{n}.$$

These representations aren’t at all unique: many Cauchy sequences of rationals represent the same real number. And in particular, for any positive real number $a$, it is possible to find a sequence of natural numbers $a \colon \mathbb{N} \to \mathbb{N}$ so that the sequence $n \mapsto \frac{a_n}{n}$ converges to $a$ in the above sense, i.e.:

$$|\frac{a_n}{n} - a | \le \frac{A}{n}$$

or equivalently

$$|a_n - n \cdot a | \le A.$$

This sequence $a \colon \mathbb{N} \to \mathbb{N}$ will encode the real number $a$ (which is why I’ve given them the same name).

The construction

Now that I’ve explained the idea let’s try to characterize the sequences of natural numbers that will correspond to non-negative real numbers without presupposing the existence of non-negative real numbers. The idea is that a sequence $a \colon \mathbb{N} \to \mathbb{N}$ will have the property that the sequence $n \mapsto \frac{a_n}{n}$ encodes some non-negative real number just when this sequence is Cauchy, which we express in the following way: there exists a constant $A \in \mathbb{N}$ so that for all $n,m \in \mathbb{N}$,

$$|\frac{a_n}{n} - \frac{a_m}{m} | \le \frac{A}{n} + \frac{A}{m},$$

or equivalently by

$$|m \cdot {a_n} - n \cdot {a_m} | \le (m + n) \cdot A.$$

Such sequences are called nearly multiplicative. Supposedly this is equivalent to the property of a sequence being nearly additive, meaning there exists a constant $A' \in \mathbb{N}$ so that for all $m,n \in \mathbb{N}$

$$|a_{m+n} - (a_m + a_n)| \le A'.$$

The non-negative reals are then equivalence classes of nearly multiplicative sequences of natural numbers, where the equivalence relation says that $a, a' \colon \mathbb{N} \to \mathbb{N}$ represent the same real number when there exists $C \in \mathbb{N}$ so that for all $n \in \mathbb{N}$

$$|a_n - a'_n | \le C.$$

This is more or less the usual equivalence relation of Cauchy sequences, except with a specified rate of convergence.

Addition and multiplication

Now that we have the non-negative reals, how do we add and how do we multiply?

Given nearly multiplicative sequences $a \colon \mathbb{N} \to \mathbb{N}$ and $b \colon \mathbb{N} \to \mathbb{N}$, their sum $a + b \colon \mathbb{N} \to \mathbb{N}$ is defined by pointwise addition: $(a+b)_n := a_n + b_n$. This is fairly intuitive.

More interestingly, their product $a \cdot b \colon \mathbb{N} \to \mathbb{N}$ is defined by function composition: $(a \cdot b)_n := a_{b_n}$. It’s a fun exercise to work out that this converges to the desired non-negative real number.

You can think of cause and effect as the ultimate story. The world is filled with one damn thing happening after another, but to make sense of it we organize it into a narrative: this happened first, and it caused that, which caused that. We tie this to “what if” stories, stories about things that didn’t happen: if this hadn’t happened, then it wouldn’t have caused that, so that wouldn’t have happened.

We also tell stories about cause and effect. Physicists use cause and effect as a tool, a criterion to make sense of new theories: does this theory respect cause and effect, or not? And just like everything else in science, there is more than one story they tell about it.

As a physicist, how would you think about cause and effect?

The simplest, and most obvious requirement, is that effects should follow their causes. Cause and effect shouldn’t go backwards in time, the cause should come before the effect.

This all sounds sensible, until you remember that in physics “before” and “after” are relative. If you try to describe the order of two distant events, your description will be different than someone moving with a different velocity. You might think two things happened at the same time, while they think one happened first, and someone else thinks the other happened first.

You’d think this makes a total mess of cause and effect, but actually everything remains fine, as long nothing goes faster than the speed of light. If someone could travel between two events slower than the speed of light, then everybody will agree on their order, and so everyone can agree on which one caused the other. Cause and effect only get screwed up if they can happen faster than light.

(If the two events are two different times you observed something, then cause and effect will always be fine, since you yourself can’t go faster than the speed of light. So nobody will contradict what you observe, they just might interpret it differently.)

So if you want to make sure that your theory respects cause and effect, you’d better be sure that nothing goes faster than light. It turns out, this is not automatic! In general relativity, an effect called Shapiro time delay makes light take longer to pass a heavy object than to go through empty space. If you modify general relativity, you can accidentally get a theory with a Shapiro time advance, where light arrives sooner than it would through empty space. In such a theory, at least some observers will see effects happen before their causes!

Once you know how to check this, as a physicist, there are two kinds of stories you can tell. I’ve heard different people in the field tell both.

First, you can say that cause and effect should be a basic physical principle. Using this principle, you can derive other restrictions, demands on what properties matter and energy can have. You can carve away theories that violate these rules, making sure that we’re testing for theories that actually make sense.

On the other hand, there are a lot of stories about time travel. Time travel screws up cause and effect in a very direct way. When Harry Potter and Hermione travel back in time at the end of Harry Potter and the Prisoner of Azkaban, they cause the event that saves Harry’s life earlier in the book. Science fiction and fantasy are full of stories like this, and many of them are perfectly consistent. How can we be so sure that we don’t live in such a world?

The other type of story positions the physics of cause and effect as a search for evidence. We’re looking for physics that violates cause and effect, because if it exists, then on some small level it should be possible to travel back in time. By writing down the consequences of cause and effect, we get to describe what evidence we’d need to see it breaking down, and if we see it whole new possibilities open up.

These are both good stories! And like all other stories in science, they only capture part of what the scientists are up to. Some people stick to one or the other, some go between them, driven by the actual research, not the story itself. Like cause and effect itself, the story is just one way to describe the world around us.

Continuing with the supplementary material for the book, from its Chapter 2. This is in reference to Galileo’s principle of relativity, a central pillar of modern science. This principle states that perfectly steady motion in a straight line is indistinguishable from no motion at all, and thus cannot be felt. This is why we don’t feel our rapid motion around the Earth and Sun; over minutes, that motion is almost steady and straight. I wrote

• . . . Our planet rotates and roams the heavens, but our motion is nearly steady. That makes it nearly undetectable, thanks to Galileo’s principle.

To this I added a brief endnote, since the spin of the Earth can be detected, with some difficulty.

• As pointed out by the nineteenth-century French physicist Léon Foucault, the Earth’s rotation, the least steady of our motions, is reflected in the motion of a tall pendulum. Many science museums around the world have such a “Foucault pendulum” on exhibit.

But for those who would want to know more, here’s some information about how to measure the Earth’s spin.

1. Tomorrow at 1:30pm US Central time, I’ll be doing an online Q&A with Collective[i] Forecast about quantum computing (probably there will also be questions about AI safety). It’s open to all. Hope to see some of you there!
   
2. Toby Cubitt of University College London is visiting UT Austin. We’ve been discussing the question: can you produce a QMA witness state using a closed timelike curve? Since QMA⊆PSPACE, and since Fortnow, Watrous, and I proved that closed timelike curves (or anyway, Deutsch’s model of them) let you solve PSPACE problems, clearly a closed timelike curve lets you solve QMA decision problems, but that’s different from producing the actual witness state as the fixed-point of a polynomial-time superoperator. Toby has a neat recent result, which has as a corollary that you can produce the ground state of a local Hamiltonian using a CTC, if you have as auxiliary information the ground state energy as well as (a lower bound on) the spectral gap. But you do seem to need that extra information.
   
   Yesterday I realized there’s also a simpler construction: namely, take an n-qubit state from the CTC, and check whether it’s a valid QMA witness, having used Marriott-Watrous amplification to push the probability of error down to (say) exp(-n²). If the witness is valid, then send it back in time unmodified; otherwise replace it by the maximally mixed state. If valid witnesses exist, then you can check that this sets up a Markov chain whose stationary distribution is almost entirely concentrated on such witnesses. (If no valid witnesses exist, then the stationary distribution is just the maximally mixed state, or exponentially close to it.) One drawback of this construction is that it can only produce a Marriott-Watrous state, rather than the “original” QMA witness state.
   
   Is there a third approach, which overcomes the disadvantages of both mine and Toby’s? I’ll leave that question to my readers!
   
3. On the theme of QMA plus weird physics, a wonderful question emerged from a recent group meeting: namely, what’s the power of QMA if we let the verifier make multiple non-collapsing measurements of the same state, as in the “PDQP” model defined by myself, Bouland, Fitzsimons, and Lee? I conjecture that this enhanced QMA goes all the way up to NEXP (Nondeterministic Exponential-Time), by a construction related to the one I used to show that PDQP/qpoly = ALL (i.e., non-collapsing measurements combined with quantum advice lets you decide literally all languages), and that also uses the PCP Theorem. I even have some candidate constructions, though I haven’t yet proven their soundness.
   
   In the past, I would’ve spent more time on such a problem before sharing it. But after giving some students a first crack, I now … just want to know the answer? Inspecting my feelings in my second year of leave at OpenAI, I realized that I still care enormously about quantum complexity theory, but only about getting answers to the questions, barely at all anymore about getting credit for them. Admittedly, it took me 25 years to reach this state of not caring.

Rachel Greenfeld and I have just uploaded to the arXiv our paper “Undecidability of translational monotilings“. This is a sequel to our previous paper in which we constructed a translational monotiling of a high-dimensional lattice (thus the monotile is a finite set and the translates , of partition ) which was aperiodic (there is no way to “repair” this tiling into a periodic tiling , in which is now periodic with respect to a finite index subgroup of ). This disproved the periodic tiling conjecture of Stein, Grunbaum-Shephard and Lagarias-Wang, which asserted that such aperiodic translational monotilings do not exist. (Compare with the “hat monotile“, which is a recently discovered aperiodic isometric monotile for of , where one is now allowed to use rotations and reflections as well as translations, or the even more recent “spectre monotile“, which is similar except that no reflections are needed.)

One of the motivations of this conjecture was the observation of Hao Wang that if the periodic tiling conjecture were true, then the translational monotiling problem is (algorithmically) decidable: there is a Turing machine which, when given a dimension and a finite subset of , can determine in finite time whether can tile . This is because if a periodic tiling exists, it can be found by computer search; and if no tiling exists at all, then (by the compactness theorem) there exists some finite subset of that cannot be covered by disjoint translates of , and this can also be discovered by computer search. The periodic tiling conjecture asserts that these are the only two possible scenarios, thus giving the decidability.

On the other hand, Wang’s argument is not known to be reversible: the failure of the periodic tiling conjecture does not automatically imply the undecidability of the translational monotiling problem, as it does not rule out the existence of some other algorithm to determine tiling that does not rely on the existence of a periodic tiling. (For instance, even with the newly discovered hat and spectre tiles, it remains an open question whether the isometric monotiling problem for (say) polygons with rational coefficients in is decidable, with or without reflections.)

The main result of this paper settles this question (with one caveat):

Theorem 1 There does not exist any algorithm which, given a dimension , a periodic subset of , and a finite subset of , determines in finite time whether there is a translational tiling of by .

The caveat is that we have to work with periodic subsets of , rather than all of ; we believe this is largely a technical restriction of our method, and it is likely that can be removed with additional effort and creativity. We also remark that when , the periodic tiling conjecture was established by Bhattacharya, and so the problem is decidable in the case. It remains open whether the tiling problem is decidable for any fixed value of (note in the above result that the dimension is not fixed, but is part of the input).

Because of a well known link between algorithmic undecidability and logical undecidability (also known as logical independence), the main theorem also implies the existence of an (in principle explicitly describable) dimension , periodic subset of , and a finite subset of , such that the assertion that tiles by translation cannot be proven or disproven in ZFC set theory (assuming of course that this theory is consistent).

As a consequence of our method, we can also replace here by “virtually two-dimensional” groups , with a finite abelian group (which now becomes part of the input, in place of the dimension ).

We now describe some of the main ideas of the proof. It is a common technique to show that a given problem is undecidable by demonstrating that some other problem that was already known to be undecidable can be “encoded” within the original problem, so that any algorithm for deciding the original problem would also decide the embedded problem. Accordingly, we will encode the Wang tiling problem as a monotiling problem in :

Problem 2 (Wang tiling problem) Given a finite collection of Wang tiles (unit squares with each side assigned some color from a finite palette), is it possible to tile the plane with translates of these tiles along the standard lattice , such that adjacent tiles have matching colors along their common edge?

It is a famous result of Berger that this problem is undecidable. The embedding of this problem into the higher-dimensional translational monotiling problem proceeds through some intermediate problems. Firstly, it is an easy matter to embed the Wang tiling problem into a similar problem which we call the domino problem:

Problem 3 (Domino problem) Given a finite collection (resp. ) of horizontal (resp. vertical) dominoes – pairs of adjacent unit squares, each of which is decorated with an element of a finite set of “pips”, is it possible to assign a pip to each unit square in the standard lattice tiling of , such that every horizontal (resp. vertical) pair of squares in this tiling is decorated using a domino from (resp. )?

Indeed, one just has to interpet each Wang tile as a separate “pip”, and define the domino sets , to be the pairs of horizontally or vertically adjacent Wang tiles with matching colors along their edge.

Next, we embed the domino problem into a Sudoku problem:

Problem 4 (Sudoku problem) Given a column width , a digit set , a collection of functions , and an “initial condition” (which we will not detail here, as it is a little technical), is it possible to assign a digit to each cell in the “Sudoku board” such that for any slope and intercept , the digits along the line lie in (and also that obeys the initial condition )?

The most novel part of the paper is the demonstration that the domino problem can indeed be embedded into the Sudoku problem. The embedding of the Sudoku problem into the monotiling problem follows from a modification of the methods in our previous papers, which had also introduced versions of the Sudoku problem, and created a “tiling language” which could be used to “program” various problems, including the Sudoku problem, as monotiling problems.

To encode the domino problem into the Sudoku problem, we need to take a domino function (obeying the domino constraints associated to some domino sets ) and use it to build a Sudoku function (obeying some Sudoku constraints relating to the domino sets); conversely, every Sudoku function obeying the rules of our Sudoku puzzle has to arise somehow from a domino function. The route to doing so was not immediately obvious, but after a helpful tip from Emmanuel Jeandel, we were able to adapt some ideas of Aanderaa and Lewis, in which certain hierarchical structures were used to encode one problem in another. Here, we interpret hierarchical structure -adically (using two different primes due to the two-dimensionality of the domino problem). The Sudoku function that will exemplify our embedding is then built from by the formula

where are two large distinct primes (for instance one can take , for concreteness), denotes the number of times divides , and is the last non-zero digit in the base expansion of :

(with the conventions and ). In the case , the first component of (1) looks like this:

and a typical instance of the final component looks like this:

Amusingly, the decoration here is essentially following the rules of the children’s game “Fizz buzz“.

To demonstrate the embedding, we thus need to produce a specific Sudoku rule (as well as a more technical initial condition , which is basically required to exclude degenerate Sudoku solutions such as a constant solution) that can “capture” the target function (1), in the sense that the only solutions to this specific Sudoku puzzle are given by variants of (e.g., composed with various linear transformations). In our previous paper we were able to build a Sudoku puzzle that could similarly capture either of the first two components , of our target function (1) (up to linear transformations), by a procedure very akin to solving an actual Sudoku puzzle (combined with iterative use of a “Tetris” move in which we eliminate rows of the puzzle that we have fully solved, to focus on the remaining unsolved rows). Our previous paper treated the case when was replaced with a power of , as this was the only case that we know how to embed in a monotiling problem of the entirety of (as opposed to a periodic subset of ), but the analysis is in fact easier when is a large odd prime, instead of a power of . Once the first two components have been solved for, it is a relatively routine matter to design an additional constraint in the Sudoku rule that then constrains the third component to be of the desired form , with obeying the domino constraints.

Since the upcoming book is basically done, it’s time for me to launch the next phase of the project — the supplementary material, which will be placed here, on this website.

Any science book has to leave out many details of the subjects it covers, and omit many important topics. While my book has endnotes that help flesh out the main text, I know that some readers will want even more information. That’s what I’ll be building here over the coming months. I’ll continue to develop this material even after the book is published, as additional readers explore it. For a time, then, this will be a living, growing extension to the written text.

As I create this supplementary material, I’ll first post it on this blog, looking for your feedback in terms of its clarity and accuracy, and hoping to get a sense from you as to whether there are other questions that I ought to address. Let’s try this out today with a first example; I look forward to your comments.

In Chapter 2 of the book, I have written

• Over two thousand years ago, Greek thinkers became experts in geometry and found clever tricks for estimating the Earth’s shape and size.

This sentence then refers to an endnote, in which I state

• The shadow that the Earth casts on the Moon during a lunar eclipse is always disk-shaped, no matter the time of day, which can be true only for a spherical planet. Earth’s size is revealed by comparing the lengths of shadows of two identical objects, separated by a known north-south distance, measured at noon on the same day.*

Obviously this is very terse, and I’m sure some readers will want an explanation of the endnote. Here’s the explanation that I’ll post on this website:

Chapter 2, Endnote 1

By the time that ancient Rome’s power was expanding, Greek scholars understood the basics of solar and lunar eclipses. Noting the relation between the phases of the Moon and the positions of the Sun and Moon, they recognized that the Moon’s light is reflected sunlight. They were aware that New Moon occurs when the Sun and Moon are on the same side of the Earth, while Full Moon occurs when they are on the opposite sides. And they knew that a lunar eclipse occurs when the Earth lies between the Sun and Moon, so that the Earth blocks the Sun’s light and casts a shadow on the Moon. This is illustrated (not to scale) in Fig. 1.

From these shadows, they confirmed the Earth was a sphere. Clearly, if the Earth had the shape of an X, it could cast an X-shaped shadow.

If Earth were a flat disk like a coin, then depending on the Moon’s location in the sky, the Earth’s shadow might be circular or might be oval. The important point is the shadow of a circular disk is not always a circular disk. You can confirm this with a coin and a light bulb.

The only shape which always creates a circular, disk-like shadow, from any angle and from any place and at any time, is a sphere, as you can confirm with a ball and a light bulb.

This is consistent with what is actually observed in eclipses, as in Fig. 4. The circular shadow of the Earth is shown especially clearly when sets of photos taken during an eclipse are carefully aligned.

Next, one you are aware that the Earth’s a sphere (and, as the Greeks also knew, that the Sun is far away ), it’s not hard to learn the size of the Earth. Imagine two vertical towers sitting on flat ground in two different cities. To keep things especially simple, let’s imagine one city is due north of the other, and the distance between them — call it “D” — is already known.

In each city, a person observes their tower’s shadow at exactly noon. (No clock is needed, because one can watch the shadow over time, and noon is when the shadow is shortest.) The end of the shadow and the tower’s base and top form the points of a right-angle triangle, as in Fig. 5, whose other angles we can call ⍺ and 𝜃. For a person squatting at the shadow’s end, the angle ⍺ is easily measured: it is the angle formed by the tower’s silhouette against the sky. Because this is a right-angle triangle, the angle 𝜃 is 90 degrees minus ⍺, so each observer can easily determine 𝜃 for their city’s tower. We’ll call their two measurements 𝜃₁ and 𝜃₂. Knowing these angles and their distance D, they know everything we need to determine the size of the Earth.

The key observation is that the difference in these angles is the same as the difference in the latitude between the two cities. To see this, examine Fig. 6 below. The paths of sunlight (the orange dashed lines in Figs. 5 and 6) are parallel to the Earth-Sun line. [This (almost-exact) parallelism is only true because the Sun’s distance from Earth is much larger than the Earth’s size — which the Greeks knew.] Meanwhile the line from the Earth’s center to the first tower (call it L1) is a continuation of the line from that tower’s base to its top. Because (a) L1 forms an angle 𝜃₁ with the sunlight, as shown in Fig. 5, and (b) the sunlight line and Earth-Sun line are parallel, as shown in Fig. 6, the intersection of L1 with the Earth-Sun line is also 𝜃₁! Similarly, the line from the Earth’s center to tower 2 forms the angle 𝜃₂ with the Earth-Sun line. As can be seen in Fig. 6, it follows that the angle between the two lines connecting the Earth’s center to the two towers is 𝜃₂ – 𝜃₁ , the difference in the noon-time sun angles as seen by the two observers!

Now, however, the observers can use the fact that they know D, the distance between the cities. In particular, the distance D is to the Earth’s circumference C just as 𝜃₂ – 𝜃₁ is to 360 degrees (or, in radians, to 2ℼ). In formulas

• D/C = (𝜃₂ – 𝜃₁)/360°

So (dividing and multiplying on both sides) the Earth’s circumference is simply

• C = D [360° / (𝜃₂ – 𝜃₁) ]

and since they know both D and 𝜃₂ – 𝜃₁ , they now know the Earth’s circumference. (Note this correctly says that if the angle were 90 degrees = 2ℼ/4, then D would be C/4.)

Eratosthenes made this measurement (in a slightly different way) around 240 B.C.E. Reports by classical historians do not quite agree on what he found, but in the most optimistic interpretation of the historical record, he was well within 1 percent of the correct answer. And why not? Once you’ve realized what you should do, this is a relatively simple measurement; it’s one that you and a distant friend could carry out yourselves.

Note: You might prefer to see the answer in radians instead of degrees; since 360° = 2ℼ radians, we can write

• C = D [2ℼ / (𝜃₂ – 𝜃₁) ] (in radians)

and since C = 2ℼR, where R is the Earth’s radius, that gives us a particularly simple formula

• R = D / (𝜃₂ – 𝜃₁) ( in radians)

Note: If the two cities are not due north-south of one another, this poses no problem. Measure tower 1’s shadow at the first city’s noon, and tower 2’s shadow at the second city’s noon on the same day; then take D not to be the distance between the two cities but instead the distance between their latitude lines. Practically speaking, we can make a triangle with one side being the distance between the cities and the other two sides aligned north-south and east-west; then D is the length of the north-south line, as in Fig. 7. With this definition of D, the formulas above are still valid.

Just a few years ago, Australian Catholic University (ACU) established a new Dianoia Institute of Philosophy. They recruited a number of researchers and made something of a splash, leading to a noticeable leap in ACU’s rankings in philosophy — all the way to second among Catholic universities in the English-speaking world, behind only Notre Dame.

Now, without warning, ACU has announced plans to completely disestablish the institute, along with eliminating 35 other academic positions in other fields. This leaves the faculty, some of which left permanent jobs elsewhere to join the new institute, completely stranded.

I sent the letter below to the Vice-Chancellor of ACU and other interested parties. I hope the ongoing international outcry leads the administration to change its mind.