Given a threshold , a -smooth number (or -friable number) is a natural number whose prime factors are all at most . We use to denote the number of -smooth numbers up to . In studying the asymptotic behavior of , it is customary to write as (or as ) for some . For small values of , the behavior is straightforward: for instance if , then all numbers up to are automatically -smooth, so

More generally, for any fixed , de Bruijn showed that

The asymptotic behavior of as is rather complicated. Very roughly speaking, it has inverse factorial behavior; there is a general upper bound , and a crude asymptotic

Lambert function

This asymptotic (2) is quite complicated, and so one does not expect there to be any simple argument that could recover it without extensive computation. However, it turns out that one can use a “maximum entropy” to get a reasonably good heuristic approximation to (2), that at least reveals the role of the mysterious function . The purpose of this blog post is to give this heuristic.

Viewing , the task is to try to count the number of -smooth numbers of magnitude . We will propose a probabilistic model to generate -smooth numbers as follows: for each prime , select the prime with an independent probability for some coefficient , and then multiply all the selected primes together. This will clearly generate a random -smooth number , and by the law of large numbers, the (log-)magnitude of this number should be approximately

The indicator of the event that divides this number is a Bernoulli random variable with mean , so the Shannon entropy of this random variable is

One could solve this constrained optimization problem directly using Lagrange multipliers, but we simplify things a bit by passing to a continuous limit. We take a continuous ansatz , where is a smooth function. Using Mertens’ theorem, the constraint (5) then heuristically becomes

This is a standard calculus of variations problem. The Euler-Lagrange equation for this problem can be easily worked out to be

Nevertheless, this demonstrates that the maximum entropy method can achieve a reasonably good heuristic understanding of smooth numbers. In fact we also gain some insight into the “anatomy of integers” of such numbers: the above analysis suggests that a typical -smooth number will be divisible by a given prime with probability about . Thus, for , the probability of being divisible by is elevated by a factor of about over the baseline probability of an arbitrary (non-smooth) number being divisible by ; so (by Mertens’ theorem) a typical -smooth number is actually largely comprised of something like prime factors all of size about , with the smaller primes contributing a lower order factor. This is in marked contrast with the anatomy of a typical (non-smooth) number , which typically has prime factors in each hyperdyadic scale in , as per Mertens’ theorem.

Hey! It escaped my notice until this week that Boaz Klartag has recently shown the existence of lattice sphere packings in dimension d of density at least c d^2 2^-d. This lower bound was stuck at d 2^-d for a long time, and was only recently improved by a factor of log d, which I learned about from Marcus Michelen‘s job talk (unfortunately for us, as the link will show you, he chose to go to Northwestern!) So it’s very cool that Klartag jumps this up a whole power of d, and by an entirely new method. As in Michelen’s work, though, the approach is probabilistic.

Here’s the idea in a nutshell. Showing the existence of a lattice packing with large density is equivalent to showing the existence of a large ellipsoid which contains no nontrivial point of the standard lattice. (Just take a linear transformation which sends the big ellipsoid to a unit sphere, and which thus has small determinant; apply it to the lattice too, and you have a dense lattice which misses the unit sphere, and Bob’s your uncle.) It turns out you can get such an ellipsoid by almost-aimless wandering! More precisely: start with some ellipsoid whose disjoint from the (nonzero elements of the) lattice, and then carry out some form of Brownian motion in the space of all ellipsoids, subject to one very important rule: every time your ellipse hits a lattice point x, you “freeze” that intersection point, and from now on the ellipsoid wanders in the one-dimension-smaller space of ellipsoids which pass through x. The process thus stops when you’ve hit about (1/2)n^2 points, because that uses up all your degrees of freedom. The game, then, is to show that there’s a slight constant drift towards increasing volume, so that in the length cn^2 lifetime of your process, the ellipsoid acquires c n^2 volume, and that’s exactly where the n^2 in the theorem comes from.

Very cool that such a charming idea actually works!

“Aristotle, so far as I know, was the first man to proclaim explicitly that man is a rational animal. His reason for this view was one which does not now seem very impressive; it was, that some people can do sums.”

Russell was one quotable individual.

It’s well known that you can construct the octonions using triality. One statement of triality is that $Spin(8)$ has nontrivial outer automorphisms of order 3. On the other hand, the octonions have nontrivial inner automorphisms of order 3. My question: can we deduce one of these facts from the other?

The second fact is perhaps not very well known. It may even be hard to understand what it means. Though the octonions are nonassociative, for any nonzero octonion $g$ the map

$$\begin{array}{rccl} f \colon & \mathbb{O} &\to& \mathbb{O} \\ & x & \mapsto & g x g^{-1} \end{array}$$

is well-defined, since $(g x)g^{-1} = g(x g^{-1})$, which one can show using the fact that the octonions are alternative. More surprisingly, whenever $g^6 = 1$, this map $f$ is an automorphism of the octonions:

$$f(x+y) = f(x) + f(y) , \qquad f(x y) = f(x) f(y) \qquad \forall x,y \in \mathbb{O}$$

and $f$ has order 3:

$$f(f(f(x))) = x \qquad \forall x \in \mathbb{O}$$

To understand this latter fact, we can look at

• P. J. C. Lamont, Arithmetics in Cayley’s algebra, Glasgow Mathematical Journal 6 no. 2 (1963), 99–106.

Theorem 2.1 here implies that an octonion $g$ with ${|g|} = 1$ defines an inner automorphism $f \colon x \to g x g^{-1}$ if and only if $x$ has order 6.

However, the result is stated differently there. Paraphrasing somewhat, Lamont’s theorem says that any $g \in \mathbb{O}$ that is not a real multiple of $1 \in \mathbb{O}$ defines an inner automorphism $f \colon x \to g x g^{-1}$ if and only if $g$ obeys

$$4 \mathrm{Re}(g)^2 = {|g|}^2$$

This equation is equivalent to $\operatorname{Re}(g) = \pm \frac{1}{2} {|g|}$, which is equivalent to $g$ lying at either a $60^\circ$ angle or a $120^\circ$ angle from the octonion $1$.

Octonions on the real line clearly define inner automorphisms. Thus, a nonzero octonion $g$ defines an inner automorphism if and only if its angle from $1$ is $0^\circ$, $60^\circ$, $120^\circ$ or $180^\circ$. In this case we can normalize $g$ without changing the inner automorphism it defines, and then we have $g^6 = 1$. Note also that $g$ and $-g$ define the same inner automorphism.

It follows that an octonion $g$ on the unit sphere defines an inner automorphism iff $g^6 = 1$, and that every nontrivial inner automorphism of $\mathbb{O}$ has order 3.

However, if you look at Lamont’s proof, you’ll see the equation $4 \operatorname{Re}(g)^2 = {|g|}^2$ plays no direct role! Instead, he really uses the assumption that $g^3$ is a real multiple of $1$, which is implied by this equation (as easily shown using what we’ve just seen).

From Lamont’s work, one can see the Moufang identities and the characteristic equation for octonions are what force all inner automorphisms of the octonions to have order 3.

Thus, an argument giving a positive answer to my question might involve a link between triality and the Moufang identities. Conway and Smith seem to link them in On Quaternions and Octonions. But I haven’t figured out how to get from the outer automorphisms of $\text{Spin}(8)$ to the inner automorphisms of $\mathbb{O}$, or vice versa!

I asked about this on MathOverflow, but I thought some people here would also be interested.

Guest post by Khyathi Komalan and Andrew Krenz

From Lawvere’s Hegelian taco to Baez’s layer cake analogy to Eugenia Cheng’s How to Bake Pi, categorists have cultivated a rich tradition of culinary metaphors and similes. A well-known example in the world of computation is Mark Dominus’s “monads are like burritos” — where a tortilla (computational context) wraps diverse ingredients (values) to create a cohesive entity (effectful value) whose burrito structure is maintained as the meal moves down the assembly line (undergoes computations).

Monads, like burritos, come in many different varieties. In computer science monads serve to streamline computational patterns such as exception handling and context management. We illustrate these two examples by analogy.

Imagine you work at a burrito truck.

If a customer orders a burrito sans rice but rice is accidentally added, it can’t be served. The Maybe monad handles exceptions such as this — when something goes wrong, it returns a special “Nothing” value rather than a flawed result, and once a failure occurs, all subsequent steps automatically preserve this state avoiding the need for repetitive error-checking.

Figure 1: The Maybe Monad illustrated with the burrito-making process

In Haskell, the parameterized type “Maybe a” has two constructors, “Just a” and “Nothing.” The former is an alias for values of type “a” whereas the latter is indicative of an error. The following Haskell code exhibits the maybe type as an instance of the monad class:

instance Maybe Monad where
return = Just
Nothing >>= f = Nothing
(Just x) >>= f = f x

the return function has type a -> Maybe a, which is suggestive of its role as the monad unit. The so-called bind operation >>= has type Maybe a -> (a -> Maybe b) -> Maybe b, and corresponds to a bare-bones Kleisli composition (see Monads: Programmer’s Definition for details).

A slight generalization allows for descriptive error messages.

Definition. Given a collection of exceptions $E$, there is an associated Either monad $((-)+E, \eta, \mu)$.

• $\eta_X:X \to X + E$ is the coproduct insertion
• $\mu_X:X + E + E \to X + E$ collapses two copies of $E$ into one
• Kleisli morphisms are computations that may fail $X \to Y + E$
• Kleisli composition automatically propagates exceptions

Of course, either monads are simply maybe monads with a set in place of the constant/singleton “Nothing” and they allow us not only to say that an error has occured, but also to indicate what that error was.

Now suppose one of your regular customers walks up to the window and orders “the usual.” Luckily you’ve recorded their preferences in a recipe book. The act of following the appropriate recipe is akin to executing computations that depend on a global read-only state. The * Reader monad * is the functional programmer’s way of incorporating this impure concept in pure functional terms.

Figure 2: The Reader Monad illustrated with the burrito-making process

Definition. Given a collection of environments $E$, there is an associated Reader monad $((-)^E, \eta, \mu)$.

• $\eta_X : X \to X^E$ turns elements into constant functions $x \mapsto \lambda e. x$
• $\mu_X : (X^E)^E \to X^E$ turns function-valued functions into functions via diagonal evaluation $f \mapsto \lambda e. f(e)(e)$
• Kleisli morphisms convert inputs into executable functions from environments to outputs $X \to Y^E$
• Composition in the Kleisli category keeps track of the (immutable) environment as computations are chained together.

Here is the same definition given as an instance of the Haskell monad class:

instance Monad ((->) r) where
return x = \_ -> x
g >>= f = \e -> f (g e) e

The seminal paper of Moggi has several other interesting examples illustrating the power of monads. Nevertheless, monads may not always suffice for all of our needs. For example, what would happen if our burrito truck suddenly exploded in popularity requiring automation of repetative processes and parallel work stations?

This is where “Arrows” enter the picture. Introduced by John Hughes in 2000, Arrows generalize strong monads. Because of this, Arrows handle more complicated computational patterns in a natural way. While monads wrap values in computational contexts (like burritos in tortillas), Arrows can represent entire preparation processes capable of coordinating multiple inputs while maintaining awareness of the broader kitchen environment.

Arrows come with three core operations that determine their behaviour; looking at their types, we see that Arrows are evocative of a lax internal hom that interacts with binary products.

class Arrow a where
arr :: (x -> y) -> a x y
(>>>) :: a x y -> a y z -> a x z
first :: a x y -> a (x,z) (y,z)

1. arr turns functions into “Arrows.” This is like incorporating a standard burrito recipe or preparation step into the food truck’s workflow — taking a simple instruction like “add beans, then cheese” and automating it within our kitchen’s setup.
2. >>> composes composable Arrows. This allows for separately automated processes to be seamlessly strung together.
3. first enacts an automated process on one burrito while simultaneously passing a second burrito through the station.

These data are subject to 9 axioms, which we eventually discuss below.

Figure 3: Arrow Operations. The three fundamental operations of Arrows enable complex workflows beyond monadic structures.

Shortly before Arrows were introduced, Power, Robinson, and Thielecke were working on Freyd categories — a categorical structure designed to model “effectful” computation. Using our simile, a Freyd category formalizes the relationship between an ideal burrito recipe (pure theory) and the real-world process of making that burrito in a particular kitchen.

A Freyd category consists of three main components:

1. A category $C$ with finite products which can be thought of as the syntax of our kitchen. In other words, $C$ is like a recipe book containing the abstract information one needs to interpret and implement in the context of an actual kitchen.
2. A symmetric premonoidal category $K$ which plays the more semantic role of our real world kitchen.

3. An identity-on-objects functor $J:C \to K$ which faithfully translates pure recipes into physical processes that work within the specific setup of the kitchen $K$.

   
   Figure 4: Freyd Category Structure. The relationship between pure recipes (category C) and real-world kitchen operations (category K), connected by the identity-on-objects functor J that preserves structure while accommodating practical constraints.

Although Arrows originated in Haskell, a highly abstract functional programming language, researchers began noticing apparent correspondences between the components of Arrows and those of Freyd categories. These two structures, developed from different starting points, seemed to address the same fundamental challenge: how to systematically manage computations that involve effects, multiple inputs and outputs, and context-awareness. Therefore, it was hypothesized that Arrows are equivalent to Freyd categories.

As a part of the Adjoint School, our group has been focusing on R. Atkey’s work, which dispells this folklore and precisely formulates the relationship between Arrows and Freyd categories. Just as Atkey asks in the title of his paper, this blog post will investigate the question of “what is a categorical model of Arrows?” The answer not only clarifies the theoretical underpinnings of these structures, but also reveals practical insights for programming language design and quantum computation models. Ultimately, we will see that there are indeed subtle differences between Arrows and Freyd categories.

Key Insights: - Monads encapsulate computational effects by wrapping values in contexts, much like burritos wrap ingredients in tortillas - Different monads (Maybe, Reader, etc…) deal with different patterns like exception handling and context management - Arrows generalize monads to handle multiple inputs and coordinate complex processes, like managing an entire kitchen rather than just making individual burritos

Beyond the Kitchen: Arrows and Freyd Categories

Formally, a monad on a category $C$ is a monoid in the category of endofunctors of $C$. Arrows, like monads, are monoids in a certain category of functors. To be more specific, the structure of an Arrow on a category $C$ can be described as a monoid in the category of strong profunctors on $C$. Let’s take a closer look at this construction.

Arrows A profunctor $P$ on a category $C$ is a functor $P: C^{\text{op}}\times C \to \text{Set}.$ Intuitively, a profunctor associates to each pair of objects a set of “generalized morphisms” between those objects.

The identity profunctor is simply $\text{id}(x, y) := C(x, y)$, which uses the hom-sets of $C$.

Composition of profunctors is defined as a coend. Given profunctors $P$ and $Q$, their composition is the following profunctor:

$$(P * Q)(x, z) = \int^y P(x, y) \times Q(y, z)$$

Notice that this formula is vaguely reminiscent of a dot product; replacing the integral with a sum over $y$, and the cartesian product with multiplication, it looks like the dot product of the row vector $P(x,-)$ with the column vector $Q(-,z)$.

operations. –> We will now unpack this data to reach a more down-to-earth description of Arrows. This resulting characterization aligns more closely with the way in which Arrows are implemented in programming languages like Haskell.

Definition. An Arrow in a cartesian closed category $C$ consists of a mapping on objects and three families of morphisms:

• A mapping on objects $\text{Ar} : \text{ob}(C) \times \text{ob}(C) \to \text{ob}(C)$

This defines the Arrow type constructor, which takes input and output types and produces an Arrow type between them.

• A family of morphisms $\text{arr} : Y^X \to \text{Ar}(X, Y)$

This operation lifts a pure function into the Arrow context, allowing regular functions to be treated as Arrows.

• A family of morphisms $\ggg : \text{Ar}(X, Y) \times \text{Ar}(Y, Z) \to \text{Ar}(X, Z)$

  This enables sequential composition of Arrows, similar to function composition but now in terms of Arrows.

• A family of morphisms $\text{first} : Y^X \to \text{Ar}(X \times W, Y \times W)$

This is perhaps the most distinctive operation. Intuitively, it allows an Arrow to process the first component of a pair while leaving the second component unchanged.

These data are subject to nine axioms which govern their interactions. To make these abstract operations more concrete, consider the following example, where $\text{Ar}(x, y) := Y^X,$ $\text{arr} := \text{id}_{(Y^X)},$ $\ggg := \text{composition},$ and $\text{first}(f) := f \times \text{id}.$ In what follows we list the Arrow laws and draw commutative diagrams based on this example.

The Arrow laws $\text{arr}(\text{id})\ggg a=a$ and $a\ggg \text{arr}(\text{id})=a$ express left and right unitality of identities under composition.

Figure 5: Arrow Laws

The Arrow law $(a \ggg b)\ggg c=a \ggg (b \ggg c),$ represents associativity of composition.

Figure 6: Arrow Laws

The Arrow law $\text{first}(a\ggg b)=\text{first}(a)\ggg \text{first}(b)$ encodes functoriality of $- \times W: C \to C$.

Figure 7: Arrow Laws

The Arrow law $\text{first}(a)\ggg \text{arr}(\pi_{1})=\text{arr}(\pi_{1})\ggg a$ express naturality of the counit $- \times W \to \text{id}_{C}$, i.e., the first projection maps.

Figure 8: Arrow Laws

The Arrow law $\text{first}(a)\ggg \text{arr}(\alpha)=\text{arr}(\alpha)\ggg \text{first}(\text{first}(a))$ asks that $\text{first}$ play nicely with associators.

Figure 9: Arrow Laws

The Arrow law $\text{first}(a)\ggg \text{arr}(\text{id} \times f)=\text{arr}(\text{id} \times f)\ggg \text{first}(a)$ is an interchange law which says $\text{id} \times g:(- \times W) \to (- \times W')$ is a natural transformation for every $g:W \to W'$ in $C$.

Figure 10: Arrow Laws

Two Arrow laws trivialise as a result of our example, so diagrams aren’t produced. The first such law is $\text{arr}(f;g)=\text{arr}(f)\ggg \text{arr}(g).$ For our example, this law trivialises, as $\ggg : = \text{composition}$ and $\text{arr} := \text{id}_{(Y^X)}.$ The second law to trivialise is $\text{first}(\text{arr}(f))=\text{arr}(f \times \text{id})$ since we have set $\text{first}(f) := f \times \text{id}.$

Freyd Categories

To understand Freyd categories, we must first define what a symmetric premonoidal category is.

Definition. A symmetric premonoidal category includes:

• An object $I$ (unit).
• Natural transformations that define how objects interact:
  • Associativity: $\alpha : (x \otimes y) \otimes z \to x \otimes (y \otimes z)$
  • Left unitor: $\lambda : x \otimes I \to x$
  • Right unitor: $\rho : I \otimes x \to x$
  • Symmetry: $\sigma : x \otimes y \to y \otimes x$
• All components are central .

A morphism $f : x \to x'$ is central if $\forall g:y \to y', \quad f \otimes y ; x' \otimes g = x \otimes g ; f \otimes y'$

Now, we can define a Freyd category, recalling the definition from the introduction.

Definition. A Freyd category consists of:

• A category $C$ with finite products.
• A symmetric premonoidal category $K$.
• An identity-on-objects functor $J : C \to K$ that:
  • Preserves symmetric premonoidal structure.
  • Ensures $J(f)$ is always central.

Arrows vs Freyd Categories: Similarities and Differences

At first glance, the definition of a Freyd category appears strikingly similar to that of an Arrow. This apparent similarity led to the folklore belief that they were equivalent structures.

A Freyd category consists of two categories $C$ and $K$ with an identity-on-objects functor $J: C \to K$, where: - $C$ has finite products - $K$ is symmetric premonoidal (with a functor $− \otimes z$) - $J$ maps finite products in $C$ to the premonoidal structure in $K$

In our culinary metaphor, this loosely translates to: - $C$: The idealized recipes (Haskell types and functions) - $K$: The real-world kitchen operations (computations represented by the Arrow type $\text{Ar}(x,y)$) - $J$: The translation process (via arr, embedding pure functions) - Composition in $K$: The sequencing of operations (via >>>) - Premonoidal structure in $K$: The ability to process pairs (via first)

Recalling the how we’ve interpreted Arrows in the cullinary setting, the apparent correspondence between Arrows and Fryed categories seemes quite natural. In fact, for many years the two concepts were thought to be two ways of speaking about the same thing among those in the programming languages community.

However, Atkey’s work revealed a crucial distinction: Arrows are more general than Freyd categories . The key difference lies in how they handle inputs:

• Freyd categories allow only a single input to computations
• Arrows support two separate inputs:
  • One may be structured (modeled using comonads)
  • This additional flexibility allows Arrows to represent computations that Freyd categories cannot

To bridge this gap, Atkey introduced the concept of indexed Freyd categories , which can model two structured inputs. The relationship can be summarized as: Arrows are equivalent to Closed Indexed Freyd Categories.

In our culinary metaphor, we can understand this relationship as follows: a Freyd category is like a restaurant that can only take one order at a time (a single input), while Arrows are like a more sophisticated establishment that can handle both individual orders and special requests that come with their own context (two inputs, one potentially structured). The closed indexed Freyd categories that Atkey identifies represent the perfect middle ground — restaurants that can efficiently manage multiple orders with specialized instructions while maintaining the core operational principles that make kitchens function. This is particularly valuable when preparing complex “quantum dishes” where ingredients might be entangled and interact with each other in non-local ways.

Figure 11: Arrows vs. Freyd Categories. Arrows support two inputs (one potentially structured) and are equivalent to Closed Indexed Freyd Categories, which generalize standard Freyd Categories that handle only single inputs.

This distinction helps explain why Arrows have proven particularly useful in domains like quantum computing, where managing multiple inputs with complex relationships is essential.

R. Atkey’s paper finds the relationship between Arrows and different constraints on Freyd categories as follows:

Figure 12: Relationship Between Structures

Key Insights: - Arrows can be defined both as monoids in categories of strong profunctors and operationally through concrete morphisms ($\text{arr}$, $\ggg$, $\text{first}$) - Freyd categories formalize the relationship between pure functions and effectful computations using symmetric premonoidal structure - Despite the folklore belief, Arrows are strictly more general than Freyd categories because they can handle two separate inputs (one potentially structured) - Arrows are equivalent to closed indexed Freyd categories, bridging the conceptual gap

Applications and Questions The main goal of our Adjoint School project was to structure effects in quantum programming languages using generalizations of monads. Relative monads are a popular generalization of monads. These monads need not be endofunctors, and they’re known to generalize Arrows as well. Since we already know how to structure quantum effects using Arrows, it follows that it should be theoretically possible to structure quantum effects using relative monads.

Arrows’ capacity to handle multiple inputs with a single, potentially structured output offers tractability that is particularly useful in quantum computing. Particles in quantum systems can be in entangled states, where the manipulation of one particle influences others in real time, irrespective of distance. This non-local interaction can be modeled through Arrows’ ability to combine several inputs while keeping track of their interrelationships.

Our group investigated the possibility of doing exactly this. The main technical issue arises from the fact that the way Arrows have been implemented in Haskell to structure quantum effects does not provide a categorical semantics for the problem.

For our ACT2025 presentation, we were able to construct a relative monad capable of handling classical control in the quantum setting, but the following questions still remain:

• Can one build a relative monad to model quantum effects?

• If so, how might an implementation of these ideas in Haskell compare to Arrow-based approaches?

The ride from burrito monads to Arrow kitchens has carried us farther than we anticipated, illustrating that even established mathematical folklore sometimes requires precise re-evaluation. As we continue to learn about these structures, we hope this post will motivate others to participate in the exploration of these tools and their use in quantum computing and beyond.

Academics tend to face a lot of impostor syndrome. Something about a job with no clear criteria for success, where you could always in principle do better and you mostly only see the cleaned-up, idealized version of others’ work, is a recipe for driving people utterly insane with fear.

The way most of us talk about that fear, it can seem like a cognitive bias, like a failure of epistemology. “Competent people think they’re less competent than they are,” the less-discussed half of the Dunning-Kruger effect.

(I’ve talked about it that way before. And, in an impostor-syndrome-inducing turn of events, I got quoted in a news piece in Nature about it.)

There’s something missing in that perspective, though. It doesn’t really get across how impostor syndrome feels. There’s something very raw about it, something that feels much more personal and urgent than an ordinary biased self-assessment.

To get at the core of it, let me ask a question: what happens to impostors?

The simple answer, the part everyone will admit to, is to say they stop getting grants, or stop getting jobs. Someone figures out they can’t do what they claim, and stops choosing them to receive limited resources. Pretty much anyone with impostor syndrome will say that they fear this: the moment that they reach too far, and the world decides they aren’t worth the money after all.

In practice, it’s not even clear that that happens. You might have people in your field who are actually thought of as impostors, on some level. People who get snarked about behind their back, people where everyone rolls their eyes when they ask a question at a conference and the question just never ends. People who are thought of as shiny storytellers without substance, who spin a tale for journalists but aren’t accomplishing anything of note. Those people…aren’t facing consequences at all, really! They keep getting the grants, they keep finding the jobs, and the ranks of people leaving for industry are instead mostly filled with those you respect.

Instead, I think what we fear when we feel impostor syndrome isn’t the obvious consequence, or even the real consequence, but something more primal. Primatologists and psychologists talk about our social brain, and the role of ostracism. They talk about baboons who piss off the alpha and get beat up and cast out of the group, how a social animal on their own risks starvation and becomes easy prey for bigger predators.

I think when we wake up in a cold sweat remembering how we had no idea what that talk was about, and were too afraid to ask, it’s a fear on that level that’s echoing around in our heads. That the grinding jags of adrenaline, the run-away-and-hide feeling of never being good enough, the desperate unsteadiness of trying to sound competent when you’re sure that you’re not and will get discovered at any moment…that’s not based on any realistic fears about what would happen if you got caught. That’s your monkey-brain, telling you a story drilled down deep by evolution.

Does that help? I’m not sure. If you manage to tell your inner monkey that it won’t get eaten by a lion if its friends stop liking it, let me know!

I’m thrilled that our paper entitled Demonstrating an unconditional separation between quantum and classical information resources, based on a collaboration between UT Austin and Quantinuum, is finally up on the arXiv. I’m equally thrilled that my coauthor and former PhD student William Kretschmer — who led the theory for this project, and even wrote much of the code — is now my faculty colleague at UT Austin! My physics colleague Nick Hunter-Jones and my current PhD student Sabee Grewal made important contributions as well. I’d especially like to thank the team at Quantinuum for recognizing a unique opportunity to test and showcase their cutting-edge hardware, and collaborating with us wild-eyed theorists to make it happen. This is something that, crucially, would not have been feasible with the quantum computing hardware of only a couple years ago.

Here’s our abstract, which I think explains what we did clearly enough, although do read the paper for more:

A longstanding goal in quantum information science is to demonstrate quantum computations that cannot be feasibly reproduced on a classical computer. Such demonstrations mark major milestones: they showcase fine control over quantum systems and are prerequisites for useful quantum computation. To date, quantum advantage has been demonstrated, for example, through violations of Bell inequalities and sampling-based quantum supremacy experiments. However, both forms of advantage come with important caveats: Bell tests are not computationally difficult tasks, and the classical hardness of sampling experiments relies on unproven complexity-theoretic assumptions. Here we demonstrate an unconditional quantum advantage in information resources required for a computational task, realized on Quantinuum’s H1-1 trapped-ion quantum computer operating at a median two-qubit partial-entangler fidelity of 99.941(7)%. We construct a task for which the most space-efficient classical algorithm provably requires between 62 and 382 bits of memory, and solve it using only 12 qubits. Our result provides the most direct evidence yet that currently existing quantum processors can generate and manipulate entangled states of sufficient complexity to access the exponentiality of Hilbert space. This form of quantum advantage — which we call quantum information supremacy — represents a new benchmark in quantum computing, one that does not rely on unproven conjectures.

I’m very happy to field questions about this paper in the comments section.

Unrelated Announcement: As some of you might have seen, yesterday’s Wall Street Journal carried a piece by Dan Kagan-Kans on “The Rise of ‘Conspiracy Physics.'” I talked to the author for the piece, and he quoted this blog in the following passage:

This resentment of scientific authority figures is the major attraction of what might be called “conspiracy physics.” Most fringe theories are too arcane for listeners to understand, but anyone can grasp the idea that academic physics is just one more corrupt and self-serving establishment. The German physicist Sabine Hossenfelder has attracted 1.72 million YouTube subscribers in part by attacking her colleagues: “Your problem is that you’re lying to the people who pay you,” she declared. “Your problem is that you’re cowards without a shred of scientific integrity.”

In this corner of the internet, the scientist Scott Aaronson has written, “Anyone perceived as the ‘mainstream establishment’ faces a near-insurmountable burden of proof, while anyone perceived as ‘renegade’ wins by default if they identify any hole whatsoever in mainstream understanding.”

 A few more interesting tidbits from the concluding half-day of the DOE ECMP PI meeting:

• Dmitri Basov showed some of the remarkable experiments enabled by layers of MoOCl2, which in the IR is an intrinsically hyperbolic optical material.  This material has unusual plasmonic properties considering its high resistivity.  These include peculiar cavity effects such as modifying superfluid density of a proximally coupled superconductor.
• Leonid Butov explained some remarkable evidence for superfluidity of indirect excitons excited in the moire bilayer of MoSe2/WSe2.  Low temperature mean free paths of these objects can exceed hundreds of microns (!).
• Cui-Zu Chang showed evidence that truly stoichiometric FeTe is actually a superconductor with a critical temperature of about 13.5 K, rather than the usual thinking that it is an antiferromagnetic metal. Apparently an extra 2% of interstitial iron is enough to kill superconductivity and induce AFM order.
• James McIver presented an example of how nonlinear optical effects in an optically driven (Floquet) Weyl semimetal seem to vary linearly with driving field - anomalously strong.
• Dmytro Bozhko showed a really neat technique, using Brillouin light scattering to map out the dispersion of phonons and magnons in YIG, and to extend this approach with a special hollow-core optical fiber to low temperatures with the motivation of probing magnon superfluidity in a particular antiferromagnetic insulator.
• Ray Ashoori used his characteristically pretty quantum capacitance measurement technique to examine the density+displacement field+magnetic field phase diagram of 5-layer rhombohedral graphene, revealing some surprising fractional Chern insulator states.
• Claudia Ojeda-Aristizabal discussed some mesoscopic transport measurements in bilayer graphene, where an adsorbed layer of spin-containing CuPc molecules seems to affect both decoherence and the trigonal warping contribution to it (related to intervalley scattering). 
• Feng Wang and You Zhou both discussed recent measurements looking at Wigner crystals and their properties in 2D TMDs, through a variety of means.
• Liuyan Zhao showed some very rich physics obtained in studies that moiré stack bilayers of the van der Waals insulating magnet CrI3.  

How, exactly, can category theory help modeling in public health? I wrote a paper about this with two people who helped run Canada’s COVID modeling, together with a software engineer and a mathematician at the Topos Institute:

• John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Eric Redekopp, A categorical framework for modeling with stock and flow diagrams, in Mathematics of Public Health: Mathematical Modelling from the Next Generation, eds. Jummy David and Jianhong Wu, Springer, 2023, pp. 175-207.

Anything you can with category theory, you can also do without it—just like you can cross the Alps without shoes. But categorical methods make public health modeling easier in a lot of ways.

The introduction lists a few of these ways. Then the paper goes on to provide details, including a long appendix showing actual Julia code for our software.

But here’s the basic idea:

Many people working on epidemiological modeling like diagrams because they provide easily understandable but informal steps towards a mathematically rigorous formulation of a model in terms of ordinary differential equations (ODEs). But ODEs are typically opaque to non-modelers—including the interdisciplinary members of the teams that typically are required for impactful models.

The tradition of modeling called System Dynamics places a premium on engagement with stakeholders, so it offers a modeling approach centered around diagrams. This approach commonly proceeds in a manner that depicts model structure using successively more detailed models.

The process starts with a ‘causal loop diagram’ illustrating causal connections and feedback loops:

It then often proceeds to a ‘system structure diagram’, which distinguishes stocks from flows but still lacks quantitative information. The next step is to construct a stock and flow diagram’:

This diagram is visually identical to the system structure diagram, but it also includes formulae (at the bottom here).

The stock–flow diagram is treated as the durable end result of this modeling process, since it uniquely specifies a system of first-order ODEs. System Dynamics modeling typically then alternates between assessing scenario outcomes resulting from numerically integrating the ODEs, performing other analyses (e.g., identifying location or stability of equilibria), and elaborating the stock-flow diagram.

While each of the 3 types of diagrams in the System Dynamics tradition is recommended by visual accessibility, the traditional approach suffers from a number of practical shortcomings:

• Monolithic models: Models are traditionally built up in a monolithic fashion, leading ultimately to a single large piece of code. Drawn as a single diagram, a model can be extremely complex. For example, here is Canada’s main model of COVID during the pandemic, put together by Nathaniel Osgood and Xiaoyan Li, made using the commercially available software called AnyLogic:

Click to enlarge. If it looks like a huge mess, that’s part of the point.

Working with a single huge model like this inhibits independent simultaneous work by multiple modelers. Lack of model modularity further prevents effective reuse of particular model elements. If elements of other models are used, they are commonly copy-and-pasted into the developing model, with the source and destination then evolving independently. Such separation can lead to a proliferation of conceptually overlapping models in which a single conceptual change requires corresponding updates in several successive models.

• The curse of dimensionality: Modelers refine simple models by ‘stratifying’ them, subdividing stocks into smaller stocks. For example, the ‘infected’ stock might be stratified into ‘infected male’ and ‘infected female’. While stratification is a key tool for representing heterogeneity, stratification commonly requires modifications across the breadth of a model—stocks, flows, derived quantities, and many parameters. When stratification involves multiple dimensions of heterogeneity, it can lead to a proliferation of terms in the ODEs. For example, rendering a model characterizing both COVID-19 into a model also characterizes influenza would require that each COVID-19 state to be replicated for each stage in the natural history of influenza. Represented visually, this stratification leads to a multi-dimensional lattice, commonly with progression proceeding along several dimensions of the lattice. Because of the unwieldy character of the diagram, the structure of the model is obscured. Adding, removing, or otherwise changing dimensions of heterogeneity routinely leads to pervasive changes across the model.

• Privileging ODE semantics: The structure of causal loop diagrams, system structure diagrams and stock-flow diagrams characterizes general state and accumulations, transitions and posited causal relations—including induced feedbacks—amongst variables. Nothing about such a characterization restricts its meaning to ordinary differential equations; indeed, many other interpretations and uses of these diagrams are possible. However, existing software privileges an ODE interpretation for stock-flow diagrams, while sometimes allowing for secondary analyses in ad hoc way—for example, identifying causal loops associated with the model, or verifying dimensional homogeneity in dimensionally annotated models. Conducting other sorts of analyses—such as computation of eigenvalue elasticities or loop gains, analysis as a stochastic transition system, or other methods such as particle filtering, particle Markov chain Monte Carlo, or Kalman filtering—typically requires bespoke software for reading, representing and analyzing stock-flow models.

• Divergence of model representations: Although the evolution from causal loop diagrams to system structure diagrams to stock-flow models is one of successive elaboration and informational enrichment, existing representations treat these as entirely separate characterizations and fail to capture the logical relationships between them. Such fragmentation commonly induces inconsistent evolution. Indeed, in many projects, the evolution of stock-flow diagrams renders the earlier, more abstract formulations obsolete, and the focus henceforth rests on the stock-flow diagrams.

What is less widely appreciated is that beyond their visual transparency and capacity to be lent a clear ODE semantics, the 3 kinds of diagrams I mentioned each have a precise mathematical structure—a corresponding grammar, as it were. This algebraic structure, called the ‘syntax’ of tehse diagrams, can be characterized using category theory. Formalizing the syntax this way lends precise meaning to the process of ‘composing’ models (building them out of smaller parts), stratifying them, and other operations. Explicitly characterizing the syntax in software also allows for diagrams to be represented, manipulated, composed, transformed, and flexibly analyzed in software that implements the underlying mathematics.

Formalizing the mathematics of diagram-based models using category theory and capturing it in software offers many benefits. Our paper discusses and demonstrates just a few:

• Separation of syntax and semantics. Category theory gives tools to separate the formal structure, or ‘syntax’, of diagram-based models from the uses to which they are put, or ‘semantics’. The syntax lives in one category, which can then be mapped to various different semantic categories using various functors. This separation permits great flexibility in applying different semantics to the same model. With appropriate software design, this decoupling can allow the same software to support a diverse array of analyses, which can be supplemented over time.

• Reuse of structure. Category theory provides a structured way to build complex diagrams by composing small reusable pieces. Diagrams are morphisms in a monoidal category, and you build bigger diagrams by composing and tensoring these morphisms. With software support, modeling frameworks can allow for saving models and retrieving them for reuse as parts of many different models. For example, in public health a diagram representing contact tracing can be reused across diagrams addressing different pathogens.

• Modular stratification. A categorical foundation further supports a structured method to build stratified diagrams out of modular, reusable, largely orthogonal pieces. This method is called taking a ‘pullback’. In contrast to the global changes commonly required to a diagram and the curse of dimensionality that traditionally arises when stratifying a diagram, categorically-founded stratification methods allow for crisply characterizing a stratified diagram as built from simpler diagrams, one for each heterogeneity or progression dimension.

Our paper goes into detail about how all this works. Elsewhere we have longer lists of what’s bad about current modeling practice, and how we hope to improve it. But I hope this helps a bit.

It was another very full day.   I had to pop in and out to attend to some things so I didn't get everything, but here are some physics items I learned:

• Dillon Fong introduced me to a technique I didn't know about before, x-ray photon correlation spectroscopy (see this paper).  You can look at time correlations of x-ray speckle near a particular Bragg spot and learn about dynamics and kinetics of transitions and materials growth.  Very cute.
• Charles Ahn presented work on high magnetic field superconductivity in Nd(1-x)Eu(x)NiO2, and I learned about the Jaccarino-Peter effect, in which an external magnetic field can counter the interaction between magnetic dopants and the conduction electrons.  This leads to "reentrant" superconductivity at high magnetic fields. 
• Danny Phelan showed that you can have two different crystal structures for La3Ni2O7, one that is stacked bilayers ("2222"), and one that is stacked monolayer/trilayer ("1313").  
• Ian Fisher talked about using the elastocaloric effect (rapidly and therefore adiabatically stretch or compress a material, leading to a change in its temperature) to identify phase transitions, since the effect is proportional to \( (\partial S/\partial \epsilon)_{T}\), the change in entropy with strain.
• Dan Dessau presented an interesting analysis of data in cuprates suggesting a form for the electronic self-energy that is called a power law liquid, and that this analysis implies that there is not a quantum critical point under the middle of the superconducting dome.
• Jak Chakhalian showed that epitaxially growing an iridate Weyl semimetal directly on top of insulating Dy2Ti2O7 spin ice leads to a dramatic anisotropic magnetoresistance at high in-plane fields that identifies interesting previously unknown physics.
• Daniel Rhodes showed some pretty work on superconductivity in T_d-MoTe2.  This material is extremely air-sensitive, and all of the device fabrication has to be done with great care in a glovebox.  This led to the following exchange.  Audience question: "It is notoriously difficult to make electrical contact to this material.  How did you do this?"  Answer: "Through tears and blood."  This was followed by a serious answer that concluded "The glovebox is always the problem."

Sunflowers are blooming, stores are trumpeting back-to-school sales, and professors are scrambling to chart out the courses they planned to develop in July. If you’re applying for an academic job this fall, now is the time to get your application ducks in a row. Seeking a postdoctoral or faculty position? Your applications will center on research statements. Often, a research statement describes your accomplishments and sketches your research plans. What do evaluators look for in such documents? Here’s my advice, which targets postdoctoral fellowships and faculty positions, especially for theoretical physicists.

• Keep your audience in mind. Will a quantum information theorist, a quantum scientist, a general physicist, a general scientist, or a general academic evaluate your statement? What do they care about? What technical language do and don’t they understand?

• What thread unites all the projects you’ve undertaken? Don’t walk through your research history chronologically, stepping from project to project. Cast the key projects in the form of a story—a research program. What vision underlies the program?

• Here’s what I want to see when I read a description of a completed project.
  • The motivation for the project: This point ensures that the reader will care enough to read the rest of the description.
  • Crucial background information
  • The physical setup
  • A statement of the problem
  • Why the problem is difficult or, if relevant, how long the problem has remained open
  • Which mathematical toolkit you used to solve the problem or which conceptual insight unlocked the solution
  • Which technical or conceptual contribution you provided
  • Whom you collaborated with: Wide collaboration can signal a researcher’s maturity. If you collaborated with researchers at other institutions, name the institutions and, if relevant, their home countries. If you led the project, tell me that, too. If you collaborated with a well-known researcher, mentioning their name might help the reader situate your work within the research landscape they know. But avoid name-dropping, which lacks such a pedagogical purpose and which can come across as crude.
  • Your result’s significance/upshot/applications/impact: Has a lab based an experiment on your theoretical proposal? Does your simulation method outperform its competitors by X% in runtime? Has your mathematical toolkit found applications in three subfields of quantum physics? Consider mentioning whether a competitive conference or journal has accepted your results: QIP, STOC, Physical Review Letters, Nature Physics, etc. But such references shouldn’t serve as a crutch in conveying your results’ significance. You’ll impress me most by dazzling me with your physics; name-dropping venues instead can convey arrogance.

• Not all past projects deserve the same amount of space. Tell a cohesive story. For example, you might detail one project, then synopsize two follow-up projects in two sentences.

• A research statement must be high-level, because you don’t have space to provide details. Use mostly prose; and communicate intuition, including with simple examples. But sprinkle in math, such as notation that encapsulates a phrase in one concise symbol.
  
  
• Be concrete, and illustrate with examples. Many physicists—especially theorists—lean toward general, abstract statements. The more general a statement is, we reason, the more systems it describes, so the more powerful it is. But humans can’t visualize and intuit about abstractions easily. Imagine a reader who has four minutes to digest your research statement before proceeding to the next 50 applications. As that reader flys through your writing, vague statements won’t leave much of an impression. So draw, in words, a picture that readers can visualize. For instance, don’t describe only systems, subsystems, and control; invoke atoms, cavities, and lasers. After hooking your reader with an image, you can generalize from it.

• A research statement not only describes past projects, but also sketches research plans. Since research covers terra incognita, though, plans might sound impossible. How can you predict the unknown—especially the next five years of the unknown (as required if you’re applying for a faculty position), especially if you’re a theorist? Show that you’ve developed a map and a compass. Sketch the large-scale steps that you anticipate taking. Which mathematical toolkits will you leverage? What major challenge do you anticipate, and how do you hope to overcome it? Let me know if you’ve undertaken preliminary studies. Do numerical experiments support a theorem you conjecture?

• When I was applying for faculty positions, a mentor told me the following: many a faculty member can identify a result (or constellation of results) that secured them an offer, as well as a result that earned them tenure. Help faculty-hiring committees identify the offer result and the tenure result.

• Introduce notation before using it. If you use notation and introduce it afterward, the reader will encounter the notation; stop to puzzle over it; tentatively continue; read the introduction of the notation; return to the earlier use of the notation, to understand it; and then continue forward, including by rereading the introduction of the notation. This back-and-forth breaks up the reading process, which should flow smoothly.

• Begin every paragraph with a topic sentence. 

• Avoid verbs that fail to relate that you accomplished anything: “studied,” “investigated,” “worked on,” etc. What did you prove, show, demonstrate, solve, calculate, compute, etc.?
  
  
• Tailor a version of your research statement to every position. Is Fellowship Committee X seeking biophysicists, statistical physicists, mathematical physicists, or interdisciplinary scientists? Also, respect every application’s guidelines about length.

• If you have room, end the statement with a recap and a statement of significance. Yes, you’ll be repeating ideas mentioned earlier. But your reader’s takeaway hinges on the last text they read. End on a strong note, presenting a coherent vision.
  
  
• Writing is rewriting, a saying goes. Draft your research statement early, solicit feedback from a couple of mentors, edit the draft, and solicit more feedback.

That was a full day.  Here are some things I learned, beyond the fact that the ballroom here is clearly kept at about 15°C by default.  (Apologies for not getting everything....)

• About 40% of the DOE ECMP program is related to 2D materials these days.
• Long Ju showed some interesting work trying to understand rhombohedral (ABC-stacked) 5-layer graphene encapsulated by hBN.  Trying to get rid of moiré effects from the hBN/graphene interfaces leads not to more robust quantum anomalous Hall response, but instead leads to very peculiar superconductivity that survives up to very large in-plane and moderately large out-of-plane magnetic fields. This happens in the same regime of charge and gate that would otherwise show QAH.  Looks like some kind of chiral superconductivity that may be topological.
• Andrea Young, meanwhile, in fewer layer rhombohedral systems, showed experiments pointing to superconductivity happening at the verge of a canting transition, where spins are reorienting.
• Eva Andrei gave a nice talk looking at the variety of states one can get when interfacing moiré systems with other moiré systems, and explaining what is meant by intercrystals.  
• Gleb Finkelstein showed how a measurement intended to look at shot noise instead became a very cute noise thermometry probe of thermal transport at the boundary between (graphene) quantum Hall currents and a superconducting electrode.
• Xiao-Xiao Zhang showed a really cute experiment, where the resonance of a drumhead made from an atomically thin film of MnPS3 convey information about magnetic transitions in that material as a function of magnetic field.
• Dan Ralph gave a nice talk about the challenges of electrically generating currents of properly oriented spins to drive magnetic switching in films magnetized perpendicular to the film plane, for spin-orbit torque memories (and fundamental understanding).
• Philip Kim gave a great overview of some remarkable results in electronic interferometers made on graphene, in which telegraph noise shows signatures
• Lu Li spoke about recent measurements showing magnetic oscillations and specific heat signatures of possible neutral fermions in a kagome lattice Mott insulator.
• Xavier Roy talked about CeSiI, a 2D material that is also a heavy fermion metal.  This and its related compounds look like a fascinating family of (unfortunately extremely air sensitive) materials. 
• Harold Hwang gave a great overview of recent work in nickelate superconductors, highlighting the similarities to the cuprates as well as the profound differences (like how electronic configurations other than d9 can also lead to superconductivity).

This week I am attending the every-two-years DOE Experimental Condensed Matter Physics PI meeting.  Previously I have written up highlights of these meetings (see here, here, here, here, here), though two years I was unable to do so because I was attending virtually.  I will do my best to hit some high points (though I will restrict myself to talking only about already published work, to avoid any issues of confidentiality).  

In the meantime, here are a couple of topics of interest from the last couple of weeks.  

• I just learned about the existence of Mathos AI, an AI product that can function as a math solver and calculator, as well as a tutor for students.  It is pretty impressive.
• I liked this historical piece about Subrahmanyan Chandrasekhar (he of the “Chandrasekhar limit”, which describes the degeneracy physics + gravitation that limits the upper size of compact stellar objects like white dwarfs and neutron stars before they collapse into black holes) and his interactions with Stephen Hawking.  It's pretty humanizing to see an intellectual giant like Chandra sending a brief letter to Hawking in 1967 asking for advice on what to read so that Chandra can understand Hawking’s work on singularities in cosmology.  Hawking’s handwritten response is clear and direct.
• In an online discussion about what people will do if Google decides to stop supporting Google Scholar, I was introduced to OpenAlex.  This seems like an interesting, also-free alternative.  Certainly worth watching.  There is no obvious reason to think that Google Scholar is going away, but Alphabet has retired many free products, and it’s far from obvious how they are making any money on this.  Anyone from Google who reads the blog, please chime in.  (Note to self:  keep regularly backing up this blog, since blogger is also not guaranteed future existence.)

Is category theory being co-opted? These authors think so:

• Esteban Montero and Brandon Baylor, Category theory is being co-opted, Holon Substack, 7 September 2025.

Category theory offers a powerful new way of viewing the world… if we embrace its lessons and are willing to think in brand new ways. But what are the chances of that? It could easily become just another way for the rich and powerful to get more rich and powerful while the world burns. Applied category theory is already heavily funded by the military and tech firms.

Luckily, the UK has injected money into category theory for developing safeguards to AI. It’s risky, but it might do some good. And some of us are using category theory to develop new software for epidemiology, and soon labor economics, trying to model aspects of the green transition.

Montero and Baylor’s article is interesting—but it ignores those initiatives. It focuses heavily on the supposed ills of set theory—but I don’t think escaping set theory is either necessary or sufficient to free up our ways of thinking.

I have plenty of other complaints about this essay:

• Much as I like applied category theory, I don’t think our work so far “demonstrates the immense practical power of categorical thinking.” I think the jury is still out, which is why I keep working hard on this.

• The focus on Buddhism makes me uneasy. Buddhism has some good ideas, but I don’t think the cause of good new mathematics should be shackled to this religion.

• It seems like a bad idea, tactically, to combine this sort of essay with a plea for funding—even if you really want funding!

Nonetheless, not enough people are talking about the remarkable transformation of category theory into an ever-more-practical discipline, and the question of whether it will be truly revolutionary, or just another tool.

So until someone writes a better essay about this, it’s worth reading this one!

I particularly like their vision of how categorical network theory could be part of a new civilization:

• Systems Engineering: For a systems engineer designing a smart grid, a logistics network, or a complex software architecture, the most important thing is not the individual components, but the rules of their interaction. Category theory provides a formal language for designing systems not by specifying parts, but by defining relationships, interfaces, and compositions.

• Ecological Thinking: In ecology, an organism cannot be understood apart from its niche—the web of relationships it has with its environment and other organisms. The categorical perspective aligns with this systemic view, seeing entities as defined by their context.

• A New Civilization: At the largest scale, this relational philosophy suggests a different way of organizing society. Instead of a collection of atomized individuals, it envisions a system of interconnected stakeholders, where the health of the whole is determined by the quality of the relationships between the parts.

We just need to work out a few details.

It’s no secret that the early twentieth century was a great time to make progress in fundamental physics. On one level, it was an era when huge swaths of our understanding of the world were being rewritten, with relativity and quantum mechanics just being explored. It was a time when a bright student could guide the emergence of whole new branches of scholarship, and recently discovered physical laws could influence world events on a massive scale.

Put that way, it sounds like it was a time of low-hanging fruit, the early days of a field when great strides can be made before the easy problems are all solved and only the hard ones are left. And that’s part of it, certainly: the fields sprung from that era have gotten more complex and challenging over time, requiring more specialized knowledge to make any kind of progress. But there is also a physical reason why physicists had such an enormous impact back then.

The early twentieth century was the last time that you could dig up a rock out of the ground, do some chemistry, and end up with a discovery about the fundamental laws of physics.

When scientists like Curie and Becquerel were working with uranium, they didn’t yet understand the nature of atoms. The distinctions between elements were described in qualitative terms, but only just beginning to be physically understood. That meant that a weird object in nature, “a weird rock”, could do quite a lot of interesting things.

And once you find a rock that does something physically unexpected, you can scale up. From the chemistry experiments of a single scientist’s lab, countries can build industrial processes to multiply the effect. Nuclear power and the bomb were such radical changes because they represented the end effect of understanding the nature of atoms, and atoms are something people could build factories to manipulate.

Scientists went on to push that understanding further. They wanted to know what the smallest pieces of matter were composed of, to learn the laws behind the most fundamental laws they knew. And with relativity and quantum mechanics, they could begin to do so systematically.

US particle physics has a nice bit of branding. They talk about three frontiers: the Energy Frontier, the Intensity Frontier, and the Cosmic Frontier.

Some things we can’t yet test in physics are gated by energy. If we haven’t discovered a particle, it may be because it’s unstable, decaying quickly into lighter particles so we can’t observe it in everyday life. If these particles interact appreciably with particles of everyday matter like protons and electrons, then we can try to make them in particle colliders. These end up creating pretty much everything up to a certain mass, due to a combination of the tendency in quantum mechanics for everything that can happen to happen, and relativity’s . In the mid-20th century these particle colliders were serious pieces of machinery, but still small enough to make industrial: now, there are so-called medical accelerators in many hospitals based on their designs. But current particle accelerators are a different beast, massive facilities built by international collaborations. This is the Energy Frontier.

Some things in physics are gated by how rare they are. Some particles interact only very faintly with other particles, so to detect them, physicists have to scan a huge chunk of matter, a giant tank of argon or a kilometer of antarctic ice, looking for deviations from the norm. Over time, these experiments have gotten bigger, looking for more and more subtle effects. A few weird ones still fit on tabletops, but only because they have the tools to measure incredibly small variations. Most are gigantic. This is the Intensity Frontier.

Finally, the Cosmic Frontier looks for the unknown behind both kinds of gates, using the wider universe to look at events with extremely high energy or size.

Pushing these frontiers has meant cleaning up our understanding of the fundamental laws of physics up to these frontiers. It means that whatever is still hiding, it either requires huge amounts of energy to produce, or is an extremely rare, subtle effect.

That means that you shouldn’t expect another nuclear bomb out of fundamental physics. Physics experiments are already working on vast scales, to the extent that a secret government project would have to be smaller than publicly known experiments, in physical size, energy use, and budget. And you shouldn’t expect another nuclear power plant, either: we’ve long passed the kinds of things you could devise a clever industrial process to take advantage of at scale.

Instead, new fundamental physics will only be directly useful once we’re the kind of civilization that operates on a much greater scale than we do today. That means larger than the solar system: there wouldn’t be much advantage, at this point, of putting a particle physics experiment on the edge of the Sun. It means the kind of civilization that tosses galaxies around.

It means that right now, you won’t see militaries or companies pushing the frontiers of fundamental physics, unlike the way they might have wanted to at the dawn of the twentieth century. By the time fundamental physics is useful in that way, all of these actors will likely be radically different: companies, governments, and in all likelihood human beings themselves. Instead, supporting fundamental physics right now is an act of philanthropy, maintaining a practice because it maintains good habits of thought and produces powerful ideas, the same reasons organizations support mathematics or poetry. That’s not nothing, and fundamental physics is still often affordable as philanthropy goes. But it’s not changing the world, not the way physicists did in the early twentieth century.

But not the Orioles, against whom he is now 0-3. He got beat by Dylan Bundy in the 2016 game, then by Grayson Rodriguez (remember Grayson Rodriguez?) in 2023, and by Tyler Wells this week. That’s only the regular season, though. The game I really remember is the 2012 AL wild card game, the first playoff game for the Orioles in more than a decade. Darvish, then an MLB rookie, started for the Rangers. And we beat him then, too. I just claimed to really remember that game, but if you asked me who won it for the Orioles? You could give me fifty guesses and I wouldn’t have come up with Joe Saunders.

I would have thought it was hard to beat every team, but in the balanced schedule / interleague schedule it’s gotten a lot easier. Of course Wikipedia has a page of everybody who’s done it. Our old friend Kevin Gausman has managed to beat all 30 teams despite having only 110 wins. Maybe even more impressive is Jameson Taillon (who I saw beat the Brewers a couple of weeks ago in my first-ever trip to Wrigley Field, a game I never got around to blogging about), who is 0-1 against the Dodgers in 4 tries, but who has beaten all 29 other teams with only 80 career wins!

Wikipedia is really an incredible triumph of human cooperation. Why does it work so well? How is it so unpolluted? How is it that when some very weird niche question like “which pitchers have beaten all 30 teams,” comes to my mind, some human being has already compiled this and put it there? I don’t know. But I’m glad it exists.

I don’t know why there are so many baseball posts right now. They’re faster to write than math posts. More math posts to come, I promise! (But there’s also one more Orioles post I want to write…)

Update: This is not the Orioles post I have planned, but I should point out that perhaps even more impressively than any of the above feats of win distribution is that the 2025 Orioles, with only 64 wins to their name so far, have already beaten 28 of the 29 other teams in baseball, getting season-swept only by the Minnesota Twins.

In response to my recent blog posts, which expressed views that are entirely boring and middle-of-the-road for Americans as a whole, American Jews, and Israelis (“yes, war to destroy Hamas is basically morally justified, even if there are innocent casualties, as the only possible way to a future of coexistence and peace”)—many people declared that I was a raving genocidal maniac who wants to see all Palestinian children murdered out of sheer hatred, and who had destroyed his career and should never show his face in public again.

Others, however, called me something even worse than a genocidal maniac. They called me a Republican!

So I’d like to state for the record:

(1) In my opinion, Trump II remains by far the worst president in American history—beating out the second-worst, either Trump I or Andrew Jackson. Trump is destroying vaccines and science and universities and renewable energy and sane AI policy and international trade and cheap, lifesaving foreign aid and the rule of law and everything else that’s good, and he’s destroying them because they’re good—because even if destroying them hurts his own voters and America’s standing in the world, it might hurt the educated elites even more. It’s almost superfluous to mention that, while Trump himself is neither of these things, the MAGA movement that will anoint his successor now teems with antisemites and Holocaust “revisionists.”

(2) Thus, I’ll continue to vote straight-ticket Democrat, and donate money to Democrats, so long as the Democrats in question are seriously competing for Zionist Jewish votes at all—as, for example, has every Democratic presidential candidate in my lifetime so far.

(3) If it came down to an Israel-hating Squad Democrat versus a MAGA Republican, I’m not sure what I’d do, but I’d plausibly sit out the election or lodge a protest vote.

(4) In the extremely unlikely event that I had to choose between an Israel-hating Squad Democrat and some principled anti-MAGA Republican like Romney or Liz Cheney—then and only then do I expect that I’d vote Republican, for the first time in my life, a new and unfamiliar experience.

Hardly for the first time in my life, this weekend I got floridly denounced every five minutes—on SneerClub, on the blog of Peter Woit, and in my own inbox. The charge this time was that I’m a genocidal Zionist who wants to kill all Palestinian children purely because of his mental illness and raging persecution complex.

Yes, that’s right, I’m the genocidal one—me, whose lifelong dream is that, just like Germany and Japan rose from their necessary devastation in WWII to become pillars of our global civilization, so too the children in Gaza, the West Bank, Syria, Lebanon, and Iran will one day grow up in free and prosperous societies at peace with the West and with Israel. Meanwhile, those who demand an actual genocide of the Jews, another one—those who pray to Allah for it, who attempt it over and over, who preach it to schoolchildren, who celebrate their progress toward it in the streets—they’re all as innocent as lambs.

Yesterday, in The Free Press, came the report of a British writer who traveled to southern Lebanon, and met an otherwise ordinary young man there … who turned out to be excited for Muslims and Christians to join forces to slaughter all the Yahood, and who fully expected that writer would share his admiration for Hitler, the greatest Yahood-killer ever.

This is what the global far left has now allied itself with. This is what I’m right now being condemned for standing against, with commenter after commenter urging me to seek therapy.

To me, this raises a broader question: how exactly do you keep your sanity, when you live on a planet filled with brain-eaten zombies?

I’m still struggling with that question, but the best I’ve come up with is what I think of as the Weinberg Principle, after my much-missed friend and colleague here at UT Austin. Namely, I believe that it’s better to have one Steven Weinberg on your side while the rest of humanity is against you, than the opposite. Many other individuals (including much less famous ones) would also work here in place of Steve, but I’ll go with him because I think most of my readers would agree to three statements:

1. Steve’s mind was more in sync with the way the universe really works, than nearly anyone else’s in history. He was to being free from illusions what Usain Bolt is to running or Magnus Carlsen is to chess.
2. Steve’s toenail clippings constituted a greater contribution to particle physics than would the life’s work of a hundred billion Peter Woits.
3. Steve’s commitment to Israel’s armed self-defense, and to Zionism more generally, made mine look weak and vacillating in comparison. No one need wonder what he would’ve said about Israel’s current war of survival against the Iranian-led terror axis.

Maybe it’s possible to wake the zombies up. Yoram Arnon, for example, wrote the following eloquent answer on Quora, in response to the question “Why are so many against freeing Palestine?”:

When Westerners think about freedom they think about freedom of speech, freedom of expression, freedom of movement, freedom of religion, freedom to form political parties, etc.

When Palestinians say “Free Palestine” they mean freedom from Jews, and from Israel’s existence. They’re advocating for the abolition of Israel, replacing it with an Arab country.

Israel is the only country in the Middle East that is free, in the Western sense of the word. If Israel were to disappear, Palestinians would fall under an autocratic regime, just like every other Arab country, with none of the above freedoms. And, of course, Israelis would suffer a terrible fate at their hands.

Pro Palestinians are either unable to see this, or want exactly that, but thankfully many in the West do see this – the same “many” that are against “freeing Palestine”.

Palestinians need to accept Israel’s right to exist, and choose to coexist peacefully alongside it, for them to have the peace and freedom the West wants for them.

Maybe reading words like these—or the words of Coleman Hughes, or Douglas Murray, or Hussein Aboubakr Mansour, or Yassine Meskhout, or John Aziz, or Haviv Rettig Gur, or Sam Harris, or the quantum computing pioneer David Deutsch—can boot a few of the zombies’ brains back up. But even then, I fear that these reboots will be isolated successes. For every one who comes back online, a thousand will still shamble along in lockstep, chanting “brainsssssss! genocide! intifada!”

I’m acutely aware of how sheer numbers can create the illusion of argumentative strength. I know many people who were sympathetic to Israel immediately after October 7, but then gradually read the room, saw which side their bread was buttered on, etc. etc. and became increasingly hostile. My reaction, of course, has been exactly the opposite. The bigger the zombie army I see marching against me, the less inclined I feel to become a zombie myself—and the clearer to me becomes the original case for the Zionist project.

So to the pro-Zionist students—Jewish of course, but also Christian, Muslim, Hindu, atheist, and everyone else—who feel isolated and scared to speak right up now, and who also often email me, here’s what I say. Yes, the zombies vastly outnumber us, but on the other hand, they’re zombies. Some of the zombies know longer words than others, but so far, not one has turned out to have a worldview terribly different from that of the image at the top of this post.

I’ll keep the comments closed, for much the same reasons I did in my last post.  Namely, while there are many people of all opinions and backgrounds with whom one can productively discuss these things, there are many more with whom one can’t. Furthermore, experience has shown that the latter can disguise themselves as the former for days on end, and thereby execute a denial-of-service attack on any worthwhile and open public discussion.

Addendum: The troll who sent the antisemitic image now says that he regrets and apologizes for it, and that he’s going to read books on Jewish history to understand his error. I’ll believe that when he actually sends me detailed book reports or other evidence, but just wanted to update.

In my last post, I wrote about all the hate mail I’ve received these past few days. I even shared a Der-Stürmer-like image of a bloodthirsty, hook-nosed Orthodox Jew that some troll emailed me, after he’d repeatedly promised to send me a “diagram” that would improve my understanding of the Middle East. (Incredibly, commenters on Peter Woit’s blog then blamed me for this antisemitic image, mistakenly imagining that I’d created it myself, and then used their false assumption as further proof of my mental illness.)

Thanks to everyone who wrote to ask whether I’m holding up OK. The answer is: better than you’d expect! The first time you get attacked by dozens of Internet randos, it does feel like your life is over. But the sixth or seventh time? After you’ve experienced, firsthand, how illusory these people’s power over you actually is—how they can’t even dent your scientific career, can’t separate you from any of the friends who matter most to you (let alone your family), can’t really do anything to you beyond whatever they induce you to do to yourself? Then the deadly wolves appear more like poodles yapping from behind a fence. Try it and see!

Today I want to focus on a different kind of message that’s been filling my inbox. Namely, people telling me to stay strong, to keep up my courage, that everything I wrote strikes them as just commonsense morality.

It won’t surprise anyone that many of these people are Jews. But almost as many are not. I was touched to hear from several of my non-Jewish scientific colleagues—ones I’d had no idea were in my corner—that they are in my corner.

Then there was the American Gentile who emailed me a story about how, seeing an Orthodox family after October 7, he felt an urge to run up and tell them that, if worst ever came to worst, they could hide in his basement (“and I own guns,” he added). Amusingly, he added that his wife successfully dissuaded him from actually making such an offer, pointing out that it might freak out the recipients.

I replied that, here in America, I don’t expect that I’ll ever need to hide in anyone’s basement. But, I added, the only reason I don’t expect it is that there are so many Americans who, regardless of any religious or ideological differences, would hide their Jewish neighbors in their basements if necessary.

I also—despite neither I nor this guy exactly believing in God—decided to write a blessing for him, which came out as follows:

May your seed multiply a thousandfold, for like King Cyrus of Persia, you are a righteous man among the Gentiles.  But also, if you’re ever in Austin, be sure to hit me up for tacos and beer.

I’m even grateful, in a way, to SneerClub, and to Woit and his minions. I’m grateful to them for so dramatically confirming that I’m not delusional: some portion of the world really is out to get me. I probably overestimated their power, but not their malevolence.

I’ve learned, for example, that there are no words, however balanced or qualified, with which I can express the concept that Israel needs to defeat Hamas for the sake of both Israeli and Palestinian children, which won’t lead to Woit calling me a “genocide apologist who wants to see all the children in Gaza killed.” Nor are there any words with which to express my solidarity with the Jewish Columbia students who, according to an official university investigation, were last year systematically excluded from campus social life, intimidated, and even assaulted, and which won’t earn me names from Woit like “a fanatic allied with America’s fascist dictator.” Even my months-long silence about these topics got me labeled as “complicit with fascism and genocide.”

Realizing this is oddly liberating. When your back is to the wall in that way, either you can surrender, or else you can defend yourself. Your enemy has already done you the “favor” of eliminating any third options. Which, again, is just Zionism in a nutshell. It’s the lesson not only of 3,000 years of Jewish history, but also of superhero comics and of much of the world’s literature and cinema. It takes a huge amount of ideological indoctrination before such things stop being obvious.

Reading the SneerClubbers’ armchair diagnoses of my severe mental illness, paranoia, persecution complex, grandiosity, etc. etc. I had the following thought, paraphrasing Shaw:

Yes, they’re absolutely right that psychologically well-adjusted people generally do figure out how to adapt themselves to the reigning morality of their social environment—as indicated by the Asch conformity test, the Milgram electric-shock experiment, and the other classics of social psychology.

It takes someone psychologically troubled, in one way or another, to persist in trying to adapt the reigning morality of their social environment to themselves.

is a pitcher named Carlos Lagrange and if his major league nickname isn’t “The Multiplier” I will be sorely disappointed.

Today I gave $10,000 to Doctors Without Borders, since they’re doing a lot of good work in Gaza. I made this gift in memory of Mariam Abu Dagga, a freelance photographer who was killed in the Nasser Hospital in the Gaza Strip on August 25th this year.

After the Israeli air force bombed this hospital, she and other journalists rushed in to check on the well-being of their colleague, Reuters journalist Hussam al-Masri. He had fact been killed.

In a ‘double tap’ attack, the Israeli air force then dropped more bombs, killing Dagga and others.

Wikipedia writes:

Dagga was known for documenting the experiences of displaced Palestinians, and of doctors who treated wounded or malnourished children, with “rare honesty and courage.” [2][3] Independent Arabia described Dagga as an “example of dedication and professional commitment,” bringing “her camera into the heart of the field.” [3] Associated Press Executive Director and Senior Vice President Julian Pace said that Dagga’s difficult journalistic work in Gaza was remarkable, particularly for her “coverage of the war’s impact on children.” [5]

Dagga won an internal award at the Associated Press for her coverage of malnourished children in Gaza. [5]

Speaking to the BBC, Palestinian journalist Hadar al-Qurd described Dagga as one of the most active female journalists in southern Gaza. [1] Al-Qurd said that other journalists relied on Dagga for the news and information that she gathered in her work. [1] Al-Qurd noted that Dagga was especially brave, always carrying her camera, and often going to the sites of airstrikes directly; she also noted that Dagga was a champion of the rights of female journalists in Gaza. [1] Other colleagues also noted Dagga’s bravery as a war correspondent. [3] Al Jazeera journalist Youmna El Sayed noted that Dagga had visibly lost a significant amount of weight during the war. [6]

Due to Israel’s repeated targeting and killing of journalists in the Gaza War, Dagga wrote her will during the conflict.

I urge you all to help Doctors Without Borders (= Médecines sans Frontiérs), who are doing good work in Gaza and other afflicted regions. They write:

MSF has over 1,300 staff working in Gaza’s hospitals, clinics, and other facilities, including our field hospital in Deir al-Balah and our clinic in Gaza City. Our teams provide surgical care, wound and burn care, malnutrition screening and treatment, maternal and pediatric care, physiotherapy, vaccination, mental health support, water and sanitation support, and care for non-communicable diseases, among other services. We are also providing rehabilitative care for war-wounded children we’ve managed to evacuate to our reconstructive surgery hospital in Amman, Jordan.

Facilities MSF has run or supported in Gaza

MSF teams have carried out lifesaving work in health facilities across Gaza, including in the Strip’s largest hospitals and our own clinics and field hospital. However, due to extremely volatile conditions on the ground—including attacks on health facilities and recurrent evacuation orders—our teams have had to move from facility to facility and continually adapt our activities. Hospitals and clinics we have supported during this war include:

• Nasser Hospital, European Gaza Hospital, and Martyrs, Beni Suhaila, Khan Younis, Al-Qarara, and Al-Attar clinics in Khan Younis
• Al-Aqsa Hospital and the MSF field hospital in Deir al-Balah    
• Al-Awda Hospital and mobile clinics in northern Gaza
• Al-Shifa Hospital, Al Helou Maternity Hospital, and our clinic in Gaza City   
• Rafah Indonesian Field Hospital, Emirati Maternity Hospital, Al-Najjar Hospital, and Al-Mawasi Health Post in Rafah

Water and sanitation activities

A lack of drinkable water, poor sanitation, and the destruction of water infrastructure have had dire consequences for people’s health in Gaza. Of more than 82,000 primary health care consultations MSF conducted in the first two months of this year, nearly a fifth were related to conditions linked with lack of water and hygiene, such as scabies and other skin conditions. This is why water distribution is an important part of MSF’s response.  

Between January and the end of April 2025, MSF teams distributed over 60 million liters of clean water and produced over 7.8 million liters through desalination. However, the already-dire water crisis in Gaza has worsened after Israeli authorities halted aid from entering the Strip on March 2, and then cut electricity on March 9, as water pumps and desalination plants require fuel and power to operate.

Humanitarian aid and medical supplies

Since Israeli authorities halted the flow of humanitarian aid and other supplies into the Strip on March 2, food, fuel, and medical stocks have been depleted. This total blockade of aid has deprived people of most basic needs and could lead to a high number of health complications and deaths.

Prior to the total siege, MSF provided over 636 tons of logistic and medical equipment from our international supply centers—as much as 30 planes or 130 trucks full. However, some supplies that are critical to our operations and the security of our staff have been difficult to transport into Gaza. These include generators, desalination stations and motor pumps, oxygen concentrators, vehicles, and equipment for communication.

Thomas Bloom’s erdosproblems.com site hosts nearly a thousand questions that originated, or were communicated by, Paul Erdős, as well as the current status of these questions (about a third of which are currently solved). The site is now a couple years old, and has been steadily adding features, the most recent of which has been a discussion forum for each individual question. For instance, a discussion I had with Stijn Cambie and Vjeko Kovac on one of these problems recently led to it being solved (and even formalized in Lean!).

A significantly older site is the On-line Encyclopedia of Integer Sequences (OEIS), which records hundreds of thousands of integer sequences that have some mathematician has encountered at some point. It is a highly useful resource, enabling researchers to discover relevant literature for a given problem so long as they can calculate enough of some integer sequence that is “canonically” attached to that problem that they can search for it in the OEIS.

A large fraction of problems in the Erdos problem webpage involve (either explicitly or implicitly) some sort of integer sequence – typically the largest or smallest size of some -dependent structure (such as a graph of vertices, or a subset of ) that obeys a certain property. In some cases, the sequence is already in the OEIS, and is noted in the Erdos problem web page. But in a large number of cases, the sequence either has not yet been entered into the OEIS, or it does appear but has not yet been noted on the Erdos web page.

Thomas Bloom and I are therefore proposing a crowdsourced project to systematically compute the hundreds of sequences associated to the Erdos problems and cross-check them against the OEIS. We have created a github repository to coordinate this process; as a by-product, this repository will also be tracking other relevant statistics about the Erdos problem website, such as the current status of formalizing the statements of these problems in the Formal Conjectures Repository.

The main feature of our repository is a large table recording the current status of each Erdos problem. For instance, Erdos problem #3 is currently listed as open, and additionally has the status of linkage with the OEIS listed as “possible”. This means that there are one or more sequences attached to this problem which *might* already be in the OEIS, or would be suitable for submission to the OEIS. Specifically, if one reads the commentary for that problem, one finds mention of the functions for , defined as the size of the largest subset of without a -term progression. It is likely that several of the sequences , , etc. are in the OEIS, but it is a matter of locating them, either by searching for key words, or by calculating the first few values of these sequences and then looking for a match. (EDIT: a contributor has noted that the first foursequences appear as A003002, A003003, A003004, and A003005 in the OEIS, and the table has been updated accordingly.)

We have set things up so that new contributions (such as the addition of an OEIS number to the table) can be made by a Github pull request, specifically to modify this YAML file. Alternatively, one can create a Github issue for such changes, or simply leave a comment either on the appropriate Erdos problem forum page, or here on this blog.

Many of the sequences do not require advanced mathematical training to compute, and so we hope that this will be a good “citizen mathematics” project that can bring in the broader math-adjacent community to contribute to research-level mathematics problems, by providing experimental data, and potentially locating relevant references or connections that would otherwise be overlooked. This may also be a use case for AI assistance in mathematics through generating code to calculate the sequences in question, although of course one should always stay mindful of potential bugs or hallucinations in any AI-generated code, and find ways to independently verify the output. (But if the AI-generated sequence leads to a match with an existing sequence in the OEIS that is clearly relevant to the problem, then the task has been successfully accomplished, and no AI output needs to be directly incorporated into the database in such cases.)

This is an experimental project, and we may need to adjust the workflow as the project progresses, but we hope that it will be successful and lead to further progress on some fraction of these problems. The comment section of this blog can be used as a general discussion forum for the project, while the github issue page and the erdosproblems.com forum pages can be used for more specialized discussions of specific problems.

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The Simons-Laufer Mathematical Sciences institute, or SLMath (formerly the Mathematical Sciences Research Institute, or MSRI) has recently restructured its program formats, and is now announcing three new research initiatives, whose applications open on Sep 1 2025:

• AxIOM (Accelerating Innovation in Mathematics) is a new, month-long research program at SLMath, designed to accelerate innovation and introduce transformative ideas into the mathematical sciences. Programs begin in Spring 2027.
• PROOF (Promoting Research Opportunities and Open Forums) is a two-week summer program designed to provide research opportunities for U.S.-based mathematicians, statisticians, and their collaborators in the U.S. and abroad, whose ongoing research may have been impacted by factors such as heavy teaching loads, professional isolation, limited access to funding, heavy administrative duties, personal obligations, or other constraints. Programs begin June-July 2026. The priority application deadline for PROOF 2026 is October 12, 2025.
• Lasting Alliance Through Team Immersion and Collaborative Exploration (LATTICE) is a yearlong program which provides opportunities for U.S. mathematicians to conduct collaborative research on topics at the forefront of the mathematical and statistical sciences. Programs begin June-July 2026. LATTICE 2026 applications are open through February 1, 2026.

(Disclosure: I am vice-chair of the board of trustees at SLMath.)

Pick a type of categorical structure: say bicategories, or monoidal categories, or whatever you like. Some of the functors between structures are equivalences, in whatever the appropriate sense might be. And some of those equivalences have one or both of these two properties:

• They’re not just essentially surjective in every dimension — they’re actually surjective in every dimension.

• They don’t just preserve the structure up to isomorphism or equivalence — they strictly preserve it.

Call an equivalence with both these properties a strict surjective equivalence. So a strict surjective equivalence is an equivalence of a very special and easy kind.

General principle: the standard notion of equivalence between structures is generated by just these very special ones. For example, two bicategories are biequivalent if and only if they can be linked up by a zigzag of strict surjective equivalences.

Why should we care? Because there are some types of structure where the right notion of equivalence isn’t clear, and this principle guides us to it. For example, it tells us the right notion of equivalence for double categories.

All this is done in my new paper:

Tom Leinster, Equivalence via surjections. arXiv:2508.20555, 2025.

I started thinking about this question during Maru Sarazola’s invited talk at Category Theory 2025 in Brno last month. She asked the question:

What is the right notion of equivalence between double categories?

and carefully went through the properties that the right notion of equivalence should have, some possible candidates, and different approaches one might take to deciding what “right” means.

The answer that Maru ultimately gave was that the right notion is “gregarious double equivalence”, proposed by Alexander Campbell in about 2020. And she gave a justification in terms of model categories, representing joint work between her, Lyne Moser and Paula Verdugo.

For the purposes of this post, it actually doesn’t matter what “gregarious double equivalence” means. What I want to talk about is the following principle, which popped into my head as Maru was speaking:

For many types of categorical structure, the natural notion of equivalence is generated, as an equivalence relation, by identifying $A$ and $B$ when there exists a strict surjective equivalence $A \to B$.

It occurred to me that this principle might give a rather different justification for why gregarious double equivalence is the right answer. And after some checking, I discovered that it does.

Let me explain.

A more concrete way to express the principle is that $A$ and $B$ are equivalent in the standard sense — whatever’s appropriate for the structures at hand — if and only if there exists a zigzag of strict surjective equivalences

$$A = A_0 \leftarrow A_1 \rightarrow \ \cdots \ \leftarrow A_n = B.$$

For any type of categorical structure I can think of, the pullback of a strict surjective equivalence is a strict surjective equivalence. So a simpler concrete condition is just that there exists a span of strict surjective equivalences

$$A \leftarrow C \rightarrow B.$$

But hold on… what do I mean by “principle”?

What I mean is that for simple types of categorical structure, where “equivalence” and “strict surjective equivalence”, we have a theorem. Here are three examples.

• Categories. We certainly know what it means for two categories to be equivalent. A “surjective equivalence” is an equivalence that’s not just essentially surjective on objects, but literally surjective on objects.

  In this case, the theorem is that categories $A$ and $B$ are equivalent if and only if there exists a span $A \leftarrow C \rightarrow B$ of surjective equivalences between them.

  (The word “strict” does nothing in this case.)

• Monoidal categories. Again, we know what monoidal equivalence is, and it’s clear what a “strict surjective equivalence” is: a strict monoidal functor that’s a surjective equivalence of categories.

  The theorem is that monoidal categories $A$ and $B$ are monoidally equivalent if and only if there exists a span $A \leftarrow C \rightarrow B$ of strict surjective equivalences between them.

• Bicategories. The pattern is the same. The standard notion of equivalence for bicategories is biequivalence. A “strict surjective equivalence”, in this setting, is a strict $2$-functor that is literally surjective on objects and locally a surjective equivalence of categories. (Or put another way, surjective on $0$-cells, locally surjective on $1$-cells, and full and faithful on $2$-cells.)

  The theorem is that bicategories $A$ and $B$ are biequivalent if and only if there exists a span $A \leftarrow C \rightarrow B$ of strict surjective equivalences between them.

Probably all these theorems are known. I included them in my paper because I couldn’t find them anywhere in the literature, not even the first one. But if you know a reference, I’d be glad to hear it.

Since the principle holds for categories, monoidal categories and bicategories, it’s reasonable to suppose that it might hold for other types of structure. And if we’re investigating some type of structure where the full notion of equivalence isn’t clear, this principle might help guide us to it.

For example, here’s a theorem on double categories, the main result of my paper:

• Double categories. Again, it’s clear what “strict surjective equivalence” should mean: a strict double functor that’s surjective on $0$-cells, locally surjective on both horizontal and vertical $1$-cells, and full and faithful on $2$-cells.

  The theorem is that double categories $A$ and $B$ are gregariously double equivalent if and only if there exists a span $A \leftarrow C \rightarrow B$ of strict surjective equivalences between them.

Even without me telling you what “gregarious double equivalence” means, the four theorems I’ve stated suggest that it’s the right notion of equivalence for double categories, because it continues the pattern we’ve seen for simpler categorical structures.

So, I agree with the conclusion that Moser, Sarazola and Verdugo had already reached! But for different reasons.

Incidentally, this must be the fastest paper I’ve ever written: just under six weeks from sitting in Maru’s talk and hearing the mathematical term “gregarious” for the first time ever to putting the paper on the arXiv. But the principle that all equivalences are generated by strict surjective equivalences was planted in my head in the late 1990s or early 2000s by Carlos Simpson. Back then, we were both working on higher category theory, and when he explained this principle, I found it very striking — so striking that I remembered it 20+ years later. There’s a bit more on that higher categorical context in the introduction to my paper.

A paper about scientific fraud has been making the rounds in social media lately. The authors gather evidence of large-scale networks of fraudsters across multiple fields, from teams of editors that fast-track fraudulent research to businesses that take over journals, sell spots for articles, and then move on to a new target when the journal is de-indexed. I’m not an expert in this kind of statistical sleuthing, but the work looks impressively thorough.

Still, I think the authors overplay their results a bit. They describe themselves as revealing something many scientists underestimate. They point to what they label as misconceptions: that scientific fraud is usually perpetrated alone by individual unethical scientists, or that it is almost entirely a problem of the developing world, and present their work as disproving those misconceptions. Listen to them, and you might get the feeling that science is rife with corruption, that no result, or scientist, can be trusted.

As far as I can tell, though, those “misconceptions” they identify are true. Someone who believes that scientific fraud is perpetrated by loners is probably right, as is someone who believes it largely takes place outside of the first world.

As is often the case, the problem is words.

“Scientific Fraud” is a single term for two different things. The two both involve bad actors twisting scientific activity. But in everything else — their incentives, their geography, their scale, and their consequences — they are dramatically different.

One of the types of scientific fraud is largely about power.

In references 84-89 of the paper, the authors give examples of large-scale scientific fraud in Europe and the US. All (except one, which I’ll mention later) are about the career of a single researcher. Each of these people systematically bent the truth, whether with dodgy statistics, doctored images, or inflating citation counts. Some seemed motivated to promote a particular scientific argument, cutting corners to push a particular conclusion through. Others were purer cases of self-promotion. These people often put pressure on students, postdocs, and other junior researchers in their orbits, which increases the scale of their impact. In some cases, their work rippled out to convince other researchers, prolonging bad ideas and strangling good ones. These were people with power, who leveraged that power to increase their power.

There also don’t appear to be that many of them. These people are loners in a meaningful sense, cores of fraud working on their own behalf. They don’t form networks with each other, for the most part: because they work towards their own aggrandizement, they have no reason to trust anyone else doing the same. I have yet to see evidence that the number of these people is increasing. They exist, they’re a problem, they’re important to watch out for. But they’re not a crisis, and they shouldn’t shift your default expectations of science.

The other, quite different, type of scientific fraud is fraud for a fee.

The cases this paper investigates seem to fall into this category. They are businesses, offering the raw material of academic credit (papers, co-authorship, citations, publication) for cash. They’re paper mills, of various sorts. These are, at least from an academic perspective, large organizations, with hundreds or thousands of customers and tens of suborned editors or scientists farming out their credibility. As the authors of this paper argue, fraudsters of this type are churning out more and more papers, potentially now fueled by AI, adding up to a still small, but non-negligible, proportion of scientific papers in total.

Compared to the first type of fraud, though, buying credit in this way doesn’t give very much power. As the paper describes, many of the papers churned out by paper mills don’t even go into relevant journals: for example, they mention “an article about roasting hazelnuts in a journal about HIV/AIDS care”. An article like that isn’t going to mislead the hazelnut roasting community, or the HIV/AIDS community. Indeed, that would be counter to its purpose. The paper isn’t intended to be read at all, and ideally gets ignored: it’s just supposed to inflate a number.

These numbers are most relevant in the developing world, and when push comes to shove, almost all of the buyers of these services identified by the authors of this paper come from there. In many developing countries, a combination of low trust and advice from economists leads to explicit point systems, where academics are paid or hired explicitly based on criteria like where and how often they publish or how they are cited. The more a country can trust people to vouch for each other without corruption, the less these kinds of incentives have purchase. Outside of the developing world, involvement in paper mills and the like generally seems to involve a much smaller number of people, and typically as sellers, not buyers: selling first-world credibility in exchange for fees from many developing-world applicants.

(The one reference I mentioned above is an interesting example of this: a system built out of points and low trust to recruit doctors from the developing world to the US, gamed by a small number of co-authorship brokers.)

This kind of fraud doesn’t influence science directly. Its perpetrators aren’t trying to get noticed, but to keep up a cushy scam. You don’t hear their conclusions in the press, other scientists don’t see their work. Instead, they siphon off resources: cannibalizing journals, flooding editors with mass-produced crap, and filling positions and slurping up science budgets in the countries that can least afford them. As they publish more and more, they shouldn’t affect your expectations of the credibility of science: any science you hear about will be either genuine, or fraud from the other category. But they do make the science you hear about harder and harder to do.

(The authors point out one exception: what about AI? If a company trains a large language model on the current internet, will its context windows be long enough to tell that that supposedly legitimate paper about hazelnuts is in an HIV/AIDS journal? If something gets said often enough, copied again and again in papers sold by a mill, will an AI trained on all these papers be convinced? Presumably, someone is being paid good money to figure out how to filter AI-generated slop from training data: can they filter paper mill fraud as well?)

It’s a shame that we have one term, scientific fraud, to deal with these two very different things. But it’s important to keep in mind that they are different. Fraud for power and fraud for money can have very different profiles, and offer very different risks. If you don’t trust a scientific result, it’s worth understanding what might be at play.

Suppose a man has already murdered most of your family, including several of your children, for no other reason than that he believes your kind doesn’t deserve to exist on earth. The murderer was never seriously punished for this, because most of your hometown actually shared his feelings about your family. They watched the murders with attitudes ranging from ineffectual squeamishness to indifference to unconcealed glee.

Now the man has kidnapped your last surviving child, a 9-year-old girl, and has tied her screaming to train tracks. You can pull a lever to divert the train and save your daughter. But there’s a catch, as there always is in these moral dilemmas: namely, the murderer has also tied his own five innocent children to the tracks, in such a way that, if you divert the train, then it will kill his children. What’s more, the murderer has invited the entire town to watch you, pointing and screaming “SHAME!!” as you agonize over your decision. He’s persuaded the town that, if you pull the lever, then having killed five of his children to save only one of yours, you’re a far worse murderer than he ever was. You’re so evil, in fact, that he’s effectively cleansed of all guilt for having murdered most of your family first, and the town is cleansed of all guilt for having cheered that. Nothing you say can possibly convince the town otherwise.

The question is, what do you do?

Zionism, to define it in one sentence, is the proposition that, in the situation described, you have not merely a right but a moral obligation to pull the lever—and that you can do so with your middle finger raised high to the hateful mob. Zionism is the belief that, while you had nothing against the murderer’s children, while you would’ve wanted them to grow up in peace and happiness, and while their anguished screams will weigh on your conscience forever, as your children’s screams never weighed on the murderer’s conscience, or on the crowd’s—even so, the responsibility for those children’s deaths rests with their father for engineering this whole diabolical situation, not with you. Zionism is the idea that the correct question here is the broader one: “which choice will bring more righteousness into the world, which choice will better embody the principle that no one’s children are to be murdered going forward?” rather than the narrowly utilitarian question, “which choice will lead to fewer children getting killed right this minute?” Zionism is the conviction that, if most of the world fervently believes otherwise, than most of the world is mistaken—as the world has been mistaken again and again about the biggest ethical questions all through the millennia.

Zionism, so defined, is the deepest moral belief that I have. It’s deeper than any of my beliefs about “politics” in the ordinary sense. Ironically, it’s even deeper than my day-to-day beliefs about the actual State of Israel and its neighbors. I might, for example, despise Benjamin Netanyahu and his ministers, might consider them incompetent and venal, might sympathize with the protesters who’ve filled the streets of Tel Aviv to demand their removal. Even so, when the murderer ties my child to the train tracks and the world cheers the murderer on, not only will I pull the lever myself, I’ll want Benjamin Netanyahu to pull the lever if he gets to it first.

Crucially, everything worthwhile in my life came when, and only when, I chose to be “Zionist” in this abstract sense: that is, steadfast in my convictions even in the face of a jeering mob. As an example, I was able to enter college three years early, which set the stage for all the math and science I later did, only because I finally said “enough” to an incompetent school system where I was bullied and prevented from learning, and to teachers and administrators whose sympathies lay with the bullies. I’ve had my successes in quantum computing theory only because I persisted in what at the time was a fairly bizarre obsession, rather than working on topics that almost everyone around me considered safer, more remunerative, and more sensible.

And as the world learned a decade ago, I was able to date, get married, and have a family, only because I finally rejected what I took to be the socially obligatory attitude for male STEM nerds like me—namely, that my heterosexuality was inherently gross, creepy, and problematic, and that I had a moral obligation never to express romantic interest to women. Yes, I overestimated the number of people who ever believed that, but the fact that it was clearly a nonzero number had been deterrent enough for me. Crucially, I never achieved what I saw for years as my only hope in life, to seek out those who believed my heterosexuality was evil and argue them out of their belief. Instead I simply … well, I raised a middle finger to the Andrea Dworkins and Arthur Chus and Amanda Marcottes of the world. I went Deep Zionist on them. I asked women out, and some of those women (not having gotten the memo that I was “problematic,” gross, and worthless) said yes, and one of them became my wife and the mother of my children.

Today, because of the post-October-7 public stands I’ve taken in favor of Israel’s continued existence, I deal with emails and social media posts day after day calling me a genocidal baby-killing monster. I’ve lost perhaps a dozen friends (while retaining hundreds more friends, and gaining some new ones). The haters’ thought appears to be that, if they can just raise the social cost high enough, I’ll finally renounce my Zionist commitments and they can notch another win. In this, they oddly mirror Hamas, Hezbollah, and the IRGC, who think that, if they can just kill and maim enough Israelis, the hated “settler-colonialist rats” will all scurry back to Poland or wherever else they came from (best not to think too hard about where they did come from, what was done to them in those places, how the Palestinian Arabs of the time felt about what was done to them, or how the survivors ended up making a last stand in their ancestral home of Israel—even afterward, repeatedly holding out olive branches that were met time after time with grenades).

Infamously, Israel’s enemies have failed to understand for a century that, the more they rape and murder, the more Zionist the hated Zionists will become, because unlike the French in Algeria or whatever, most of the Zionists have no other land to go back to: this is it for them. In the same way, my own haters don’t understand that, the more they despise me for being myself, the more myself I’ll be, because I have no other self to turn into.

I’m not opening the comments on this post, because there’s nothing here to debate. I’m simply telling the world my moral axioms. If I wrote these words, then turned to pleading with commenters who hated me because of them, then I wouldn’t really have meant the words, would I?

To my hundreds of dear friends and colleagues who’ve stood by me the past two years, to the Zionists and even just sympathetic neutrals who’ve sent me countless messages of support, but who are too afraid (and usually, too junior in their careers) to speak up in public themselves: know that I’ll use the protections afforded by my privileged position in life to continue speaking on your behalf. Know that I’m infinitely grateful, that you give me strength, and that if I can give you a nanoparticle of strength back to you, then my entire life wasn’t in vain. And if I go silent on this stuff from time to time, for the sake of my mental health, or to spend time on quantum computing research or my kids or the other things that bring me joy—never take that to mean that I’ve capitulated to the haters.

To the obsessive libelers, the Peter Woits and other snarling nobodies, the self-hating Jews, and those who’d cheer to see Israel “decolonized” and my friends and family there murdered, I say—well, I don’t say anything; that’s the point! This is no longer a debate; it’s a war, and I’ll simply stand my ground as long as I’m able. Someday I might forgive the Gentiles among you if you ever see the light, if you ever realize how your unreflective, social-media-driven “anti-fascism” led you to endorse a program that leads to the same end as the original Nazi one. The Jews among you I’ll never forgive, because you did know better, and still chose your own comfort over the physical survival of your people.

It might as well be my own hand on the madman’s lever—and yet, while I grieve for all innocents, my soul is at peace, insofar as it’s ever been at peace about anything.

Update (Aug. 29): This post was born of two years of frustration. It was born of trying, fifty or a hundred times since October 7, to find common ground with the anti-Zionists who emailed me, messaged me, etc.—“hey, obviously neither of us wants any children killed or starved, we both have many bones to pick with the current Israeli government, but surely we at least agree on the necessity of defeating Hamas, right? right??“—only to discover, again and again, that the anti-Zionists had no interest in such common ground. With the runaway success of the global PR campaign against Israel—i.e., of Sinwar’s strategy—and with the rise of figures like Mamdani (and his right-wing counterparts) all over the Western world, anti-Zionists smell blood in the water today. And so, no matter how reasonable they presented themselves at first, eventually they’d come out with “why can’t the Jews just go back to Germany and Poland?” or “the Holocaust was just one more genocide among many; it doesn’t deserve any special response,” or “why can’t we dismantle Israel and have a secular state, with a Jewish minority and a majority that’s sworn to kill all Jews as soon as possible?” And then I realize, with a gasp, that we Jews really are mostly on our own in a cruel and terrifying world—just like we’ve been throughout history.

To say that this experience radicalized me would be an understatement. Indeed, my experience has been that even most Israelis, who generally have far fewer illusions than we diaspora Jews, don’t understand the vastness of the chasm that’s formed. They imagine that they can have a debate with outsiders similar to the debates playing out within Israel—one that presupposes basic factual knowledge and the parameters of the problem (e.g., clearly we can’t put 7 million Jews under the mercy of Hamas). The rationale for Zionism itself feels so obvious to them as to be cringe. Except that, to the rest of the world, it isn’t.

We’re not completely on our own though. There remain decent people of every background, who understand the stakes and feel the weight of history—and I regularly hear from them. And whatever your criticisms of Israel’s current tactics, so long as you accept the almost comically overwhelming historical case for the necessity of Jewish self-defense, this post wasn’t aimed at you, and you and I probably could discuss these matters. It’s just that the anti-Zionists scream so loudly, suck up so much oxygen, that we definitely can’t discuss them in public. Maybe in person sometime, face to face.

You can cut a hole in a cube that’s big enough to slide an identical cube through that hole! Think about that for a minute—it’s kind of weird.

Amazingly, nobody could prove any convex polyhedron doesn’t have this property! It’s called ‘Rupert’s property’.

Until this week.

This week Steininger and Yurkevich proved there is a convex polyhedron that you can’t cut a hole in big enough to slide the entire polyhedron through the hole. It has 90 vertices, and apparently 240 edges and 152 faces.

To prove that no such hole is possible, they had to do a computer search of 18 million different holes, plus use a lot of extra math to make sure they’d checked enough possibilities:

• Jakob Steininger and Sergey Yurkevich, A convex polyhedron without Rupert’s property.

To celebrate their discovery, they gave this polyhedron a silly name. Since this polyhedron lacks Rupert’s property, they called it a ‘noperthedron’.

Why is this property called ‘Rupert’s property’? Wikipedia explains:

In geometry, Prince Rupert’s cube is the largest cube that can pass through a hole cut through a unit cube without splitting it into separate pieces. Its side length is approximately 1.06, 6% larger than the side length 1 of the unit cube through which it passes. The problem of finding the largest square that lies entirely within a unit cube is closely related, and has the same solution.

Prince Rupert’s cube is named after Prince Rupert of the Rhine, who asked whether a cube could be passed through a hole made in another cube of the same size without splitting the cube into two pieces. A positive answer was given by John Wallis. Approximately 100 years later, Pieter Nieuwland found the largest possible cube that can pass through a hole in a unit cube.

Here Greg Egan shows how Rupert’s property works for the cube:

Here he shows how it works for the regular octahedron:

And finally, here’s a video by David Renshaw showing 26 polyhedra with Rupert’s property… and 5 polyhedra that might lack it: