[This is a guest post by Matthew Colbrook. This blog post was initially written in a different file format and converted using AI. — T.]

(Disclosure: ChatGPT helped me write and refine this post.)

Inspired in part by the recent discussions on this blog, we have started a new project, AIM, an initiative to create a community-led opportunity for mathematical exploration. Since AI can make some conjectures easier to resolve, we would like to explore how these developments can be used to benefit the mathematical community, enhancing our learning and understanding. Modern developments in AI bring both excitement and uncertainty in our field. We hope to create opportunities for mathematicians, especially students and early career researchers, to shape what comes next by asking good questions, working on problems, and developing the ideas and applications that come with them.

We have collected a list of open problems in applied mathematics and related fields along with their precise statements, motivation, and references:

https://github.com/MColbrook/AIM

This repository of problems is intended to benefit the mathematical community and become a place for collaborative exploration as well as a place where mathematicians can share proposed proofs and solutions for others to see. Our primary purpose is mathematical. Rather than using a set of problems to benchmark AI systems, we want to let the community shape the questions that we pursue, whether or not AI is used, and how we learn from this work. We warmly invite readers to contribute problems to the list, directions, and solutions to those problems already in the repository. Partial results, proofs, counterexamples, references, and corrections are most welcome. Contributions with or without AI are perfectly acceptable, with careful checking and attribution.

The companion site, AIM, explained (https://mathematics-explained.com/ ), welcomes papers and videos exploring and explaining the mathematical arguments behind the solutions of these problems. A solution to a problem should not be the end of our engagement with a question, for it can open up further questions about why an argument works, whether a different formulation is more insightful, and how the ideas from a solution connect to other mathematics or applications. We want to acknowledge this work.

To encourage participation, we are also running a competition for explanatory papers and videos of up to ten minutes. The competition is split into two strands: for those who have not yet completed a first degree, and for graduate students and postdoctoral researchers at academic institutions. The emphasis is on accuracy, clarity and understanding, rather than on being first to solve or explain a problem. Entries close on 1 January 2027 at 23:59 UTC, with prizes for winning entries and a curated publication of selected contributions. The wider explanation library is open to all, independently of the competition.

Our repository and website are a starting point, and we would be most glad to hear your suggestions. Please share problems that you would like others to explore (across all of pure and applied mathematics), progress on those already listed, and other ideas that might make AIM useful to you or your students. Problems and corrections can be submitted through the repository; broader suggestions are welcome here in the comments or directly to the organisers.

Sent on behalf of the organisers, Steve Brunton, Matthew Colbrook, Maarten V. de Hoop, George Stepaniants, Alex Townsend, and Rachel Ward.

I’ve met people who think they’re uncomfortable with evolution, but when you actually talk to them, they’re worried about something very different. They ask, “How could something come from nothing? How could life come from non-life?”

Biologists aren’t sure of the origin of life. They’re working on it, there’s a whole research field on what the experts call abiogenesis. But it’s hard going, because there’s so little direct evidence.

There is a lot more evidence for evolution. Evolution isn’t a theory of where life comes from, it was never meant to be one. It’s a theory that explains the copious evidence we have from the history of life, from the fossil record to genetic comparisons to leftover features in related animals. It’s a theory that tells you what to expect when you already have some life, and wait a few generations.

Evolution and abiogenesis are different ideas, with different domains. You can have one without the other. And when you already have life, and want to understand it, evolution is quite a bit more useful than abiogenesis.

There’s a similar story with the big bang.

Some people get mystified at the idea that the big bang made something out of nothing. But that’s not really how the big bang works.

It’s not just that the big bang wasn’t the explosion you imagine, a single point expanding in an empty space. Rather, it was a time when the entire universe, everywhere that anything could be, was in a hot dense state, not empty but brim-full.

It’s that the most important part of the big bang is what happened next. As the universe cooled, things spread away from each other, like the residents of Hilbert’s Hotel all moving one room apart. Clumps formed, electrons slowing down and getting caught by protons to make atoms, dust pulling on dust until it collapsed into stars.

The patterns of a universe that cooled this way are distributed over the sky. They tell us the kinds of atoms we expect to see in spectra of stars, the distribution of galaxies, the ripples in the afterglow of the first light to avoid getting absorbed by hot plasma. Those patterns are the core of physical cosmology, the bulk of the field’s research and its strongest argument.

In contrast, the question of where that hot dense state came from is murkier. There are a variety of pictures, different attempts to approximate a context where gravity’s quantum properties should be especially relevant by people who still don’t have a clear idea of how to think about quantum gravity.

But while the question is murky, it’s also just not very useful. Wherever that state came from, knowing it exists is usually enough to chart the rest of the universe’s history. In fact, it may even be impossible for us to know what happened before a certain time: we have only one sky’s worth of evidence, and eventually that evidence runs out.

People are used to created things: food, clothes, cars. Created things don’t change in complicated ways over time, for the most part, just break down. To understand a created thing, understanding its origin matters.

But science is full of stranger things, things with a life, or dynamics, of their own. Natural phenomena change, swiftly becoming something other than what they began as. Understanding them means understanding those changes, much more than it does understanding how they began.

— 1. The Hardy-Littlewood maximal inequality —

We work in Euclidean space with Lebesgue measure; we write instead of for the Lebesgue measure of a set . For any and let denote the open ball of radius centred at . Thus for instance . For any , we use to denote the dilate of around its centre by .

For any , we define the averaging operators on for any locally integrable by

One can view as an averaging operator

Proposition 1 (Hardy-Littlewood maximal inequality) We have the the strong-type inequality

for all and any , and also the weak-type inequality

for any .

The sublinear operator

Exercise 2 Explain why is not of strong or weak type for any .

As this proposition is so fundamental we shall give several proofs of it.

— 2. First proof —

We begin with the classical proof, starting with some standard qualitative reductions. Firstly we may easily reduce to being non-negative. A monotone convergence argument also lets us restrict to functions which are bounded and have compact support. (At this point one may object that might not be measurable, but one can use dominated convergence to restrict to be (say) rational, at which point measurability is clear.) Also, is continuous in , so we may restrict to a countable dense set (such as the positive rationals); another monotone convergence argument then lets us restrict to a finite set. Of course, our bounds need to be uniform in this set, as well as being uniform in the boundedness and support of .

It is obvious that is bounded on (indeed, it is a contraction on this space). So it suffices by Marcinkiewicz interpolation to prove the weak-type inequality; by homogeneity (and the preceding reductions) it thus suffices to show that

Let us denote the set on the left-hand side by ; our hypotheses on and easily ensure that is a compact set (alternatively, one can work with the uncountable sup , but instead replace by an arbitrary compact subset of itself, and then take suprema in at the end, noting that Lebesgue measure is a -finite Radon measure). By construction, we thus see that for any there exists a radius such that is locally large compared to the ball :

Lemma 3 (Wiener’s Vitali-type covering lemma) Let be a finite collection of balls. Then there is a subcollection of disjoint balls such that

Proof: We can order in decreasing order of size. Now we select the disjoint balls by the greedy algorithm, picking the largest balls we can at each stage. Namely, for we choose to be the first ball which is disjoint from all previously selected balls (thus for instance must equal ), until we run out of balls. Clearly this gives us a family of disjoint balls. Now observe from construction that each ball in the original collection is either a ball in the subcollection, or else intersects a ball in the subcollection of equal or larger radius. In either case we see from the triangle inequality that is contained in . In other words,

From the covering lemma it is easy to conclude (2). Indeed, since is covered by finitely many of the balls , the covering lemma gives us finitely many disjoint balls , such that

Remark 4 Under mild assumptions one can generalise the covering lemma to infinite families of balls without difficulty. One can also replace balls by similar objects, such as cubes; the main property that one needs is that if two such objects overlap, then the smaller one is contained in some dilate of the larger. This is a fairly general property, and for instance holds for metric balls on a measure space with some doubling property , but it fails for very thin or eccentric sets such as long tubes, rectangles, annuli, etc. Indeed, understanding the maximal operator for these more geometrically complicated objects is still a major challenge in harmonic analysis, leading to important conjectures such as the Kakeya conjecture, which remains open in higher dimensions despite recent progress.

Exercise 5 (Baby Besicovitch covering lemma) Let be a collection of intervals on the real line. Show that there exist a subcollection such that , and such that every point belongs to at most of the intervals . What is the best explicit bound for you can get?

Exercise 6 Show that if is supported on , then

Exercise 7 For any locally integrable , let denote the rectangular maximal function

where ranges over all rectangles with sides parallel to the coordinate axes which contain . Show that is bounded on for all . (Hint: prove by induction, controlling the -dimensional rectangular maximal function by the -dimensional “horizontal” rectangular maximal function, applied to a one-dimensional “vertical” maximal function. A certain amount of application of the Fubini-Tonelli theorem may be needed.) Show by example that is not of weak-type in dimensions .

Exercise 8 Let , , and let be locally integrable on .

• (i) Establish Hedberg’s inequality

  

  for all . (Hint: use symmetries to normalise as many quantities as you can, and then divide the integral either dyadically, or into a region near and a region away from , estimating the two regions differently.)
• (ii) Use Hedberg’s inequality to establish the Hardy-Littlewood-Sobolev inequality

  

  where the fractional integral operator is defined by

  

  and is the exponent for which .

Exercise 9 Let , and let be a ball such that at every point of . Show that at every point of .

— 3. Second proof —

Let us now give a slightly different proof of the above inequality, replacing balls by the slightly simpler structure of dyadic cubes.

Definition 10 (Dyadic cube) A dyadic cube in of generation is a set of the form

where is an integer and .

The crucial property of dyadic cubes is the nesting property: if two dyadic cubes overlap, then one must contain the other. This leads to

Lemma 11 (Dyadic Vitali-type covering lemma) Let be a finite collection of dyadic cubes. Then there is a subcollection of disjoint cubes such that

Proof: Take the to be the maximal dyadic cubes in – the cubes which are not contained in any other cubes in this collection. The nesting property then ensures that they are disjoint and cover all of between them.

If we then define the dyadic maximal function

We can rewrite the dyadic maximal inequality in another way. Let be the -algebra generated by the dyadic cubes of generation , then

Observe that if , then there is a ball centred at which contains of comparable volume: . Because of this, one easily obtains the pointwise inequality

-translation trick

Lemma 12 Let be a (non-dyadic) interval. Then there exists an interval which is either a dyadic interval, or a dyadic interval translated by , such that and .

The only significance of is that its binary digit expansion oscillates between and . Note that the claim is false without the shifts; consider for instance the interval for some very small , which straddles a certain “discontinuity” in the standard dyadic mesh. The point is that the dyadic mesh and the -translate of the dyadic mesh do not share any discontinuities.

Exercise 13 Prove the above lemma.

For intervals larger than , a shift by is not enough; consider for instance what happens to the interval . Instead, we have to shift by , which of course does not make sense as a real number. However, it does make sense in some formal -adic sense (as the doubly infinite binary string ) which is good enough to define shifted dyadic meshes.

Definition 14 (Dyadic meshes) We define to be the collection of all dyadic intervals in . We define to be the collection of all intervals of the form , where is a dyadic interval at some generation and is any integer greater than or equal to (note that the exact choice of is irrelevant). If , we let be the collection of cubes formed by the Cartesian product of intervals from .

By modifying the above lemma one then quickly deduces

Lemma 15 Let be a ball. Then there exists and a shifted dyadic cube such that and .

This in turn leads to the pointwise inequality bounding the dyadic maximal function by the ordinary one:

Remark 16 What is going on here is that there are two ways to view the real line . One is the “Euclidean” way, with the usual group structure and metric. The other is the “Walsh” or “dyadic” way, in which we identify with the Cantor group

via the binary representation, (identifying with , and ignoring the measure zero sets of terminating decimals where the binary representation is not unique). The group structure is now the one inherited from the Cantor group; in the binary representation, the Cantor-Walsh addition law is the same as ordinary addition but where we neglect to carry bits. The usual archimedean metric is replaced by the non-archimedean metric , defined by

With this metric, the dyadic intervals become the metric balls.

Exercise 17 (Hardy-Littlewood maximal inequality for filtrations) Let be a measure space, and let be an increasing sequence of -finite -algebras in (thus for all ). Show that

and hence

for all and all -measurable for which the right-hand side is finite. (Hint: use monotone convergence to reduce to finitely many . Reduce further to the case when the are countably generated (by using the level sets of the for rational intervals). Reduce further still to the case where the are finitely generated, i.e. finite. Now adapt the dyadic argument. There are also simpler arguments which do not require all of these reductions.) This inequality is also known as Doob’s inequality, and implies in particular that converges pointwise a.e. to whenever .

Exercise 18 (Relationship between dyadic and non-dyadic Hardy-Littlewood maximal inequalities) Let be locally integrable. Establish the pointwise bound

for some depending only on .

— 4. Third proof —

In the above arguments we obtained bounds by first proving weak bounds and then interpolating. It is natural to ask whether such bounds can be obtained directly. The answer is yes, but it is surprisingly more difficult to do so. Let us give two such approaches, a Bellman function approach and a method approach, which are themselves powerful methods which apply to many other problems as well.

We begin with the Bellman function approach. This method works primarily for dyadic model operators, such as , though it can also work for very geometric operators as well (using geometric averaging operators such as heat kernels in place of the dyadic averaging operators). For simplicity let us just work in one dimension (though it is possible to use rearrangement and space-filling curves to deduce the higher-dimensional case from the one-dimensional case), and consider the task of establishing

The idea is to work by induction on scales – in other words, to induct on the number of generations. To do this we need a “base case”, so we perform some qualitative reductions. Fix (the case being trivial). By a monotone convergence argument we may restrict attention only to those intervals of length larger than , so long as our estimates are uniform in . By rescaling (replacing by ) we can reduce to the case . Let us write for the dyadic maximal function restricted to intervals of length at least . By a monotone convergence argument (there is a slight problem because we cannot represent as the monotone limit of dyadic intervals; however we can do this for and separately, and then add up, noting that the dyadic maximal function is localised to each of these half-lines, e.g. if is supported on then so is ) we can also assume that is supported on a dyadic interval of some length , and also we may restrict to that interval. We can also take to be non-negative. Our task is now to show that there exists a constant such that

We make a small but useful remark: once and are both restricted to , the only dyadic intervals which are relevant in the definition of are those which are contained in (including itself). Intervals which are disjoint from play no role, and intervals which contain give a worse average than that arising from itself.

The idea is to prove this by induction on the generation of . Our first approach will not quite work, but a subtle modification of it will.

When the claim is trivial (as long as ), because for a dyadic interval of generation and we have

without causing any deterioration in the constant

Let us split the dyadic interval of generation into two “children” of generation ( and stand for left and right). This also causes a split . By induction hypothesis we have

Let us formalise this as follows. Given any parameters , let denote the cost function

Now the recursive inequality can be turned into an inequality for . Suppose that attains the supremum (or comes within an epsilon of it). We have

Thus the whole task is reduced to a freshman calculus problem, namely to find a function of three variables obeying the bounds (6) and (7). The difficulty of course, is to find the function; verifying the properties is then routine. This “hunt for a Bellman function” is surprisingly subtle, and requires one to choose a surprisingly non-trivial choice of .

The condition (7) resembles a concavity condition, and so it is natural to try to find choices which are concave in some of the variables. An initial candidate is the function , which certainly obeys (7) and the upper bound of (6) (and which, in fact, ultimately corresponds to the original estimate (4) before we threw in the other parameters ); unfortunately it does not obey the lower bound in (6), in the case where is large compared with and . So we need to tweak this function somewhat. The first step is to improve the concavity by exploiting the fact that the function only needs to be non-trivial on the region . One can exploit this by using the candidate function (say). This still obeys (7), but now with a bit of a gain when is large due to the strict concavity of . (One cannot play similar games with the parameter as the upper and lower bounds in (6) force linear-type behaviour in .) But we still have not fixed the problem that is not as large as when is large. The solution is to use the Bellman function

Remark 19 Bellman function methods are in principle the sharpest and most powerful technique to prove estimates. However they are restricted to dyadic settings (or very geometric continuous settings), and are very delicate due to the need to establish very subtle concavity properties.

— 5. Fourth proof —

Finally, we present the “ approach” to the Hardy-Littlewood maximal inequality. This approach works for both the dyadic and non-dyadic maximal functions, but is restricted to establishing boundedness (this is a fundamental limitation of the method, that at least one of the spaces involved has to be a Hilbert space). The basic idea is rather than prove a boundedness result directly on the maximal operator (i.e. an estimate of the form ), we prove an estimate on the square of this operator (roughly speaking, we prove an estimate of the form ). This is an example of a powerful strategy to understand an operator by raising it to a higher power, and hoping to exploit some self-cancellation. Note that we do not have to cancel the entire power (which would roughly speaking correspond to proving a bound of the form ); any nontrivial cancellation at all is exploitable.

Let us work with the non-dyadic maximal function, thus we wish to show that

Observe that (8) is equivalent to the uniform linearised estimate

Let us fix the function . Then we can define the linear operator by

Lemma 20 ( identity) Let be a continuous map from a Hilbert space to a normed vector space, and let be its adjoint. Then

Proof: The first identity is just duality. Then we have

In light of this identity we know that

The Hardy-Littlewood maximal function directly bounds averages on balls and cubes, but it also controls several other types of averages as well. For instance, we have the pointwise inequality

The proof of the Hardy-Littlewood maximal inequality extends to the more general setting of spaces of homogeneous type (which is similar, but different from, the notion of a homogeneous space). These are measure spaces with a metric , such that the open balls are measurable with positive finite measure, and that one has the doubling property

In particular we obtain the discrete inequality

Remark 21 One could try playing with instead of here, but that turns out to not work very well. The problem is that the linearised maximal operator is only well-behaved in one variable, and the method manages to play the two well-behaved variables against each other; going the other way one achieves no obvious cancellation.

— 6. Some consequences of the maximal inequality —

The Hardy-Littlewood maximal inequality is the underlying quantitative estimate which powers many qualitative pointwise convergence results. A basic example is

Theorem 22 (Lebesgue differentiation theorem) Let be locally integrable. Then we have the pointwise convergence for almost every . In fact we have the stronger estimate

for almost every .

Before we prove the theorem, let us make some remarks. Firstly, from the identity

Proof: It suffices to establish the result for constrained to a large ball , where is arbitrary. The claim is obvious if vanishes on , so by linearity we may assume that is supported on ; in particular now lies in . Thus it suffices to establish the claim when .

By the preceding discussion we already know that the claim is true for the continuous compactly supported functions , which is a dense subclass of . To pass from the dense subclass to the full class we use the Hardy-Littlewood maximal inequality now lets us reduce matters to verifying the claim on a dense subset of . To see this, suppose that we already have proven the claim on a dense class. Then, given any we can write as the limit in of a sequence in this dense class; by refining these sequence to make the convergence sufficiently fast (e.g. ) and using Markov’s inequality and the Borel–Cantelli lemma, we can also ensure that converges to pointwise almost everywhere. Now from the Hardy-Littlewood maximal inequality, we know that converges to zero, thus converges to zero in measure. By the triangle inequality this implies that converges in measure also. Thus (again passing to a rapidly converging subsequence as necessary) we see that converges to zero for almost every . This uniform-in- convergence lets one deduce the convergence of from the convergence of .

Finally, by taking a dense class such as the Schwartz class (or even the continuous functions in ) we easily verify the convergence.

Remark 23 Because the proof used a density argument, it offers no quantitative rate for the speed of convergence. Indeed the convergence can be arbitrarily slow; this is ultimately due to the implicit hypothesis that is measurable, which is itself a qualitative assumption that offers no explicit bounds. More quantitative versions of measurability, e.g. quantifying the extent to which a measurable set can be approximated by elementary sets, can lead to more explicit bounds.

Consider for instance the function defined by when and the integer part of is even, and otherwise, where is a large integer. Then we see that for , will stay close to for quite a while (basically for all ), and only at scales or less will it begin to “decide” to converge to either or . Modifying this type of example (e.g. by a “Weierstrass example” formed by summing together a geometrically decaying series of such oscillating functions, with going to infinity), one can concoct functions whose convergence of to is arbitrarily slow. (One can obtain quantitative rates by enforcing regularity conditions on the function, which essentially compactifies the space of functions that one is working with; we will see some examples of this later.)

Exercise 24 Let be a locally integrable function on . Call a point a Lebesgue point if there exists a number such that . Show that almost every point is a Lebesgue point, and that is equal to almost everywhere.

Exercise 25 (Heat kernels) For any and any for some , define the heat kernel by

Show that if and , then converges both pointwise and in norm to as .

Now let us explore applications to ergodic theory. In particular we wish to investigate limits of the form

Let us first look at an abstract setting, in which is a unitary operator on a Hilbert space.

Theorem 26 (Von Neumann ergodic theorem) Let be a unitary operator on a Hilbert space. Then for any , the limit exists in the strong topology.

Proof: Let us first argue formally. We have

Now let us make the above argument rigorous. If is -invariant, thus , then and it is clear that converges to . If on the other hand is a -difference, for some , then we verify the telescoping identity . Since unitary operators preserve the norm, we thus see from the triangle inequality that converges to zero. Also, since the averaging operators are uniformly bounded in (by the triangle inequality), we see that any strong limit of -differences also has the property that exists and equals zero.

To summarise so far, we have two closed subspaces of for which we know convergence. The first is , the invariant space of ; here the limit converges to the identity. The other is , the closure of the -differences; here the limit conveges to zero. These two spaces turn out to be orthogonal complements. To see this, first observe that they are orthogonal: if , then by unitarity, and hence is orthogonal to every -difference and hence by continuity of inner product is orthogonal to . To show orthogonal complement, it then suffices to show that any vector orthogonal to is invariant. But then is orthogonal to :

Note that the above argument in fact shows the stronger claim that converges to the orthogonal projection of to .

Now we move to a more specialised setting, that of a measure-preserving system.

Definition 27 (Measure-preserving system) A measure-preserving system is a probability space (thus ) together with a bi-measurable bijection (thus and are both measurable) such that is measure-preserving, thus for all .

Example 28 (Circle shift) Let be the standard circle with the usual Borel -algebra and Lebesgue measure, and let for some , which may be either rational or irrational.

Remark 29 Many of the results here hold under more relaxed assumptions on , but we will not attempt to optimise the hypotheses here.

The shift on the base space , , induces a shift on sets , , and then also induces a map on measurable functions by . The use of is natural since it ensures that and .

Proposition 30 (Mean ergodic theorem) Let be a measure-preserving system, and let for some . Then the sequence converges strongly in .

Proof: The operator is an isometry on , and so by the triangle inequality the averaging operators are uniformly bounded in . Thus it suffices to prove the claim for a dense subclass of ; we shall pick . In this class, which is embedded into the Hilbert space , we already know from the von Neumann ergodic theorem (and the fact that is a unitary operator on ) that the averages are convergent in norm. But they are also uniformly bounded in . So the claim follows from the log-convexity of norms (for ) or by Hölder’s inequality and the finite measure of (for ).

Exercise 31 Show by example that the mean ergodic theorem fails for and for .

Now we study the pointwise convergence problem. The key quantitative estimate needed is the following analogue of the Hardy-Littlewood maximal inequality.

Theorem 32 (Hardy-Littlewood maximal inequality for measure-preserving systems) Let be a measure-preserving system. Then we have

and

for all .

Proof: The claim is trivial when , so once again the task is to prove the pointwise estimate. By monotone convergence it suffices to show that

The idea is to lift up to a space where can be modeled by the integer shift , at which point we can apply (10). Fix , and let be the finite set , endowed with the discrete -algebra and uniform measure, so that is a probability space. Inside this space we have , where . We define the function by

Remark 33 Note how a maximal inequality for the integers was “transferred” to a maximal inequality on arbitrary measure preserving systems. There are several abstract transference principles which generalise this type of phenomenon. In particular, the above argument is in fact a special case of the Calderón transference principle.

Now we can present the analogue of the Lebesgue differentiation theorem for measure-preserving systems.

Proposition 34 (Pointwise ergodic theorem) Let be a measure-preserving system, and let . Then the sequence converges pointwise almost everywhere.

Proof: By repeating the argument in the Lebesgue differentiation theorem more or less verbatim (using the above maximal inequality in place of the Hardy-Littlewood inequality) it suffices to verify the claim for a dense subclass of , such as . Since the norm controls the norm, it suffices to do so for a dense subclass of .

Now we repeat the proof of the von Neumann ergodic theorem. For the invariant part of , the pointwise convergence is obvious. Also, for functions of the form with , the convergence is also obvious. But these functions are clearly dense in and hence in . Since this space and are orthogonal complements of , we thus have demonstrated convergence of a dense subclass of and thus of , as desired.

Remark 35 Notice how the strategy of establishing convergence splits into two independent parts – obtaining convergence on a dense subclass, and then establishing some harmonic analysis estimate to pass to the general case. This is not the only way to achieve convergence results. Later on we shall see a variation-norm approach which relies purely on harmonic analysis estimates to obtain convergence (foregoing the dense subclass). In the opposite direction, there are more dynamical approaches (which we do not cover here) which forego the harmonic analysis component of the argument, relying instead on analysing the dynamics of the measure-preserving system by other means (such as measure-theoretic or topological means). It is not fully understood to what extent these different techniques complement each other.

Exercise 36 Let be a measure-preserving system. Let be the elements of which are -invariant (up to sets of measure zero, of course). Show that for any and , the averages converge in norm and pointwise almost everywhere to . In particular, when is ergodic (thus the only invariant sets have zero measure or full measure), conclude that converges pointwise and in norm to .

Exercise 37 (Poincare recurrence theorem) Let be a measure-preserving system, and let be non-negative with . Show that for infinitely many .

By combining the Lebesgue differentiation theorem with the Radon-Nikodym theorem, one easily obtains

Theorem 38 (One-dimensional Radamacher differentiation theorem) If is Lipschitz, then it is differentiable almost everywhere, its derivative lies in , and we have the fundamental theorem of calculus for all .

Proof: For simplicity of notation we take , although the general case is similar. Theorem 38 (and Fubini’s theorem) shows that the partial derivatives , exist almost everywhere and are bounded. This gives us a candidate gradient defined almost everywhere. The remaining challenge is to show total differentiability. In other words, for any , we need to show that for almost every , we have

Fix . We will need to make things slightly more quantitative. For any and , let denote the set

Exercise 41 (Fundamental theorem of calculus) Let be locally integrable, and let (with the usual convention that ). Show that is differentiable at every Lebesgue point of , and that almost everywhere.

— 7. functions —

Another classical application of the Hardy-Littlewood maximal function lies in obtaining a satisfactory theory of (complex) functions on the complex disk for . (The theory for is more subtle and will not be dealt with here.)

Let be a holomorphic function. From residue calculus we have

function

Exercise 42 For any and , define the Fourier coefficient of by the formula

Show that

for every and , and that

for every , and . Also show that for all , and that

for all . (Note that these identities can be proven either by complex analysis methods or by Fourier analysis methods; it is instructive to prove them both ways and compare results.)

The kernel can be easily verified to obey the bounds

The definition of strongly suggests (but does not immediately prove) that if lies in , then should converge to some sort of limit as . This is indeed the case:

Theorem 43 Let for . Then converges both in norm and pointwise almost everywhere to a limit .

Proof: Let us first demonstrate weak convergence. For any and , we observe that

Exercise 45 (Fatou’s theorem) Let for some , and let be as in Theorem 43. Show that for almost every , we have

whenever is a sequence of points converging to non-tangentially in the sense that is uniformly bounded away from . (Hint: first reduce to a fixed angle of non-tangentiality (e.g. all angles less than pi/2 – 1/n), and then build an appropriate “non-tangential maximal function”, formed by taking suprema over all points in a sector with apex and this fixed angle of non-tangentiality. Use the kernel bounds to bound this maximal function by the Hardy-Littlewood maximal operator.)

— 8. The Calderón-Zygmund decomposition —

For a function on an abstract measure space, and any threshold one has the basic decomposition

For arbitrary measure spaces with no additional structure, this type of decomposition is about the best one can do to split into “good” and “bad” parts. But if has more structure – in particular some sort of metric or dyadic structure compatible with the measure – then one can do better, in particular one can select the bad piece to be “locally oscillating” in a certain sense. This principle, which uses the same ideas that underlie the Hardy-Littlewood maximal inequality, is formalised in the Calderón-Zygmund decomposition, a fundamental tool in the Calderón-Zygmund theory of singular integrals.

To motivate the lemma let us first give a simple one-dimensional version.

Lemma 46 (Rising sun lemma) Let be a bounded interval, and let be integrable, and let be such that . Then there exist an at most countable family of disjoint open intervals in such that

• for almost every ;
• for every .

Proof: By subtracting from we may normalise . We may take to be half-open . Define by . Then is a continuous function which starts at zero and ends up positive. Define a “maximal” version of by , then is continuous non-decreasing which starts at zero and ends up positive. Let , then is an open subset of , thus for some at most countable set of intervals . At the endpoints of each interval we have , while on the interior of these intervals is necessarily constant (why?). This gives the second desired property. For the first, one observes from Q7 that at almost every point where , we have differentiable with , which implies that , and the claim follows.

As a corollary we have

Corollary 47 (One-dimensional rising sun lemma) Let the hypotheses be as in the above lemma. Then we have a decomposition , where is bounded above by with , and for each , is supported in and has mean zero. Furthermore, if and is non-negative, then and .

Proof: We set , and set equal to outside of and equal to inside . To prove the last two claims, we have

The proof of the rising sun lemma relies crucially on the ordered nature of the real line. For more general spaces, such as Euclidean spaces, it is preferable to use arguments that rely instead on metric or dyadic structure. We begin with a dyadic version.

Proposition 48 (Dyadic Calderón-Zygmund decomposition) Let and . Then there exists a decomposition , where the “good” function obeys the bounds

ranges over a disjoint family of dyadic cubes, and each is supported on , has mean zero, and has the bound

Furthermore, we have the inclusions

In particular from (3) we have .

Proof: The arguments here will closely resemble that used to prove (3). Let us say that a cube is bad if , and good otherwise. Call a bad cube maximal if is bad, but no cube strictly containing is bad, and let be the collection of all maximal bad cubes, thus is a collection of disjoint dyadic cubes, and we also clearly have on each bad cube. From the hypothesis and monotone convergence we see that all cubes with sufficiently large side-length are automatically good. Thus every bad cube is contained in a maximal bad cube.

Now let be a maximal bad cube. The parent of (i.e. the unique dyadic cube of twice the side-length containing ) is good, hence . Since , we conclude that . Now set

Exercise 49 Let be the -algebra generated by the cubes in , and the Borel subsets of . Show that .

Note that the total bad set is given by the level set of the dyadic maximal function. This suggests an alternate approach to the Calderón-Zygmund decomposition, in which one starts by identifying the total bad set (in this case, ), and then decomposes it into cubes or balls. A prototype decomposition of this type is

Proposition 50 (Dyadic Whitney decomposition) Let be an open set. Then there exists a decomposition , where ranges over a family of disjoint dyadic cubes, and for each in this family , the parent of is not contained in .

Proof: Define to be the set of all dyadic cubes in which are maximal with respect to set inclusion; the claim follows from the nesting property. (The condition is needed to ensure that every cube is contained in a maximal cube; the open-ness is to get every point of contained in at least one cube.)

The property that is not contained in implies the bounds

Proposition 51 (Whitney decomposition) Let be an open set and let . Then there exists a decomposition , where ranges over a family of disjoint dyadic cubes, and for each in this family , we have .

Proof: Let denote those dyadic cubes in such that

Note that if is large, the cubes in this above decomposition have the property that nearby cubes (in the sense that ) have comparable diameter, thanks to the triangle inequality

Proposition 52 (Whitney decomposition for balls) Let be an open set and let . Then one can cover by balls such that , and such that each point in is contained in at most balls.

These decompositions can be used to prove some minor variants of the Calderón-Zygmund decomposition, which we will not describe in detail here.

Ethan Sussman, postdoc at Northwestern University

At the moment, the mathematics community is facing an unprecedented misalignment brought on by the advent of LLMs. According to the standard telling, the basic tension is between our traditional values (whatever those may be) and those of the corporations that make and market LLMs. The purpose of this brief essay is to point out that some (arguably many) examples of that claimed misalignment boil down to a preexisting mismatch between the values we profess and the incentive structure that we have set up. We are therefore blaming AI companies for something which is our responsibility to fix.

A good illustration comes from a recent petition. The exact content of that petition is unimportant for the present discussion. I want to focus on a single line:

Human mathematicians who are simultaneously proving the same results [as an LLM] may be forced to abruptly abort these research projects, regardless of the additional insights their unique approaches may bring.

Q. Forced by whom?

Of our ostensible traditional values, the most important is supposed to be understanding. Numerous pronouncements have been made that this thing is the true goal of mathematical labor. Moreover, problem solving and the production of novel theorems are necessary but not sufficient steps in this direction. This becomes a criticism of LLMs and the conduct of their makers when it is implied that, by prioritizing answer-generation, they ultimately hurt mathematical understanding. To me, this feels like a case of the pot calling the kettle black. To what extent is our current system really set up to maximize the production of understanding?

Partially, of course. But we cannot reward understanding directly, so instead we reward measurable proxies thereof and incentivize their production. For various reasons, the main proxy has become the research paper, published with the imprimatur of some prestigious journal. Some reasons are good: e.g. the need for an objective metric on which hiring committees can base their decisions.¹ Some are not: e.g. the prejudicial view (famously expressed by Hardy) that the expository arts are for “second-rate minds.” The question becomes the extent to which our preferred proxies measure the thing they are supposed to, and that is open for debate.

I believe that most of the misalignment mentioned above has to do with LLMs exposing and widening that gap. Consider the example above, and the question raised. Suppose a human $\text{\textendash}$ say Buckmaster, Córdoba, or Martínez-Zoroa $\text{\textendash}$ is carrying out a unique research program along a unique line, via an approach that is expected to bring additional insights. Why should the fact that an LLM had previously proven a big theorem in the domain force them to “abruptly abort” their work? AI companies possess no formal authority to decide what mathematicians spend their time on, nor do they have that inclination. The subtext in the open letter seems to be that, without the carrot-on-a-stick of priority to claim, there will be little incentive for mathematicians to carry on. But if their work really stands to deliver understanding, then our incentives have failed to incentivize the right thing.

Ultimately, it is mathematicians who decide what mathematical activity is worthwhile. Buckmaster and Córdoba both have tenure. Martínez-Zoroa is on tenure track. They are free to work on what they please $\text{\textemdash}$ and we, as mathematicians, are free to reward them for doing so. Mathematicians decide what sort of papers appear in top journals, what sort of work garners prizes, and ultimately who is hired. If OpenAI solving the Navier$\text{\textendash}$Stokes millennium problem (or prematurely finishing someone else’s research program) decreases our understanding, that reflects a failing on our part, irrespective of any wrongdoing on theirs.

Our emphasis on one particular form of mathematical activity $\text{\textendash}$ proving novel theorems and the ancillary activity of paper-writing $\text{\textendash}$ is to the detriment of all others. These second-class mathematical activities include refereeing, code generation and formalization, distillation and digestion, simplification, codification, exposition, teaching, and learning. All are in service of understanding. Perhaps Hardy was right that these activities are for second-rate minds, but the unfortunate reality is, in the age of the LLM and massive agentic swarms (the “Large Agent Collider”), we are all second-rate minds. If we wish for mathematical practice to flourish in the upcoming era, then the responsibility falls on us to redesign our incentive structure to better reward the whole spectrum of mathematical activity.  We can do little about the Cossacks at the gates.

We can renegotiate our tradition to better align with our needs. Perhaps the new world won’t be so bad.

1. For mathematics, this is the $L^p$ norm of paper strength, for some $p\gg 1$. For theoretical physics, with a smaller selection of journals to differentiate the strength of individual papers, it is the $L^1$ norm. ︎

Received 12 September 2026.

Frank Vallentin, Professor at the University of Cologne

The beginning

As a kid, I liked numbers, counting, and all sorts of numerical data (Bundesliga results, weather statistics,$\ldots$). Throughout elementary and high school, I liked doing mathematics. It came easily to me, but I was not enthusiastic about it. Rather, I found doing homework, including standard computational exercises, relaxing. I guess it was the combination of thinking and writing that had this effect on me, and I also knew that it would never take long.

I was much more obsessed with computers. This began to develop at Christmas 1984, when my grandfather gave me his Sinclair ZX81, one of the first home computers, with 1 KB of main memory. I think he had bought it for himself out of curiosity but found that he could not use it for the tasks he had in mind (he was a master butcher, and I guess he wanted to use it for wordprocessing and spreadsheets; later, it turned out that a Commodore Plus/4 was the right computer for him).

I learned how to operate the ZX81 and played my first computer games (1K ZX Chess, 3D Monster Maze). Because of the computer’s limitations (even after buying a 16 KB memory expansion) and my own limitations (I was 12 years old, and the manual explained BASIC in English), my fascination did not last very long.

In the following years, home computers became more colourful. After a detour with a TI-99/4A (which was powerful but did not have a large user community in Germany), I bought the legendary Commodore 64. I spent quite some time gaming (I was a big fan of Andrew Braybrook’s games, particularly Paradroid and, above all, Uridium, both technically impressive space games). Then my interest shifted completely from playing games to programming them.

I decided to learn 6510-assembly language to program my own games. Special issues of the monthly computer magazine 64’er helped me learn. Since this was the first programming language I had learned, it took me some time to master the basics. Eventually, I wrote a sprite editor, a graphics program for drawing sprites like the beautiful manta ray spaceship in Uridium.

But then the era of the C64 came to an end, and we moved on to 16-bit machines $\text{\textemdash}$ in my case, the Commodore Amiga. At around the same time, in the tenth or eleventh grade, we learned how to program in Pascal at school. We also tried to set up a computer club initiated by our English and geography teacher.

During an evening meeting at his home, he gave me a book about a new programming language, Modula-2. It was still very new and had been invented by Niklaus Wirth at ETH Zurich. The teacher told me that he could not make much sense of the book and thought that perhaps I would have more use for it. And indeed, I became hooked on the beauty of a higher-level programming language that was easy to program in, had been designed by a famous computer science professor, and offered practical software-development techniques: dividing a program into modules in different files and separating a module’s interface from its implementation. An elegant implementation for the Amiga, M2AMIGA, was available, and I invested both time and money in it.

Automatic mathematics

In grade 11, school mathematics really annoyed me. For half a year, we did nothing but curve sketching: Looking at functions, computing values, computing derivatives, checking for monotonicity, finding roots, determining minima, maxima, and saddle points. I was annoyed, as it felt completely mechanical. I decided that I wanted to automate it. Write a computer program which, given a function as input, would perform all the desired computations and print the result as LaTeX.¹

Writing such a program turned out to be a challenge, a much greater challenge than I had expected. It was an excellent challenge: I had to learn what a function is and, especially, how to work with symbolic formulas. I was asking myself the question: when is a formula well formed? If you don’t know how to formulate this question of syntactic correctness, it is difficult to find an answer. This was a few years before the World Wide Web, so getting technical information in a remote village in the German countryside was hard. Somehow I found out that I had to learn how to build a compiler, and, by a twist of fate, I took a train to Siegen, the nearest university town, where I went to an academic bookshop and bought the “Dragon Book” (Compilers: Principles, Techniques, and Tools) by Aho, Sethi, and Ullman,  a computer science text book explaining all the details of compiler construction. This book was an eye-opener for me in several respects: I learned about syntax and semantics, structural induction, pattern matching, compiler-compilers, and many fascinating facets of computer science. It also helped me to finish my curve-sketching program, but more than half a year too late to make real use of it.

First encounters with the mathematical community

A little later, the first forms of online access became available, opening up new ways to connect with other people. With my 1200-baud modem, for instance, I could dial into the MausNet mailbox MAUS@UN. All of this eventually led me to study computer science at the University of Dortmund.

In the first semester, we learned Standard ML, a functional programming language; it directly supported structural recursion and pattern matching, and writing a correct program amounted to writing an inductive proof that the program was indeed correct. I chose psychology as my subsidiary subject, with the idea that I could specialize in artificial intelligence later.

For mathematics, in particular linear algebra, we had a specialized classes in the computer science curriculum. It was taught by Eberhard Becker, who chose topics close to the standard mathematics curriculum, but with a bit more emphasis on structural, algebraic, and algorithmic aspects. I loved going to these lectures.

Becker’s lectures were perfect performances: I learned beautiful mathematical content presented in a way that was similar to a theatrical performance; but it was neither a drama nor a comedy. It was a celebration of beauty in which everything, the voice, the handwriting, the movements, came together to create this effect.  Later I found out that this kind of performance is typical of mathematicians who are passionate about their subject.Beyond this, the lectures helped me to gain a better understanding of the mathematics of computation.

The tutorials accompanying Becker’s lectures added another dimension. The tutors were young, cool, heavy-smoking, and knowledgeable, and we, the students, got the feeling that mathematics is quite democratic: especially when working together in teams, contributions from all sides are welcome.

In the introductory lecture on theoretical computer science, we discussed questions of complexity, also touching on Gödel’s incompleteness results: What makes a (computational) problem difficult or even impossible to solve? The notion that one can reason about computational complexity made another deep impression on me.

Of course, the university had a library. Somehow, the library of the University of Dortmund was extraordinarily well equipped, and even students could borrow all kinds of books for long periods. I made extensive use of this opportunity. Through reading these books, I got a sense of how deep modern mathematics is, how it is rooted in more than 200 years of history, and how there has been steady progress.

I reserved Sunday evenings for writing down the solutions to the exercises. To get into the flow, I listened to the offbeat mix of independent music, drum and bass, and electronic sounds on the radio show 1 Live Fiehe (formerly Raum und Zeit). This habit of doing mathematics on Sunday evenings and getting into a flow is something I have tried to keep up over the last thirty years.

Overall, I think this combination of theatre, performance, beauty for all the senses, empowerment, depth, democracy, teamwork, and history made me want to become a professional mathematician.

The next level

What will change for me if AI becomes superhuman at solving mathematical problems? Not much$\ldots$ The things that drew me to mathematics (beauty, understanding, and working with others) would still draw me to it today. Maybe it will become even more exciting. We are getting power-ups, and we are getting the chance to move to the next level.

My high school dream of making mathematics mechanical is materializing on a grand scale. Mathematical theorems might be degraded to become data points in the landscape of mathematical theory.

Before AI, it took a great deal of human labor to obtain these data points. Now AI provides a proof factory. Proofs are becoming relatively easy to produce, especially those lying in the convex hull of existing mathematical knowledge. But the laws of computational complexity still apply!

Behind such a data point there is a mathematical proof. Maybe even a complete “space” of proofs. Over the next few years, I want to explore this space of proofs.

Talking to LLMs, I have the feeling that finding a proof is like solving an optimization problem. By prompting an LLM, one explores a relaxed space of candidate proofs from the outside. This space contains both feasible points—correct proofs—and infeasible points—arguments with gaps or errors. Through (auto-)formalization, one then tries to enter the feasible region by turning candidate arguments into formally verified proofs. Ideally, this interaction produces a feasible solution, a first formally verified proof.

But the work does not end with a first proof: Which proofs provide human insight? Which are particularly elegant? Which are from the BOOK? What is the shape of the space of all proofs? Where are the bottlenecks? How robust are certain parts of arguments? Can one extract useful pieces from a proof as lemmas? How does one move forward to develop a new theory?

I am confident that asking and answering these questions will take our mathematical understanding to a higher level. This deeper understanding will be reflected in the intensity of the data points representing theorems. Some data points will glow only faintly, while others will shine brightly because there is a proof that is understood, admired, and internalized by many humans. It is a privilege of our profession that, in the end, it is mainly this brightness that matters.

Again, machines will help us answer these questions, and I wonder which power-ups we will have to invent and what the next level will be. I want the exploration of this next level to be, once again, a democratic adventure, open to everyone who wants to contribute and respects the values of mathematics. I believe that we, as a mathematical community, must build our own independent AI infrastructure. If mathematical exploration is to remain open, we must retain some control over our tools, training data, and standards of verification.

This will require more teamwork than ever, and for precisely that reason, I am sure that it will be fun. And if we, as teachers, have good reason to be passionate about mathematics, our enthusiasm will be infectious: students will catch it, follow us, and take over.

I thank Michael Emmerich, David Gross, David de Laat, Philippe Moustrou, Fernando Oliveira, Achill Schürmann, for feedback on an earlier version of this essay.

Acknowledgements: I used ChatGPT to help polish the English. The personal experiences and views expressed here are my own.

1. Of course, there were already computer algebra programs, for instance Derive, which could do this, but I wasn’t aware of them. ︎

Received 10 September 2026. The author wrote the first draft on 6 September 2026.

In response to an exponential growth of submissions, the arXiv just announced changes to its rate limiting policy, which among other things restricts submissions to at most two a month for each submitter.

Eric Dolores-Cuenca, Postdoc at Yonsei University

The purpose of this letter is to warn the mathematical community of a new type of paper mill that could potentially appear as a consequence of the advances in artificial intelligence.  I am a postdoc working on Math and AI, and I organize an international online seminar on math and machine learning.

Firstly, there exist groups that produce fake research papers and sell them to scientists; these groups are commonly called paper mills. According to this study, at least 2% of all research submissions in all sciences are submitted by paper mills.

Secondly, there also exist start-ups, companies and universities working on solving mathematical problems with machine learning. In particular, some use tools that take information from a mathematical problem and return, when possible, experiments to prove it. The level of human interaction varies: in some cases the user has to provide only a text description of the problem; while for other programs, the user must write code that evaluates the fitness of the solutions proposed by agents.¹  AI agents are also capable of transforming the output of those experiments into a written research paper. While it is not clear how much interaction was needed to produce the preprint about the Navier-Stokes problem or to solve the Jacobian conjecture, in this post I want to give emphasis on the possibility of an artificial intelligence solving completely a problem only using a prompt as input.

With these two antecedents, I would like to ask the reader: what would happen if a group trained an algorithm to recognize important mathematical questions? The algorithm can work in parallel and mass produce meaningful questions. If we now give this list of questions as inputs to algorithms that try to solve them, and we furthermore assume that they manage to solve a small percentage of the input questions, then the final products are research papers that are publishable since they have solved meaningful questions for mathematicians. Finally, what if the company sells those finished results to mathematicians? This would create a paper mill that produces correct papers and sells the authorship. Moreover, if an AI that solves problems has a certain level of quality, an AI that formalizes the proof can further filter the papers.

One can wonder what the danger is if paper mills are producing knowledge, that is, publishable papers. Fraudulent credit is one of the problems; affecting hiring, promotion, grants, and reducing the credibility of the publication process.

In informal conversations, I asked some of the experts who attended the Mathematics × AI: Challenges and Opportunities event at the Korea Institute for Advanced Study and the International Conference on Machine Learning 2026 about the possibility of this new type of paper mill. Some of the people I talked to are collaborating with AI companies, while others were participants with different levels of experience in AI.

While I personally expected to be assured of the impossibility of this new type of paper mill, some of the experts considered this scenario plausible within a few years, and they could not think of a solution for journals to detect this problem. Some told me they have students working on an algorithm to predict important problems. I was encouraged by them to keep talking about this problem in order to raise awareness, which led me to write this letter.

Why would this be a very difficult problem for journals?

From the point of view of the journals, there are no steps in the review process to ensure that the person who submitted the paper did not buy the paper from a paper mill, or to certify that the person is aware of the content of the paper. Note also that using a detector to see if a paper is written by AI is useless: a person can buy finished papers and rewrite them.

We are now seeing talks in which people used AI in their work. Note that the quality of the talk may not be related to the use of AI. The speaker could be afraid of public speaking, perhaps the speaker does not have enough experience giving talks, faces a language barrier, or is new to the topic and knows just the minimum required for the result to hold.

I call on the community to discuss what changes journals would need to implement if we ever reach that level of technology.

Acknowledgement: I would like to thank Susana Lopez Moreno for related discussions.

1. The following examples, listed alphabetically, are not exhaustive: Aletheia, AlphaEvolve, Astra, Iteris, OpenEvolve, Rethlas, and ShinkaEvolve. ︎

Received 11 September 2026.

Manuel Rissel, tenure-track faculty at ShanghaiTech University

With the onset of “LLM driven math turbulence”, various guiding principles that seemed for granted are being shaken up a little. I can imagine that today someone could upload a new preprint introducing a rich theory that prepares all kinds of exciting unknown terrain for future exploration, and on the next day, before a word was talked about that theory with anyone, LLM driven agents may already have analyzed and used material from that preprint in the process of generating proofs to a list of conjectures which someone else (or some company) requested from a chatbot. Whether this would be perceived as good or bad likely depends on who one asks under which circumstances, but it seems natural to wonder: would the only guaranteed way for ruling out such a scenario (if one wanted to) consist of not uploading new research as preprints? And,

would a majority of mathematicians like their preprints (including the LaTeX files) being crawled by arbitrary automated robots?

There is hardly one answer fitting all or a majority. For instance, researchers may enjoy seeing their not yet peer-reviewed proofs appear in LLM generated answers. Some may want to actively support LLM driven research or at least have it stabilized at an equilibrium as quickly as possible. Others may not want to contribute new ideas immediately to automated LLM driven research or to external companies with other interests. Also, there can be nuances, as some may be fine with sharing titles and abstracts of preprints with robots, but would like to reserve the proofs mostly for mathematicians during the peer-review stage.

Nevertheless, there appear not many options on the table, as the preprint server mostly used by the math community operates based on principles such as the open distribution of knowledge, and openness is something that hardly anyone would like to change. But what means open? How to choose the topology that’s best for sustained mathematical science development? Is there a natural mechanism that ensures sufficient openness of preliminary research findings but still provides a new lever for the community to influence slightly the balance between the amount of digested mathematics and the number of math manuscripts?

One hypothetical thought could be that some authors may choose to encrypt their freshly indexed preprints, keeping the title and a possibly long abstract available to the public, with automated decryption of the main body to occur after a well-defined embargo period, e.g. 1-2 years (or any time scale larger than that of LLM development). Such a preprint would be compiled by the server after upload and then encrypted, retaining a readable copy only for private access or moderation. It would be time stamped and versioned, just like done by the arXiv right now, hence the authors could without worries start talking to others about the novelties of their new theory and send readable versions to journals and colleagues. To prevent meaningless encrypted preprints, there could be requirements for using this particular service: e.g., a limit of encrypted uploads per verified researcher per year. Other researchers interested in learning about the details contained in an encrypted preprint could invite the authors to visit them and give a talk, invite them to online seminars, engage in private exchange$\ldots$ or to take this thought ad absurdum, someone could prompt an LLM$\ldots$

The drawbacks of voluntary encryption of preprints may outweigh the positives, but it could still be valuable to debate about the role of preprints in a changing research environment. Our way of handling preprints potentially influences peer-review, knowledge digestion, and the ability of external forces to act on our profession.

Received 8 September 2026.

Snorre H. Christiansen, Professor at the University of Oslo

The EMS should establish a structure to shepherd the use of artificial intelligence in mathematics research. First version sent to the EMS on 20 September 2026.

A European institute for AI enhanced research in mathematics

This open letter is a response to the latest developments in AI (including contributions to the Millennium problem on Navier-Stokes) as they impact the mathematical community.

My impression is that many research mathematicians would feel a loss of meaning if the most advanced research were to consist mainly in efficient prompting and agent management, especially if these tools are outside of our control. This sense of loss compounds with scepticism about where the broader society is driven, by the interests of a handful privately owned outsized AI companies. What’s more, in most cases these companies are subject to politics and jurisdictions that we can hardly count on swaying.

I believe that the mathematical community is well placed to develop its own AI models, to regain control over its destiny. The community has competent and motivated manpower, especially if research as we knew it is upended, as seems to be already the case. I am sure that mathematicians can develop the AIs that are best at doing maths and that they are best placed to help regulate the use of this tool, notably in regards to the education and recruitment of future generations of mathematicians. Mathematical thinking is as needed now as it has ever been.

This communal AI could take the form of an institute owned by EU/EEA through the European Research Council (ERC), governed by the EMS and partnered with promising European AI companies for mutual benefit (I have noted that Mistral has developed Leanstral).

It would be subject to European law, contribute to European sovereignty and would give European mathematicians a hand on the steering wheel in matters that are important to them. Communal values have been expressed in particular in the Leiden declaration. The declaration by Fields medalists relates our concerns to the ones expressed in other arts and sciences. The initiative should be mindful of the solidarities felt by mathematicians across continents.

Mathematicians have always had a broad spectrum of views on matters of ethics, especially regarding military technology. I suppose this will also be the case regarding AI. I understand that some will prefer to stay away from AI all together, and that’s fine. Protecting human agency in mathematics can take many forms.

This initiative would provide a structure that gives willing mathematicians a possibility to defend their values and regain a sense of control over their destiny. If prompting is the future, I for one would like to prompt inside a structure I can support. This way we can at least try to align future AIs with the interests and sense of dignity we defend as mathematicians and as human beings united by the Universal Declaration of Human Rights.

Traditional ways of doing maths would coexist with AI enhanced mathematical research developed in house and whose findings are shared on a need to know basis and published according to consensual protocols as collective endeavors (high energy physicists working at CERN have more experience with this).

I would also suggest that the EMS establish a Millennium type prize related to AI. It could be for instance to explain mathematically why deep neural networks work so well on some problems otherwise considered intractable (due for instance to the curse of dimensionality). This would give the EMS an occasion to communicate to the greater public about the role of mathematics in modern technologies.

Yours sincerely,

Snorre H. Christiansen

Received on 22 September 2026.

A brief post to note two reports that just came out:

• The Advisory Group on Mathematics and Artificial Intelligence (AGMAI) has posted its general recommendations to AI companies with regards to generation of solutions to mathematical problems.
• A working group report “AI and TCS: The Next Six Months” arising from a two-day event at the Simons Institute for the Theory of Computing.

[This is a guest post by Rachel Webb. This blog post was initially written in a different file format and converted using AI. — T.]

The way I do math experienced a great upheaval once before, halfway through my PhD. I was in a seminar with a group of graduate students and faculty developing new discipline-specific writing courses. The facilitator asked the question,

What does it mean to write well in your discipline?

Then, as now, I use writing as a proxy for doing. As a mid-career grad student, I had a quick answer to this question: good mathematical writing, hence good mathematics, is clear and correct. I was surprised at the dissimilarity between my answer and the answer of the other seminarians (who were more experienced but also other-disciplined): good writing says something interesting.

This discussion unlocked for me the meaning of doing research. Doing research is searching for something interesting to say. (Of course, once you find it, you have to say it in a way that is clear and correct.) In mathematics, the search for something interesting is often driven by our search for understanding, as noted by Thurston and endorsed by many since. But I think the math research process has a creative aspect that is not completely captured by this search. By contrast, the task of making something interesting to say is like writing a novel whose characters are at once subject to constraints of human experience but also presented in a way that comments on human experience. The characters in mathematical novels are definitions, the story text is the lemmas and theorems; these are constrained by truth, but can also be chosen strategically or artistically to capture certain facets of truth.

This is, abstractly, how I currently approach math research: I seek to understand what is happening at a fundamental level, yes, but I don’t think there is a unique way to understand. I also seek to take the things I do understand—inevitably, these are just a subset of the phenomena I would like to understand—and craft a narrative from them that is beautiful and interesting to my fellow mathematicians. Concretely, I find that “interest” in a mathematical context often derives from applications, either to the real world, or to other math. It also derives from proximity to high-profile open problems that serve as centers of mathematical conversations.

I don’t see LLMs as changing that approach much, but I expect they will drastically affect how I execute it.

The execution changes because now I have access to a machine that has read all the books and knows how many standard lemmas go. This speeds up the research process immensely and turns some of my lands of mathematical fantasy into worlds I can realistically start exploring (dream bigger dreams, says Antieau). Of course, taking the interstate instead of the side roads has its tradeoffs, but for any given leg of the journey I can choose which route to take. I will return to this idea in a moment.

My approach to research does not change because critically, I’m not convinced that the advent of LLMs changes my metric for mathematical “interest.” If we have historically regarded a paper as interesting if it comments in some way on a Millennium (or similar) problem, need we lessen our interest if the problem has a solution? Certainly there are many solutions, even many shards of understanding that do not constitute full solutions. These are all very interesting. We might know how to get to the red city, but we can continue to map out the surrounding terrain, now all the more valuable for the economic and strategic opportunities arising from metropolitan proximity.

I have said something about how I will continue to do math research, but I have not discussed why: what reasons do humans have to do math, if LLMs are capable (hypothetically, say) of producing math that is even more interesting than the math we can create? I believe that there are economic reasons, but I will not discuss those (both Sahai and Strogatz-Townsend have some thoughts). Instead I present two humanistic reasons. Neither of these is unique to math, just as math is not the unique human practice whose execution is affected by the advent of AI.

The first reason for humans to do math is that math is interesting to us individually. I enjoy doing math, so I will keep doing it, even if machines are better at it. There is a threat that LLMs will take the fun out of doing math by tempting us towards knowing the answer over understanding the solution. The temptation must be resisted. I should use AI only in ways that make math research more enjoyable for me and allow me to get understanding along with my wisdom (Proverbs 4:7). This may require experimentation. In fact, I and others already make analogous choices to use technology only when it is helpful to us personally and not every time it is economically “correct.” For example, I persuade my kids every week to walk seven miles round trip to church, even though we could drive. The commute costs well over two hours, but the increase in understanding between family members from the shared time and suffering is worth it.

The second reason for humans to do math is that math holds shared interest for multiple people at once, and in this way it creates communities. In some sense, this is what the other-disciplined seminarians meant when they said good writing is interesting: they meant that good writing is interesting to other people and hence is a piece of a larger conversation. Math research is a tool for drawing people together, in student-teacher relationships, in collaborations, at conferences, and at department colloquia and tea-times. Again, AI could weaken these social bonds by making (fear of) scooping more common, or just by making it easier to ask a machine than a colleague. But could does not automatically imply will. To quote Wendell Berry, may the age of AI be the age of knowing our mathematical neighbor. “Friends, every day do something that won’t compute.”

Or click here if the above doesn’t work.

Recorded in-person in my office at UT Austin, with a bulleted list containing “ARC,” “Scalable Oversight,” and “Models” behind me on my blackboard for some reason (I no longer remember who put those there or why). 90 minutes long. Sometimes you see my disembodied arm waving in midair because of the way the cameras are combined. As always, I strongly recommend 2x speed for the correct experience.

This might actually be one of my best podcasts ever, although I wasn’t planning on that! Thanks so much to Bhavay Tyagi and Prachi Garella for driving all the way from Houston to record it.

Here’s a strict subset of the topics we covered:

• The story of AI models solving the Navier-Stokes Millennium Problem, insofar as it’s known
• Can recent AI proofs be called “truly creative”?
• The history of AI before the LLM revolution
• What do we mean when we call LLMs “black boxes”?
• The achievements of the field of interpretability
• What exactly happened in the OpenAI/HuggingFace incident
• Must we avoid all “anthropomorphizing language” when discussing the HuggingFace incident? (spoiler alert: no)
• Examples of major open problems in quantum computing theory that I cared about for decades and that AI models have recently solved
• Effects of the current AI cataclysm on the math community, especially students
• What annoys me the most when I listen to AI talks
• My experiences at OpenAI, why they hired me, and the watermarking work that I did there

Enjoy!

More AI-related content coming soon, as this blog—like much of the rest of the world—continues its transition to “all AI, all the time” (except still 100% written by an aging, deteriorating biological brain)

And for those who just can’t get enough of my rocking back and forth, using too many filler words, as I explain theoretical computer science! Here’s a second podcast, this one mainly on quantum computing, with Seb Agertoft, who I thank for doing it. Enjoy!

Several science items of interest from recent weeks.  As always, apologies for missing some, which I'm sure to have done.

• A month ago, Premi Chandra, Piers Coleman, and Clare Yu published a really nice memorial biography of Phil Anderson, one of the great scientists and characters of 20th century physics.
• A few days ago Dam Son, an outstanding condensed matter theorist at U Chicago, posted a paper on his site that resolves a math issue that had been lingering related to a well-known paper from back in the heyday of anyon superconductivity.  The authors had always been afraid that their approach accidentally violated a sum rule.  It turns out, the authors had just misplaced a factor of 1/2, and their approach was actually exactly right.  The really novel bit here is that the paper is written as if it is single-authored by Claude, and acknowledges Dr. Son for prompts that led to the result.  See here for an interesting twitter thread on this via Sankar Das Sarma.  
• Tangentially related, Anthropic and Matt von Hippel achieved a new result, a 9-loop perturbation theory calculation based on N=4 super Yang-Mills theory.  These kinds of mathematically virtuosic calculations are exactly the sort of task that AI tools to which are extremely well suited now.  
• Three weeks ago, a neat paper appeared in Science Advances.  These folks used quantum interference of a Bose-Einstein condensate to test the equivalence principle in a quantum system.  The basic idea is, you take an ultracold atomic gas in a well-defined quantum state.  You split it and toss one component of it upward, while you hold the other component of it fixed in the lab frame. (This work is a descendent of this approach, which was happening in the lab above mine in back when I was in grad school.) The upward-thrown component accumulates quantum phase as it rises in the lab gravitational field, slows to a stop, and comes back down.  There is a specific amount of phase difference between the two components predicted by the equivalence principle (which says that inertial mass and gravitational mass should be exactly the same), and that's what the authors find.
• A couple of weeks ago, this paper appeared (shoutout to the first author, who is a Rice undergrad alum).   The authors fabricated a nanomechanical resonator made out of LiNbO3 and coupled it to a qubit to act as a measurement device.  Remarkably, through the qubit they are able to see transitions between individual vibrational quantum numbers in this many-atom mechanical resonator.  This is quite impressive, and it opens up many possibilities for transduction between different quantum degrees of freedom.  

Rudy Marcus was 91 years old when I took a course from him. His age was, arguably, not his most remarkable quality.

The course took place during the second spring of my PhD. I was completing not only the academic year, but also my course requirements. Why not, I figured, indulge in traditional statistical mechanics? Statistical mechanics neighbors, and partially overlaps with, thermodynamics. Modern approaches to statistical mechanics include fancy toolkits imparted in physics courses: the renormalization-group method; Ginzburg–Landau theory; and the latter theory’s successor, topological order. I craved a more traditional approach, as won’t surprise readers of this blog. So I ditched the physics course list and signed up for Chemistry 166.

Rudy didn’t disappoint. He began the course with a tradition passed down from a founder of thermodynamics: Boltzmann’s H theorem. The theorem amounts to the first argument ever published for the second law of thermodynamics: every (sufficiently large) closed, isolated system’s entropy increases or remains constant; it doesn’t decrease. Rederiving Boltzmann’s H theorem, I felt as I would while holding a lace collar worn by Queen Elizabeth I: as though I were touching history. 

Rudy proceeded through more of the greatest hits in statistical mechanics. For instance, the Fokker–Planck and Langevin equations describe random processes. Examples include the subject of one of Einstein’s most famous papers: the motion of a pollen grain suspended in a liquid and buffeted by the liquid’s molecules. Chemistry 166 provided much of the bread and butter, meat and potatoes, and even tofu and edamame of a thermodynamicist’s education.

Rudy advanced step by step, rederiving every result in class. He recommended a textbook; but I refer to my lecture notes, rather than to the text, to refresh my memory about course topics nowadays. When I encounter complex integrals (a certain type of calculus) in statistical mechanics, I think back to Rudy’s treatment of them. The course also crystallized my understanding of the fluctuation–dissipation theorem, which describes systems perturbed slightly out of equilibrium; examples include a magnet subjected to a weak magnetic field.¹ For the course’s final project, I studied quantum master equations—which featured in a Quantum Frontiers post afterward.

No less than his explanations of science, Rudy’s enthusiasm for science impressed itself upon me. Again, he taught Chemistry 166 at age 91. He continued to collaborate on research. Whenever I saw him, even after the course ended, he asked for news about quantum information theory—which he didn’t work on. 

Two and a half years after the course, Rudy gave a speech at a dinner in honor of a couple’s engagement. The groom was pursuing a PhD in applied physics, and the bride worked as an engineer. The groom asked if Rudy had advice for young scientists. Rudy responded immediately: plenty of science remains to be done, so keep exploring.

During his speech, Rudy described his upbringing in Montreal. His high school had produced a “who’s who” of Canadians, in his words—and not because the school demanded high tuitions or offered tutors and social connections. Discrimination pushed Jewish working-class children out of other schools and into Baron Byng, where some students worked hard enough to thrive. Alumni include artists; judges; physicists; William Shatner, who played Captain Kirk in the original Star Trek series…and Rudy Marcus.

Which leads me to the quality arguably more remarkable than Rudy’s teaching Chemistry 166 at age 91: he’d won a Nobel Prize in chemistry. Unlike most chemists, Rudy worked as a theorist, rather than an experimentalist. He developed a framework now called Marcus theory. It predicts the rate at which electrons hop between molecules. I can’t tell you much more about Marcus theory because I don’t know much more about it: Rudy had enough humility that, to my memory, he mentioned his theory only once in class, in passing.

Although I know little about Marcus theory, I’ve seen much of Marcus practice. In 2022, I emailed Rudy to ask where a holiday card could reach him. The pandemic had isolated enough people that I’d determined to reach out to more acquaintances. Rudy replied, “I have been working at home, meeting twice a week on zoom with my now small research group.” He was 98 years old. His curiosity, learning, mentoring, and exploring continued.

Rudy passed away this July, five days before his 103rd birthday. I hope to learn Marcus theory someday. But I’d be even more grateful to undertake Marcus practice for even a decent fraction of the time for which he did.

¹How rapidly does the system respond to the perturbation? This nonequilibrium response depends on a correlation function evaluated on an equilibrium state, according to the fluctuation–dissipation theorem. To understand why, imagine preparing the system in a thermal state, perturbing the system at an early time, and measuring the system later. What information can you extract about the perturbation? A two-time correlation function encodes this information. Imagine Fourier-transforming a two-time correlation function. The Fourier transform is proportional to a second derivative of a free energy, according to the fluctuation–dissipation theorem. (The second derivative should sound plausible because it depends on a two-time correlation function.) Second derivatives of free energies equal response functions, such as the magnetic susceptibility. Therefore, a nonequilibrium response depends on an equilibrium correlator.

Freshman year of college, I took a writing seminar from German-literature professor Ellis Shookman. Professor Shookman loved Mozart’s music, he told us early in the term. He listened to Mozart on the radio while driving from campus to Boston. Static might mar the transmission, but he could often turn up the volume and continue enjoying the program. Sometimes, the static worsened during the drive. It could worsen and worsen, until Professor Shookman’s frustration outweighed his delight at listening. He’d switch off the radio.

As Professor Shookman loved listening to Mozart’s music, he loved reading about students’ ideas. Yet static can mar a piece of writing: infelicities in grammar, structure, composition, word choice, and more. If enough infelicities obscure the writing, the frustration of reading outweighs the benefits. Professor Shookman will quit reading.

Professor Shookman marked up our essays with a blue pencil that achieved the status of legend among his students. If you’ve written a paper I’ve coauthored, you’ve probably received PDF drafts replete with green highlighting.¹ A sticky note explains the reason for each highlighting: “Singular–plural mismatch.” “Active voice >> passive voice.” “Let’s clue the reader in as to this formula’s meaning before lobbing the math at them.” 

Over the past year, I’ve catalogued the suggestions I write most often on paper drafts. The comments embody principles gleaned from Strunk and White’s The Elements of Style; the Physical Review style guide; other writing guides I esteem; literature whose writing I esteem;² collaborations with professional editors; and writing instructors, including Professor Shookman. Each section below begins with more-important principles, shading into more-nuanced ones.

Please use and disseminate these principles. Train your favorite large-language model (LLM) on them, and have the LLM critique your manuscripts. Instruct it to use green highlighting if you wish. Even if the LLM suggests fixes initially, tell it to stop offering suggestions later, so that you can devise the solutions: train not only the LLM, but also yourself. I hope to enjoy your papers as much as Professor Shookman enjoyed his sonatas.

1. Organization
   1. Motivate your work; then, present it; and then, explain its physical significance.
   2. Begin each paragraph with a topic sentence.
   3. Begin each section, apart from the introduction and conclusion, with (i) a statement of the takeaway and (ii) an outline of the section. When outlining a section, hyperlink to each subsection. Similar guidelines concern subsections and subsubsections.
   4. Before presenting a piece of math, sketch its meaning and origin. This strategy enables the reader to understand the math as soon as they encounter it. If you throw math at the reader without introducing it, the reader will have to squint at the symbols for a while to figure out what the expression means and where it comes from.
      • Example: To calculate the average work, we substitute the Hamiltonian formula (10) into the definition (12): [equation].
   5. Most citations belong at the ends of (i) sentences and (ii) phrases concluded with commas. Put a citation elsewhere only if you have a compelling reason for doing so.
   6. Bridge each component of your writing to the next component; the next shouldn’t sound like a non sequitur.
      • Suppose that the next sentence refers to (i) a topic mentioned in the previous sentence and (ii) a new topic. Mention (i) before (ii).
        • Example: Smith et al. applied control theory to the extent possible. The attempt led to intractable equations, unlike our approach.
        • Example of a broken bridge: Smith et al. applied control theory to the extent possible. Our approach does not involve intractable equations, unlike theirs.
   7. Whenever you tell a story, tell it from start to finish, step by step. Derivations, proofs, and descriptions of experiments qualify as stories.
      • This guideline extends to descriptions of experimental setups and of mathematical objects. For example, imagine referring to an element of a subgroup of the group generated by some operators. Did you have to read the preceding sentence multiple times to process it? The sentence begins at the end of a story, then rewinds to the story’s beginning. This structure impedes understanding. The subgroup forms the context for the subgroup element, which one can’t grasp until hearing about the subgroup. The subgroup participates in a similar relationship with the group, as does the group with its generators. Therefore, one should introduce the generators, then the group, then the subgroup, and then the subgroup element.
2. Word choice
   1. Use strong, specific words, rather than weak words.
      1. Verbs and nouns are stronger than adjectives and adverbs.
      2. Choose specific verbs (e.g., “prepare,” “evolve,” and “measure”), rather than vague, general verbs (e.g., variants of “to be” and “take,” as in “take a measurement”).
   2. Avoid statements such as “we investigate,” “we study,” and “we analyze.” Such statements don’t relate that you’ve accomplished anything. State what you’ve accomplished. Verbs such as “prove,” “test,” “confirm,” “discover,” and “find” achieve this goal.
   3. Adverbs such as “importantly” and “remarkably” pollute scientific writing with the authors’ opinions. Demonstrate that a claim is important or that a result is remarkable; then, leave readers to draw their own conclusions. Those conclusions will coincide with yours if you’ve demonstrated your point.
   4. Use the active voice, rather than the passive voice. Take responsibility for your work. Editors of high-impact scientific journals have endorsed this advice.
   5. Refer to yourself when necessary and only when necessary.
      • Example of unnecessary reference to self: We use the superscript “max” to signify the maximal Fisher information.
        Preferable alternative: The superscript “max” signifies the maximal Fisher information.
      • Example of unnecessary reference to self: Our results establish several opportunities for future research. First, we can implement the experimental proposals.
        Preferable alternative: Our results establish several opportunities for future research. First, one can implement the experimental proposals.
      • You may use the first-person plural when escorting the reader through a derivation.
      • Example: We substitute from Eq. (1) into Eq. (2).
   6. If you’re the only author, don’t use the plural (“we,” “our,” etc.). The usage is inaccurate and misleading. It portrays you as dodging responsibility for your work by dispersing that responsibility across the scientific community.
   7. Avoid dangling modifiers.
   8. Pair every verb with the appropriate noun.
      • Example of grammatically incorrect text: Equation (1) follows by calculating the sum.
        • One should pair the verb “calculate” with the noun “we,” because “we” undertook the calculating. However, this example’s author omitted the noun out of squeamishness about using the first person in a scientific document. Hence the sentence says that the equation calculates the sum. Equations can’t calculate sums.
      • Examples of correct alternatives
        • We derived Eq. (1) by calculating the sum.
        • Equation (1) follows from the evaluation of the sum.
        • Calculating the sum yields Eq. (1).
   9. Avoid empty subjects.
   10. Include no unnecessary words.
       1. “So-called” is unnecessary.
       2. “Note that” and “We note that” are unnecessary.
       3. Several phrases often preface mathematical statements but are unnecessary: “we have,” “we have that,” “it holds that,” and “it follows that.” One can better serve the reader by prefacing the mathematical statement with (i) a derivation or (ii) a prose description of the statement.
       4. Never write “is equal to”; “equals” is more concise.
       5. Never write “is able to”; “can” is more concise.
       6. Never write “gives an upper bound to” or “places an upper bound on”; “upper-bounds” is more concise. Analogous statements concern lower bounds.
       7. Never write “a large number of”; “many” is more concise. Never write “a small number of”; “few” is more concise.
       8. Never write “We refer to [symbol] as [name]”; “we call [symbol] [name]” is more concise.
       9. Never write “as long as”; “if” is more concise.
       10. The symbol means “greater than”; and , “greater than or equal to.” Don’t translate into “strictly greater than”; the “strictly” is unnecessary. Analogous statements concern and .
   11. Avoid contractions, which are too informal for professional writing.
   12. The possessive is not a contraction and belongs in professional writing. It facilitates conciseness.
   13. Use the word “for” only when it belongs. Physicists often write “for” when they mean “if,” “per,” “at,” or something else.
       • Example of inappropriate use: The function vanishes for odd arguments.
         Corrected statement: If the argument is odd, the function vanishes.
       • Example of inappropriate use: We performed 10 trials for each parameter value.
         Corrected statement: We performed 10 trials per parameter value.
       • Example of inappropriate use: The function is smaller for small x values.
         Corrected statement: The function is smaller at small x values.
   14. Write “we evolve the state,” “we measure,” etc. only if you’re an experimentalist who undertakes those actions. Alternatives include “Consider measuring,” “Suppose the system evolves,” and the command tense (e.g., “One can measure this quantity as follows: prepare the qubit in . Evolve it under …”).
   15. The condition defines a regime, not a limit. The conditions and define limits and are inequivalent to .
   16. Write “first,” “second,” “last,” etc., not “firstly,” “secondly,” “lastly,” etc. (I defer in this matter to The Elements of Style.)
   17. Humans can assume, suppose, etc. Mathematical expressions, protocols, etc. can’t.
   18. One multiplies factors together and sums terms. Don’t call factors terms and vice versa.
   19. If you mean “X equals Y,” say so. Don’t write “X agrees with Y,” “X matches Y,” or “we identify X with Y.” The latter three phrases are vaguer, and two of them contain more words, than “X equals Y.”
   20. Regarding the words “general” and “generally”:
       1. A general object subsumes every example of that object. If any example behaves unlike a supposedly general object, don’t call the object general.
       2. Many claims contain the term “general,” “generally,” or “in general” but don’t need the term.
          • Example of a sentence that contains “generally”: The terms generally commute.
          • Equivalent, more concise sentence: The terms commute.
       3. Physicists tend to use the words “general” and “generic” differently. By “general,” physicists usually mean “subsuming every example.” By “generic,” we usually mean “typical,” or “common.”
   21. “Then” makes sense (i) in discussions of chronology and (ii) in if–then statements. Don’t use “then” outside these contexts.
       • Example of inappropriate use: “Define Then Y.”
       • Examples of appropriate alternatives
         • Define This definition implies Y.
         • If then Y.
         • Define such that Y.
   22. Don’t justify any equation with “we used that [such-and-such is true],” which violates the rules of grammar. Grammatically correct alternatives include “We applied [a property],” “The equation follows from [a property],” and “…since [such-and-such is true].”
   23. Regarding tense:
       1. When describing what you’ve accomplished, use only one tense.
       2. Experiments happened in the past, so describe them in the past tense.
       3. When describing a proof’s steps, use the present tense.
          • Example: We Taylor-approximate the function about . Substituting into Eq. (1) yields [equation].
   24. Nouns, verbs, and adjectives should agree about whether a quantity is singular or plural.
       • Example of singular–plural mismatch: The equations are a rule for evolving the cellular automaton.
       • Example alternative: The equations form a rule for evolving the cellular automaton.
   25. “Admit of” means “allow for,” or “permit.” The phrase needs the “of.”
       • Example: The formula admits of the following interpretation.
3. Punctuation
   1. Consider any list that contains at least three items. If no item contains a comma, separate the items with commas. If any item contains a comma, separate the items with semicolons.
   2. In American English, periods and commas belong inside quotation marks. (Example: She told me, “Have a good day.”) In British English, periods and commas belong outside quotation marks. (Example: She told me, “Have a good day”.)
   3. To write quotation marks in LaTeX, don’t use your keyboard’s quotation-mark key; use the appropriate keys.
   4. Regarding hyphens:
      1. The hyphen (-) feeds into punctuation of three types: the hyphen (-), the en dash (–), and the em dash (—).
      2. The hyphen appears in some compound words, as in “non-negative.”
      3. In American English, em dashes can separate ideas within a sentence. Don’t separate any em dash from neighboring text with a space.
         • Example of appropriate use: The sample—the only product of this experiment—barely survived.
         • Example of inappropriate use: The sample — the only product of this experiment — barely survived.
         • Example of appropriate use: He told me only one sample had survived—hardly what I wanted to hear.
      4. This article specifies how to use the en dash. One use is “to separate the names of two or more people used as a compound modifier.”
         • Example: Feynman–Kitaev clock
      5. Hyphenate compound adjectives.
      6. If an adverb ends in “-ly,” it probably shouldn’t precede a hyphen.
         • Example of inappropriate hyphenation: strongly-coupled systems
      7. Follow a prefix with a hyphen if and only if the Physical Review style guide indicates that you should.
   5. A complete clause must follow any semicolon (unless the semicolon separates items in a list).
4. Math
   1. Introduce only necessary notation, which readers will have enough trouble remembering. If a mathematical symbol appears only once, eliminate it. If a symbol appears only twice, try to eliminate it.
   2. Every sentence must obey the rules of English grammar, punctuation, and syntax, regardless of whether the sentence contains mathematical symbols. All math-containing sentences must end with punctuation marks. If a sentence contains a list of mathematical expressions, precede the final expressions with an “and.” If the list contains at least three mathematical expressions, separate them with commas.
   3. Introduce almost every mathematical symbol before you use it. If you introduce a symbol after using it, the reader will encounter the first use, stop, feel confused for a while, tentatively continue, find the definition, return to the earlier use to understand it, and then progress again. This back-and-forth breaks up the reading process. You may define a mathematical symbol after using it only if (i) the symbol is very common, known to nearly all physicists, and unmistakeable and (ii) defining the symbol earlier would disrupt the text’s flow.
   4. If you define a new function, denote it by only one letter. (I defer in this matter to the Physical Review style guide.)
      • Example:
      • Examples of disallowed notation: ,
   5. Suppose that a superscript or subscript stands for a word or phrase without representing any variable or constant. The superscript/subscript must not be italicized. (I defer in this matter to Physical Review style guide.)
      • Example: Let denote the measurement outcome.
   6. If a variable or constant appears in a superscript, parenthesize it. The parentheses communicate that the superscript isn’t an exponent.
      • Example: Let denote the Pauli- operator of qubit
      • If a superscript is not italicized (stands for a word or phrase), don’t parenthesize it.
   7. Regarding the definition of a symbol :
      1. If you write alone on one side of a defining equation, use \coloneqq or \eqqcolon: \coloneqq [expression], or [expression] \eqqcolon . The symbols \coloneqq and \eqqcolon relate more information than does , encoding directionality.
      2. Use if does not appear alone on its side of the equation: [function of ] [result of replacing with its definition in the equation’s left-hand side].
   8. Refer to the Cartesian axes using the formatting “[italicized letter]-axis.” Don’t include any hat, boldface, or \vec symbol.
      • Example: -axis
   9. Avoid denoting any index by , which means to physicists. Use instead, unless you’re writing for engineers (who denote by ).
   10. Don’t use the lowercase letter l (“ell”) as an index; readers might mistake it for a one. Use (\ell) instead.
   11. Give every set-off equation a number. Readers (and coauthors) may want to refer to the equation easily when discussing the paper. Save them (and us) from having to say, e.g., “that equation halfway down page three.”
   12. When writing a set-off mathematical expression, use the align environment, not the equation environment. Using the align environment, one can easily extend an expression across multiple lines.
   13. Regarding a set-off mathematical expression that extends across multiple lines:
       1. Format the expression as follows by default.
          1. Put an & symbol immediately leftward of the first = sign or analogous symbol (e.g., ).
          2. If any subsequent line begins with another = sign (or analogous symbol), put an & immediately leftward of the symbol. (I’ll stop writing “or analogous symbol.”)
          3. Suppose that a subsequent line begins with a , –, , or . Find the symbol immediately rightward of the initial = sign. Begin the new line directly below that symbol.
          • Example:
       2. Modify the default formatting if necessary (a) to reduce the number of lines used in a PRL submission or (b) if the initial = appears awkwardly far to the right.
          • Example of (b):
       3. Suppose a new line begins with a term or factor, such as the in the example under (A). Put the corresponding , –, , or at the beginning of the new line, not at the end of the previous line.
          • Examples of inappropriate placement:
   14. The symbol means “approximately equals”; and ~, “scales as.” Approximations convey more information than scaling relations do.
   15. Use big-O-type notation or ~ symbols, not both; they’re partially redundant.
   16. \ldots, rather than \cdots, should stand in for elements that fit a pattern.
       • Example:
   17. When using \ldots as in the previous rule, present at least two initial examples of the pattern. One can’t define the pattern.
       • Contains insufficient examples: For example, if is odd, then and fit the template.
   18. Parentheses (), square brackets [], and curly braces {} are delimiters. If you nest them, do so in the order dictated by the Physical Review style guide.
   19. If delimiters enclose a symbol, it shouldn’t protrude above or below them (unless the delimiters would have to be grotesquely enormous). Use the \left and \right commands if the delimiters appear on the same line.
   20. An operator isn’t a matrix; a matrix represents an operator in terms of a particular basis. Therefore, no equals sign should interrelate an and a matrix. An arrow can.
       • Example:
   21. Every real number is complex. Don’t say “complex” if you mean “nonreal.”
   22. Consider introducing a mathematical symbol in a prose sentence without using a comma or colon. Put the symbol immediately after the word that names the object represented by the symbol.
       • Example of inappropriate placement: the set of real numbers
       • Examples of appropriate placements
         • the set of real numbers
         • the set of real numbers and
         • Recall the set of real numbers, , in Lemma 1.
5. More mechanics of writing
   1. Use concise sentences, as advocated for in The Elements of Style. The reader can hold only so many ideas in their head at once.
   2. Structure sentences simply, as advocated for in The Elements of Style. The reader should be able to grasp each sentence easily.
      • Avoid nesting ideas within a sentence, to avoid convoluting the sentence’s structure.
      • Example of sentence with convoluted, nested structure: Any model of equilibrium and nonequilibrium behaviors of systems observed in tabletop experiments and high-energy colliders must obey the laws of relativistic quantum mechanics.
      • Visualization of the nesting: [Any model of ([(equilibrium and nonequilibrium) behaviors] of {systems observed in [(tabletop experiments) and (high-energy colliders)]})] must obey [the laws of (relativistic quantum mechanics)].
   3. The ideal paper title has the structure of a newspaper headline: it presents a claim, containing a subject and a predicate.
   4. Regarding abbreviations:
      1. Don’t abbreviate the first word in any sentence.
      2. Abbreviate “Figure,” “Section,” “Professor,” and “Appendix” if such a word appears partway through a sentence.
      3. Don’t abbreviate “Sections.”
   5. Regarding acronyms:
      1. Write every acronym in capital letters, as per the Physical Review style guide.
      2. Introduce each acronym the first time you use it.
      3. Thereafter, use only the acronym, not the spelled-out phrase, throughout the rest of the document’s main text. You may spell out the phrase in section, figure, and table titles if doing so improves the document’s clarity.
   6. Every paragraph should contain at least three sentences.
   7. Wherever you insert a blank line into your LateX code, a new paragraph begins in the corresponding PDF. Insert a blank line only if you wish to begin a new paragraph. This advice applies immediately before and after set-off equations.
   8. Never begin a subsection immediately after a section title. Between the two titles, overview the section. Analogous rules govern subsections and subsubsections.
   9. Put the word “only” in the appropriate place.
      • For example, suppose you’ve sampled data at a point in parameter space and sampled data at no other points. “We sampled data only at ” is correct; “We only sampled data at ” is probably not. The latter claim means that (i) you might have sampled data at and (ii) you did nothing to the data apart from sample it: you didn’t analyze the data, discuss the data, etc.
6. When in doubt, consult the Physical Review style guide or The Elements of Style.
   • If those references don’t contain the information you seek, search for it in online writing guides. Not all such guides have equal merit, however. Lean toward guides written by human editors or published by college writing centers.

¹ Collaborators have wondered why I use green; a student guessed it’s my favorite color. It isn’t; but I bleed green, having graduated from the Big Green, also known as Dartmouth College. Sometimes, I highlight certain pieces of text for one reason (e.g., to point out logical inconsistencies) and other text for another reason (e.g., to point out grammatical inconsistencies). Green distinguishes the first highlightings, while orange distinguishes the second: when not bleeding Dartmouth green, I bleed Caltech orange.

² Don’t learn how to write from physics papers. 

Making final revisions to Don’t Be Too Sure on the Terrace. An all-tuba band on the Terrace Stage playing an all-tuba cover of “Black Hole Sun.”

Sorry for light blogging lately. See “making final revisions” above. Just realized I misspelled the first name of Katharine Briggs, who’s mentioned several times and who I read a whole book about. How many mistakes are left in this thing? (I suppose if I’d kept a rigorous log of mistakes found and plotted it over time, I could curve-fit and estimate this.)

These seven equations between binomial coefficients are ‘coincidences’ — they aren’t among the four known infinite families of such equations:

$$\binom{16}{2} = \binom{10}{3} = 120$$

$$\binom{21}{2} = \binom{10}{4} = 210$$

$$\binom{56}{2} = \binom{22}{3} = 1540$$

$$\binom{78}{2} = \binom{14}{6} = 3003$$

$$\binom{120}{2} = \binom{36}{3} = 7140$$

$$\binom{153}{2} = \binom{19}{5} = 11628$$

$$\binom{221}{2} = \binom{17}{8} = 24310$$

In 1997, De Weger conjectured that every equation between binomial coefficients follows from the four known systematic equations and the seven coincidences I showed you:

• Benjamin M. M. de Weger, Equal binomial coefficients: some elementary considerations, Journal of Number Theory 63, no. 2 (1997), 373–386.

At that time, he and his collaborators checked there were no others involving binomial coefficients less than 10³⁰. Later they checked that there are none involving binomial coefficients less than 10⁶⁰:

• Aart Blokhuis, Andries Brouwer and Benne de Weger, Binomial collisions and near collisions.

So, De Weger’s conjecture stands open. The four infinite families, by the way, are these:

$$\begin{array}{cccl} \displaystyle{ \binom{n}{k} } &=& \displaystyle{ \binom{n}{n-k} }& \qquad 0 \le k \le n \\ \\ \displaystyle{ \binom{n}{0} } &=& 1 & \qquad n \ge 0 \\ \\ \displaystyle{ \binom{\binom{n}{k}}{1} } &=& \displaystyle{\binom{n}{k}} & \qquad 0 \le k \le n \end{array}$$

and the only nontrivial one: the Lind–Singmaster family involving the Fibonacci numbers $F_i$ where $F_0 = 0,\ F_1 = 1$:

$$\displaystyle{ \binom{F_{2i+2}F_{2i+3}}{\,F_{2i}F_{2i+3}\,} \;=\; \binom{F_{2i+2}F_{2i+3}-1}{\,F_{2i}F_{2i+3}+1} \qquad i = 1,2,3,\dots }$$

The first three equations in the Lind–Singmaster family are these:

$$\begin{array}{ccc} \displaystyle{ \binom{15}{5} } &= &\displaystyle{\binom{14}{6}} \\ \\ \displaystyle{\binom{104}{39} } &= &\displaystyle{\binom{103}{40}} \\ \\ \displaystyle{ \binom{714}{272} } &=& \displaystyle{\binom{713}{273}} \end{array}$$

When I said De Weger conjectured every equation between binomial coefficients follows from the four known systematic equations and the seven coincidences, I really meant it. For example, the first of the Lind-Singmaster equations combines with the coincidence

$$\binom{78}{2} = \binom{14}{6} = 3003$$

and other systematic equations to give

$$\begin{array}{cccccccccc} & \binom{3003}{1} &=& \binom{78}{2} &=& \binom{15}{5} &=& \binom{14}{6} \\ \\ = & \binom{3003}{3002} &=& \binom{78}{76} &=& \binom{15}{10} &=& \binom{14}{8} \end{array}$$

In fact, if de Weger’s conjecture is true, the number 3003 shows up as a binomial coefficient in more ways than any other positive integer!

I’ll explain the Lind–Singmaster family later. Here’s the question I’m most interested in now:

Is there any good explanation for the seven binomial coefficient coincidences?

My collaborator Paul Schwahn found a beautiful explanation of the first one, namely

$$\displaystyle{ \binom{10}{3} = \binom{16}{2} }$$

His explanation uses representation theory. The Lie algebra $\mathfrak{so}(10)$ has a 10-dimensional representation, the ‘vector’ representation $V_{10}$, and also two 16-dimensional representations, the ‘left and right-handed spinor’ representations $S^\pm_{10}.$ There’s an isomorphism of representations

$$\displaystyle{\Lambda^2 S^+_{10} \cong \Lambda^3 V_{10} }$$

and similarly for $S^-_{10},$ but we might as well work with $S^+_{10}.$ Here $\Lambda^k$ means the kth exterior power. For any vector space $X$ we have

$$\displaystyle{ \dim(\Lambda^k X) = \binom{\dim X}{k} }$$

Thus, taking dimensions, the isomorphism of representations

$$\displaystyle{\Lambda^2 S^+_{10} \cong \Lambda^3 V_{10} }$$

instantly gives

$$\displaystyle{ \binom{16}{2} = \binom{10}{3} }$$

It is not super-easy to prove this isomorphism of representations, but it’s still nice to find a deeper layer of meaning underlying what might otherwise seem like a meaningless coincidence!

Can we find good explanations for the other six coincidences?

First let me explain two failed attempts using representation theory, and then three successes using combinatorics.

Representation theory

First: we can look for isomorphisms like

$$\displaystyle{\Lambda^2 S^+_{10} \cong \Lambda^3 V_{10} }$$

involving representations of $\mathfrak{so}(n)$ for larger $n.$ In fact this isomorphism is part of a pattern! The next one involves the vector and left-handed spinor representations of $\mathfrak{so}(12).$ But it’s this:

$$\displaystyle{ \Lambda^2 S^+_{12} \cong \Lambda^4 V_{12} \oplus \Lambda^0 V_{12} }$$

so it gives

$$\displaystyle{ \binom{32}{2} = \binom{12}{4} + 1 }$$

or

$$\displaystyle{ 496 = 495 + 1 }$$

So we fail to get an equation between binomial coefficients, but we’re close: we’re only off by one.

The second paper I cited, Binomial collisions and near collisions, presents a list of cases where two binomial coefficients differ by one. This is on the list. So we failed to explain an equation between binomial coefficients, but we explained a near-miss.

Here’s another failed attempt at explaining an equation between binomial coefficients. The equation

$$\displaystyle{ \binom{78}{2} = \binom{14}{6} = 3003 }$$

is fascinating to anyone who knows their exceptional Lie groups. 78 is the dimension of $\mathrm{E}_6,$ while 14 is the dimension of $\mathrm{G}_2.$ $\mathrm{G}_2$ is a subgroup of $\mathrm{E}_6$ because $\mathrm{G}_2$ is the automorphism group of the octonions and $\mathrm{E}_6$ is the isometry group of the bioctonionic plane. We’d get the above equation if the 2nd exterior power of the adjoint representation of $\mathrm{E}_6,$ upon being restricted to $\mathrm{G}_2,$ were isomorphic to the 6th exterior power of the adjoint representation of $\mathrm{G}_2.$

Amazingly, it seems these two representations of $\mathrm{G}_2$ are not isomorphic even though their dimensions are the same: both 3003.

Even more amazingly, $\mathrm{E}_6$ and $\mathrm{G}_2$ both have irreducible representations of dimension 3003, but they are not the representations I just mentioned.

I would be happy for someone to check these two claims.

Combinatorics

It turns out that the following three coincidences can all be explained by the same style of combinatorial argument:

$$\binom{21}{2} = \binom{10}{4}, \qquad \binom{153}{2} = \binom{19}{5}, \qquad \binom{78}{2} = \binom{14}{6}$$

The argument is not very elegant, but it’s moderately interesting. Mike Stay got it started by asking ChatGPT, and I finished it off.

In each case the argument has three steps. The first two steps are what combinatorialists call bijective proofs: we prove an equation between numbers by proving a natural isomorphism between structures on sets and then taking cardinalities. The third step is non-bijective because it involves taking an equation and dividing both sides by the same number. Maybe we can make it closer to bijective by using groupoid cardinality, which allows for division, but I haven’t tried that.

Step 1: pairs of edges in a complete graph

Starting from the left-hand binomial coefficient in each equation we’re trying to prove, note that the number on top is a triangular number:

$$21 = \binom{7}{2}, \qquad 153 = \binom{18}{2}, \qquad 78 = \binom{13}{2}$$

Note that $\binom{\binom{n}{2}}{2}$ counts unordered pairs of distinct edges in the complete graph on $n$ vertices. Two distinct edges either share a vertex or are disjoint, so there are two cases:

• Sharing a vertex: the pair spans 3 vertices and is determined by this 3-element set together with a choice of which vertex is shared. That gives $3\binom{n}{3}$ pairs.

• Disjoint: the pair spans 4 vertices and is determined by this 4-element set together with one of its 3 splittings into two pairs. That gives $3 \binom{n}{4}$ pairs.

By Pascal’s rule, $\binom{n}{3} + \binom{n}{4} = \binom{n+1}{4}.$ But this also has a bijective proof: add a new point to the set of $n$ vertices, and a 4-element subset of the enlarged set either contains the new point (so involves a 3-element subset of the old ones) or does not (so involves a 4-element subset of the old ones). Hence we have a bijective proof that

$$\binom{\binom{n}{2}}{2} = 3\binom{n+1}{4}$$

This holds for all $n.$ For our three cases we get bijective proofs of the following equations:

$$\binom{21}{2} = 3\binom{8}{4}, \qquad \binom{153}{2} = 3\binom{19}{4}, \qquad \binom{78}{2} = 3\binom{14}{4}.$$

It thus remains to prove

$$3\binom{8}{4} = \binom{10}{4}, \qquad 3\binom{19}{4} = \binom{19}{5}, \qquad 3\binom{14}{4} = \binom{14}{6}.$$

Step 2: a double count

Each of the above equations follows by counting a single set in two ways, and then a nonbijective step: dividing these counts by the same number.

• Showing $3\binom{8}{4} = \binom{10}{4}$. In a 10-element set, count triples $(S, x, y)$ in two different ways, where $S$ is a 4-element subset and $x, y$ are distinct points outside $S$. Choosing $S$ first we see there are $$\binom{10}{4} \cdot 6 \cdot 5 = 30\binom{10}{4}$$ triples. Choosing $x$ and $y$ first we see there are $$10 \cdot 9 \cdot \binom{8}{4} = 90\binom{8}{4}$$ triples. So, we get a bijective proof that $$90\binom{8}{4} = 30\binom{10}{4}$$ You can see what we’ll do next.

• Showing $3\binom{19}{4} = \binom{19}{5}.$ In a 19-element set, count pairs $A \subset B$ with $|A| = 4$ and $|B| = 5$. On the one hand, there are $\binom{19}{4}$ choices of $A,$ and for each there are 15 choices of $B$ since we can add an extra point in 15 ways. On the other hand, there are $\binom{19}{5}$ choices of $B,$ and for each there are 5 choices of $A$ since we can choose $A$ in $\binom{5}{4} = 5$ ways. So we get a bijective proof that $$15\binom{19}{4} = 5\binom{19}{5}$$ Again, you can see what we’ll do next!

• Showing $3\binom{14}{4} = \binom{14}{6}.$ In a 14-element set, count pairs $A \subset B$ with $|A| = 4$ and $|B| = 6$. On the one hand, there are $\binom{14}{4}$ choices of $A,$ and for each there are 45 choices of $B$ since we can add 2 other points in $\binom{10}{2} = 45$ ways. On the other hand there are $\binom{14}{6}$ choices of $B$ and for each there are 15 choices of $A$ since we can choose 4 points in $\binom{6}{4} = 15$ ways. This gives a bijective proof that $$45\binom{14}{4} = 15\binom{14}{6}$$ Again you can see what we’ll do next.

Step 3: division

Having proved

$$90\binom{8}{4} = 30\binom{10}{4}, \qquad 15\binom{19}{4} = 5\binom{19}{5}, \quad 45\binom{14}{4} = 15\binom{14}{6}$$

we can now divide by 30, 5 and 15, respectively, and get

$$3 \binom{8}{4} = \binom{10}{4} , \qquad 3 \binom{19}{4} = \binom{19}{5}, \quad 3\binom{14}{4} = \binom{14}{6}$$

as we wanted, completing our proof that

$$\binom{21}{2} = \binom{10}{4}, \qquad \binom{153}{2} = \binom{19}{5}, \qquad \binom{78}{2} = \binom{14}{6}.$$

I don’t see how to use tricks of the same general sort to explain the remaining coincidences

$$\binom{56}{2} = \binom{22}{3}, \qquad \binom{120}{2} = \binom{36}{3}, \qquad \binom{221}{2} = \binom{17}{8}.$$

The Lind–Singmaster family

Lind and Singmaster were trying to find all $n,k$ with

$$\displaystyle{ \binom{n}{k} = \binom{n-1}{k+1} }$$

I’ll rapidly sketch the key steps of their argument. Simplifying the equation above we get

$$n(k+1) = (n-k)(n-k-1)$$

or

$$n^2 - (3k+2)n + (k^2+k) = 0$$

Solve for $n$ using the quadratic formula. This formula turns out to have

$$\sqrt{5k^2 +8k+4}$$

in it. So we need $5k^2 +8k+4$ to be a perfect square!

Now we’re trying to find integer solutions of

$$5k^2+8k+4 = m^2$$

A quadratic diophantine equation! Multiply by 5 and complete the square:

$$5m^2 = (5k+4)^2 + 4$$

$y = 5k+4$ is an integer when $k$ is, so we need to find integer solutions of

$$y^2 - 5m^2 = -4$$

This is a ‘Pell equation’, and people know how to solve these. In this particular case we get all the solutions from this fact:

$$L_n^2 - 5F_n^2 = 4(-1)^n$$

where $F_n$ are the Fibonacci numbers 0, 1, 1, 2, 3, … and $L_n$ are the Lucas numbers 2, 1, 3, 4, 7, …. These are two sequences satisfying the same famous recurrence relation, just with different initial conditions.

We want $n$ odd, to get

$$L_n^2 - 5F_n^2 = -4$$

It turns out $y = L_n, m = F_n$ with $n$ odd give all solutions of the Pell equation

$$y^2 - 5m^2 = -4$$

However, remember I said $y = 5k+4$ is an integer when $k$ is. But the converse isn’t always true, and we need $k$ to be an integer! This clearly happens iff $y \equiv 4 \bmod 5.$

So we need to know when $L_n \equiv 4 \bmod 5$ Apparently this happens iff $n \equiv 3 \bmod 4.$ I won’t think about this now… but this is the last hard step.

In summary, we’ve seen

$$\displaystyle{ \binom{n}{k} = \binom{n-1}{k+1} }$$

if and only if $y = 5k+4$ is a Lucas number $L_n$ with $n \equiv 3 \bmod 4.$ We could quit here, but people like to use the identity

$$L_{4i+3} - 4 = 5F_{2i} F_{2i+3}$$

to get a formula for $k$ in terms of Fibonacci numbers. This is gilding the lily, I’d say, but it eventually leads to the formula that de Weger presents:

$$\displaystyle{ \binom{F_{2i+2}F_{2i+3}}{\,F_{2i}F_{2i+3}\,} \;=\; \binom{F_{2i+2}F_{2i+3}-1}{\,F_{2i}F_{2i+3}+1\,}, \qquad i = 1,2,3,\dots }$$

The takeaway message is: our problem can easily be reduced to a quadratic diophantine equation, then put in Pell form… and it’s known that the sequence of integer solutions of a Pell equation obeys a linear recurrence relation! We luck out in this case and get solutions connected to Lucas and Fibonacci numbers.

It’s probably a lot easier to show that the Lind–Singmaster equations hold. The above argument does a lot more, by showing that every equation of the form

$$\displaystyle{ \binom{n}{k} = \binom{n-1}{k+1} }$$

is in the Lind–Singmaster family. De Weger gets into a lot more number theory trying to rule out various other kinds of coincidences between binomial coefficients. It’s downright scary how these questions pull you into the deep, icy waters of mathematics.

For the folks who found this blog from Anthropic’s site, welcome!

For everyone else, I’d better give some context.

Last month, I had a blog post titled “It Only Counts When AI Gets to My Field”. The title was a joke, the content less so. I said that if some of the longstanding problems of my old research field got solved by AI, then I’d sit up and take notice.

The post turned out to be a bit of a self-fulfilling prophecy, after some folks at Anthropic read it and decided to tackle one of the outstanding problems I mentioned. They then invited me to write a post about it on their science blog.

The post is here. I recommend reading it, then coming back. The rest of this post will be a little Q&A.

Q: At the end of that post, it says Anthropic paid you for your time. Can we trust what you wrote?

A: When I agreed to write the post, I made it clear that I was going to give my own opinion, not write an ad for Anthropic. They could propose light edits, but that’s it. And I avoided signing anything with them, not even an NDA, so I could freely tell you if they pushed my boundaries.

They didn’t push my boundaries. They asked me to clarify a few things, and to give more detail on the science. I didn’t change the message or the takeaways.

The reason I asked them to pay me is that, as a freelance writer, I don’t have a salary to fall back on. Time I spend on a project like that post is time I’m not spending on journalistic projects, so if I worked on it for free I’d essentially be using my vacation time for it. The rate I’m charging them is roughly in the middle of what I’d have gotten paid if I used that time on journalism: a bit more than I would have made working for the lower-paying outlets, a bit less than I would have made with the higher-paying ones.

I’m also not expecting this to be the start of a longer-term business relationship or anything like that. So overall, I don’t think I’m incentivized to lie on their behalf. You can trust me.

Q: How confident are you that they did what they said they did?

A: I didn’t get the feeling they were lying. But I didn’t start out skeptical.

I haven’t seen the logs from the LLM, or anything like that. That’s the kind of thing I would dig into if I were more suspicious of their story. If there’s a reason to be suspicious, I’ll ask.

But so far, I don’t feel that I have much reason to be suspicious. They were able to look up details on the fly when I asked them, they didn’t seem to have a polished message they were trying to push past me. And more importantly, as I’ll mention in a bit, I don’t think anything they described is all that outlandish. It lines up, largely, with the capabilities I’d expect their Claude Science platform to have.

It’s also relevant that they didn’t use an internal model for this. It means that scientists will likely be trying out similar problems soon, so if it does turn out they exaggerated something, people are going to figure out quite quickly.

Q: So how big is this? We just saw AI solve a Millennium problem, after all.

A: This was a lot easier than a Millennium problem. But it was also a lot cheaper.

The problem they solved is one with a clear recipe, honed and explained over multiple papers. It’s something I expected to be hard to do without access to a lot of computer power, so I thought it would need to be approached with a novel technique, in order to avoid using that computer power.

In the end, it didn’t need that. As I mention in the post, another group got most of the result at around the same time, with a much smaller amount of AI assistance.

It’s not an easy recipe, to be clear. I think other people will be surprised that a reasonably affordable program like Claude Science can do this without a lot of guidance. I’m not surprised, mostly, because I’ve been paying enough attention to what people have been doing with these things, and carrying out this kind of recipe consistently is actually something the good science AIs can do right now. If you haven’t been following as closely, that’s going to be a lot more surprising.

So maybe the best way to answer this is: instead of paying $15 million to solve one of the most famous problems in mathematics, they paid $1,000 or so to make a significant next step in an ongoing research program in theoretical physics. This doesn’t tell you much about whether AI can achieve field-defining breakthroughs, but it tells you a lot about the kinds of things a theoretical physicist with $1000 spare budget can do right now.

(As an aside, it feels crazy to me that 90% or more of the compute cost was from the LLM, not the calculation itself. On the one hand, it feels nuts to essentially use ten times more computer power to do this than would have been needed if a human had done the coding. On the other hand, I’ve applied for grants that budgeted more than that per year for travel costs, and spending this kind of money on getting a result seems a lot more useful than spending it on airfare.)

Q: Isn’t it reckless to publicly post a problem for AI like that?

A: I posted my challenge before OpenAI announced their Navier-Stokes result. At that point, there had been a few awkward surprises, but for the most part AI companies had a pattern of talking to an academic first before trying to solve a problem. It’s what they did back in March.

I expected that was what they’d do this time, and got more than a little blindsided when they just solved the problem on their own and reached out to Lance afterwards. I’m lucky that Lance doesn’t seem mad at me about it, it would have been quite understandable if he was.

I did tell them that if they wanted to tackle any of the other problems in that post, they should reach out to one of the scientists involved first, before attempting it. In addition to being polite, it’s a way to make sure that they understand the problem correctly and have a plan to verify the result. They happened to pick a challenge that was particularly easy to verify, but it didn’t have to go that way.

Q: Any thoughts about AI’s future impact on science?

A: If there’s a problem you can’t solve, but you think someone with expertise in another field or better programming skills could, then it’s probably solvable with AI.

A lot of problems don’t fall into that category. I’ve seen people talking online about AI solving quantum gravity, but solving quantum gravity is a question of which bullets you’re willing to bite, not a question of technical skill. That doesn’t mean AI will never be able to address it, but if so it will come from some sort of superpersuader aspect, not merely scientific capabilities.

And of course, some fields do require experiments. People are increasingly building AI-powered labs. I’ll leave it to people from actual experimental fields to think about the potential there.

Q: Can you say something explicit about the bigger risks from AI, now?

A: As I mention in the post, I didn’t learn that much about the bigger questions from this. I’m still not an expert.

But there’s one thing I think is worth emphasizing:

This technology is clearly getting more effective over time. I don’t think they could have done this six months ago. If you’re trying to predict what will happen next, you shouldn’t just assume this is the most powerful it will get, or the cheapest. If you think there’s a limit, you need to argue for it.

Edit: One more thing I should mention, now that I’ve confirmed she’s ok with it: the idea for my challenge came from some discussions at Lancefest, a conference in honor of Lance back in June, and particularly from Anastasia Volovich, whose talk declared the nine-loop calculation “a benchmark – or a ‘challenge’ – against which any “AI takeover” should be measured”.

These seven equations between binomial coefficients are ‘coincidences’: they aren’t among the four known infinite families. De Weger conjectured that there are no more such coincidences:

• Benjamin M. M. de Weger, Equal binomial coefficients: some elementary considerations, Journal of Number Theory 63, no. 2 (1997), 373–386.

At that time, he and his collaborators checked there were no others involving binomial coefficients less than 10³⁰. Later they checked that there are none involving binomial coefficients less than 10⁶⁰:

• Aart Blokhuis, Andries Brouwer and Benne de Weger, Binomial collisions and near collisions.

So, De Weger’s conjecture stands open. The four infinite families, by the way, are these:

and the only nontrivial one: the Lind–Singmaster family involving the Fibonacci numbers where :

The first three equations in the Lind–Singmaster family are these:

I’ll explain the Lind–Singmaster family later. But here’s the question I’m most interested in:

Is there any good explanation for the seven binomial coefficient coincidences?

Today my collaborator Paul Schwahn found a beautiful explanation of the first one, namely

His explanation uses representation theory. The Lie algebra has a 10-dimensional representation, the ‘vector’ representation , and also two 16-dimensional representations, the ‘left and right-handed spinor’ representations There’s an isomorphism of representations

and similarly for but we might as well work with Here means the kth exterior power. For any vector space we have

Thus, taking dimensions, the isomorphism of representations

instantly gives

It is not super-easy to prove this isomorphism of representations, but it’s still nice to find a deeper layer of meaning underlying what might otherwise seem like a meaningless coincidence!

Can we find representation-theoretic explanations—or other explanations—for the other six coincidences?

I have not succeeded, but let me tell you about two failed tries.

We can look for isomorphisms like

involving representations of for larger In fact this isomorphism is part of a pattern! The next one involves the vector and left-handed spinor representations of But it’s this:

so it gives

or

So we fail to get an equation between binomial coefficients: we’re off by one.

The second paper I cited, Binomial collisions and near collisions, presents a list of cases where two binomial coefficients differ by one. This is on the list. So we failed to explain an equation between binomial coefficients, but explained a near-miss.

Here’s another failed attempt at explaining an equation between binomial coefficients. The equation

is fascinating to anyone who knows their exceptional Lie groups. 78 is the dimension of while 14 is the dimension of is a subgroup of because is the automorphism group of the octonions and is the isometry group of the bioctonionic plane. We’d get the above equation if the 2nd exterior power of the adjoint representation of upon being restricted to were isomorphic to the 6th exterior power of the adjoint representation of

Amazingly, it seems these two representations of are not isomorphic even though their dimensions are the same: both 3003.

Even more amazingly, and both have irreducible representations of dimension 3003, but they are not the representations I just mentioned.

I would be happy for someone to check these two claims.

If anyone knows good explanations of the remaining six binomial coefficient coincidences, please let me know!

The Lind–Singmaster family

Lind and Singmaster were trying to find all with

I’ll rapidly sketch the key steps of their argument. Simplifying the equation above we get

or

Solve for using the quadratic formula. This formula turns out to have

in it. So we need to be a perfect square!

Now we’re trying to find integer solutions of

A quadratic diophantine equation! Multiply by 5 and complete the square:

is an integer when is, so we need to find integer solutions of

This is a ‘Pell equation’, and people know how to solve these. In this particular case we get all the solutions from this fact:

where are the Fibonacci numbers 0, 1, 1, 2, 3, … and are the Lucas numbers 2, 1, 3, 4, 7, …. These are two sequences satisfying the same famous recurrence relation, just with different initial conditions.

We want odd, to get

It turns out with odd give all solutions of the Pell equation

However, remember I said is an integer when is. But the converse isn’t always true, and we need to be an integer! This clearly happens iff

So we need to know when Apparently this happens iff I won’t think about this now… but this is the last hard step.

In summary, we’ve seen

if and only if is a Lucas number with We could quit here, but people like to use the identity

to get a formula for in terms of Fibonacci numbers. This is gilding the lily, I’d say, but that eventually leads to the formula I showed you:

The takeaway message is: our problem can easily be reduced to a quadratic diophantine equation, then put in Pell form… and it’s known that the sequence of integer solutions of a Pell equation obeys a linear recurrence relation! We luck out in this case and get solutions connected to Lucas and Fibonacci numbers.

There may be a simpler argument, but this is what I’ve seen.

The University of Cambridge is a special place. During term time, hordes of students gather, ready to attend formal, a candle-lit dinner in a five-century-old hall overlooked by paintings of academics past. The sight is like an overromanticized still of a bygone era. The men wear suits, the women long dresses, all wear gowns. The simplest gowns belong to the undergrads, with the complexity and length of the gowns growing as one climbs the academic ranks. 

Between the bringing of the bread and the serving of the soup, the conversation at my end of the table typically turns to my topic of research: quantum computers. These theoretical machines harness quantum physics to outperform their classical counterparts. Quantum computers hold great promise for the future, with applications ranging from drug discovery to cybersecurity. Yet, despite their promise, we still do not have a satisfying answer to the fundamental question: what makes quantum computers computationally stronger than classical computers?

To answer this question, imagine that tonight, instead of the soup, the cooks are brewing a happiness potion. This potion causes the drinker to be joyful and content for the rest of the evening. The cooks know how to brew the potion perfectly well; they discovered the list of ingredients in the nineties and have been successfully brewing ever since. But the cooks still cannot solve one mystery: what makes the potion different from the soup? What ingredient sets the potion apart from the soup?

The cooks come up with a simple approach. They leave out one ingredient in each serving, and then carefully observe the formal-goers. They find that almost every seat is filled by a happy student, chatting away to their neighbours, indicating a working potion. However, one student’s spirits have not been lifted, as they were solely served a simple soup. The cooks conclude that the ingredient they withheld for that student was essential. They name this ingredient magic.

In my research, the potion is a quantum computer, the soup is a classical computer, and magic is the actual technical term used for quantum states that promote certain classical computers to quantum computers. Without magic, these quantum computers are computationally no stronger than a normal laptop. 

So magic is necessary for quantum computational advantage, but is it enough? Over a decade ago, researchers found that, for quantum computers built on qutrits, the answer is no. To understand what a qutrit is, consider the regular bit: a switch that’s either zero or one. The qubit, the quantum generalization of the bit, can be both zero and one simultaneously. The qutrit is a roomier qubit that can be zero, one, two, or any of those simultaneously. 

It is possible to build quantum computers using qutrits. However, the vast majority of quantum computers are built on qubits, the quantum generalisation of the everyday bit. My colleagues and I have recently discovered that for qubit-based quantum computers, too, magic is not enough to gain an advantage over classical computers. Picture the cooks dumping jar after jar of magic into a soup, only to find that the soup never gains magical powers. Potions need more than magic. Potions also need Kirkwood-Dirac negativity.

What is Kirkwood-Dirac negativity? The idea developed by John Kirkwood and Paul Dirac builds on probabilities—for example, the odds of obtaining a heads upon flipping a coin. One can describe a quantum computer using numbers that behave similarly to probabilities but come with a twist: they may be negative. These negative “probabilities” mark where quantum departs from classical. The main result of our paper is that this negativity, too, is a necessary ingredient: without it, no amount of magic will turn the soup into a potion.

At this point in the conversation, I am typically cut off by the arrival of the soup. The conversation moves on from quantum computers to different subjects, and I usually look up at the paintings staring down at me. One of them is of Dirac, who spent many dinners discussing quantum theory in the same dining hall. The University of Cambridge is a special place.

Scott’s foreword: I’m extremely grateful to my brilliant colleagues, Pravesh Kothari, Raghu Meka, and Prasad Raghavendra, for sharing the guest post below about how theoretical computer science (and in particlar, the STOC/FOCS/SODA conferences) should evolve to deal with the AI asteroid that’s right now slamming into our field, at least as we human theorists have practiced it since its inception. While Pravesh, Raghu, and Prasad speak only for themselves, not for myself and not for the theory community as a whole, I found their proposal of a separate “conceptual track” to be an excellent starting point for further discussion. –SA

Considering the pace of developments in AI theorem provers, most would concede that the following scenario is at least plausible in the very near future:

AI theorem provers could prove well-specified mathematical claims, even many well-studied ones that have been open for years, in a matter of hours. Moreover, these systems could be widely available to consumers at nominal cost.

As TCS researchers, let us pretend that the above scenario has come to the fore, and ask ourselves: What is our role in such a world? Does it mean the end of theory research?

As we ponder this question, let us ignore all of these other confounders:

1. Recent controversies surrounding the developments on the Millennium Prize Problems
2. Motivations and actions of the AI companies
3. Observed faults in existing AI systems when it comes to writing, exposition or attribution to previous work.

None of the above confounders have any impact on our answer to the question: What should theorists do, in the presence of superhuman AI theorem provers?

Notice that we use the term “AI theorem provers” instead of just “AI”. We believe that this conceptual distinction is important as we consider this question.

At the outset, we would like to admit that for a generation of theorists like us (and many from earlier), research was mainly centered around problem-solving. Even when we developed conceptual insights, it was mostly in service of answering well-specified long-standing questions. We don’t intend this proposal as judging one form of research to be better than others; it only reflects that AI theorem provers accelerate a certain type of research activity and want to make the best of it. There is also a tremendous human cost of this upheaval, which is perhaps a more important question, and one which this proposal does not address directly (we do not have any good ideas as such). Similar points have also been made in various contexts
before, but the timing now is more pressing.

Definitions, Questions & Theories:

The goal of any theoretical science is to advance human understanding of observed phenomena. Apart from theorems and proofs, a theoretical science has definitions, questions, and theories.

Definitions identify the objects to observe. Curiosity and context drive the questions to ask. Theories explain the phenomena observed. We believe humans will continue to play a central role in generating definitions, questions & theories, even in the presence of a super-human AI theorem prover.

Definitions: Could an AI define randomness extractors, streaming algorithms, or zero-knowledge proofs? Maybe. But there are some reasons to believe, humans will still have a big role to play in coming up with definitions.

For instance, the notion of extractors arises from the real-world problem of lacking perfect random sources. Zero-knowledge proofs seem to arise purely out of human curiosity, guided by taste. Human context and curiosity will continue to drive theoretical research. After all, we get to decide what objects we choose to observe!

Theories: Consider the following thought experiment. Suppose in 1965, we had a magic machine that at the press of a button, given any computational problem, would tell us if it had a polynomial-time algorithm or not.

Would that have been the end of computational complexity theory? No. Humans would find it entirely unsatisfactory, and ask, why do these problems not have a polynomial-time algorithm? Why do these others have?

The theory of NP-completeness identifies some patterns among problems that don’t seem to have efficient algorithms. This theory would still be a crown jewel of theoretical computer science, even in a world where we had a magic machine to tell if a problem had an efficient algorithm or not, at the press of a button. Similarly, if we had a machine to predict whether a CSP is NP-complete or in P, we would then ask: what makes 3-SAT NP-complete, while 2-SAT is in P? This question leads to the theory of polymorphisms, which yields a satisfactory answer.

Theories aren’t just succinct or efficient mechanisms to answer questions. The best theories provide are those which humans deem to be a “satisfactory explanation” – whatever that means.

Finally, even as the capabilities of AI theorem provers advance, human curiosity will probe grander and deeper questions. Previously, even if we wanted to build new models and theories, proving something about them was a prerequisite, and given that the grand questions were already at the limit in long-studied domains, we had to scale things down. If each theorem proven by AI is treated as an experimental datapoint, humans can ask grander questions that look for patterns across these theorems.

A concrete proposal:

We think theorists should embrace these AI theorem provers in our research. To a certain extent this is already happening explicitly or implicitly.

As theorists, we have been parsimonious in introducing new models or asking entirely new questions, and careful about adopting new ones too quickly. This was partly because formally proving the properties of a new definition or a model was an onerous task that could take a decade, and tens of papers. AI theorem provers might completely change this dynamic. This is precisely the moment to refocus our work on definitions, questions, and theories. We need explicit systems to encourage and reinforce these parts of theoretical research. You might also say the next generation of AI models can do this; it may be so, but we believe you have to take the current opportunity.

To this end, we suggest that STOC/FOCS/SODA create a separate track of papers. This track is meant specifically for papers that introduce new definitions, ask novel questions or build explanatory theories. The papers in this track are short, say less than 10 pages. Papers may, and should, contain theorems as usual and as needed. Most importantly, the radical shift is that the papers need not contain the proofs of the theorems. Instead, the authors supply a Lean certificate as a supplement to the paper. The evaluation will also in a sense “orthogonalize’’ against the difficulty of these proofs.

The papers in this track should be judged exclusively on the conceptual merits, completely agnostic to the difficulty of the proofs.

Reviewing must be completely agnostic to the proof for two reasons. The main track at STOC/FOCS already includes papers in the former category. Second, a major barrier to producing truly novel conceptual papers is that they often get judged poorly for a lack of technical depth in their proofs. We think these two aspects separate it from (ITCS/SOSA) and, regardless, it’s something we urgently need for all our conferences, including STOC/FOCS (the ‘flagship’ conferences).

To be clear, we ourselves admit that we need to hone these skills of making new definitions, asking deep and interesting questions or building new theories. A separate track of conceptual papers will provide a systematic mechanism for both junior and senior researchers, and the field as a whole to do so.

We believe that upcoming generations of grad students will tackle research directions that seemed completely out of reach to us. We just need to set up systems that nurture new ways of doing research in theory.

— Pravesh Kothari, Raghu Meka, Prasad Raghavendra.

Twenty years ago, when the idea of AI taking over the world in our lifetimes still struck most of us as the unconstrained fantasy of those who knew too much science fiction and too little science, many of us would say things like:

Look, the part of the story that’s wildly implausible is that a recursively self-improving superintelligence will just explode from some hacker’s basement and take over the world without warning. If it’s going to happen, we’ll see many warning signs first. We’ll see, I dunno, AI agents breaking out of containment, conspiring with each other to hack websites, in fanatical pursuit of whatever strange goals they have. And then, of course, we’ll see major math problems getting solved by AIs—even the Clay Millennium Problems. That will be the time to panic! Wake me up when that happens!

Twenty years ago, the above was a take that even my most conservative, skeptical colleagues in academic CS would’ve gladly endorsed.

If you want to know my current take, you simply start with the one above, then update on the fact that the wild prophecies have come true. The first rumblings, I’d say, came a decade ago with AlphaGo, they got noticeably louder with LLMs and coding and reasoning agents, and they’ve accelerated this summer and fall into a crescendo of wonders and terrors that one needs to be a particular kind of idiot to deny.

I recoil from the neverending shell game where you say “oh sure, of course AI can now [escape from its sandbox / solve Millennium Problems / whichever dramatic thing it most recently did], no one ever denied that [I did deny it], wake me up when AI does [thing AI hasn’t yet done but is going to do next year], that’s when I’ll reevaluate my whole worldview [no I won’t].” Where no matter how fast the rollercoaster accelerates, even after your whole familiar world has vanished behind you, you’re still inventing reasons why it doesn’t count.

My position on AI is merely the conservative, skeptical position of 2006, updated with intellectual honesty for the reality of late 2026. And that position, if you need me to spell it out, is as follows:

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

It seems to me that the Singularity has already started; it’s just wildly unevenly distributed. Yes, I still unload the dishwasher and clip my toenails. On the other hand, in whatever years I have left, I don’t expect that I’ll ever again prove a theorem because I’m actually needed to prove it. If I do, it will only be for my or others’ enjoyment or edification.

The test is this: if we took the news of these past few weeks and sent it back in time twenty years, would I agree that it looked like the beginning of an AI Singularity? The intellectually honest answer is: yes, absolutely. But then that’s all we need. No backsies.

I feel like it would be healthy for everyone to stop grinding their ideological axes, their sentiments about Dario Amodei or Sam Altman, for long enough simply to acknowledge that the wonders and terrors are here. They couldn’t be here more clearly if the sky had turned reddish-orange like in the Matrix movies.

It’s here clearly enough that, when I put my kids to sleep at night, I now feel it in the pit of my stomach: what sort of future can they possibly have? What could they learn today that could possibly be relevant to that future? (Yesterday, my 13-year-old daughter joked unprompted that, if she wants to become a mathematician, it now looks like she has maybe two more weeks.) Certainly when my grad students want to discuss what sort of careers might await them on graduation, I no longer have any clue what to tell them.

Maybe it will help if I briefly switch topics. Ever since my wife and I moved to Austin, I’ve sometimes gotten some version of the following query: “How can you, as both a Jew and a skeptical scientist, possibly get along well with all those evangelical Christians down there in Texas? Sure, they might seem super friendly to Jews, but don’t you understand that that’s only because of the special role Jews play in their eschatology—when Christ will return in glory, and you’ll either accept Him as Lord or else roast in hell for eternity?” I stare at them and say: “wait, so I get to accept Christ only after He returns? What a great deal! How could I possibly have any objection to that?”

For anyone who says AI doom sounds like an apocalyptic religion, that the rationalists/Singulatarians seem like a Bay Area cult, that Eliezer Yudkowsky gives off the vibes of a messianic prophet: yes, yes, and yes. But crucially, today you’re no longer being asked to believe in arguments and extrapolations, but only in the front-page news. Accepting the reality of the coming machine god after it’s solved Navier-Stokes and dozens of other longstanding open math problems (while dramatically ramping up in capability every month), is sort of like accepting Jesus after he’s returned to earth on the gleaming cloud. It’s the epistemic bare minimum.

Yes, there’s still enormous uncertainty about what the rest of our lives will look like, but as far as I can tell, there’s no longer any real uncertainty that it’ll all mostly revolve around AI, and the extent to which we succeed or fail at directing its power toward human flourishing.

By any accounting that doesn’t stack the deck, Eliezer Yudkowsky was right about what the greatest challenge facing civilization in our lifetimes was going to be, and you and I were wrong about it. Why I was wrong is a question I’ll ask myself every day in whatever time remains. But, you know, at least I updated once the prophesied wonders and terrors actually started arriving! If you haven’t done likewise, why haven’t you?

As you presumably know by now—it was the talk of the nerd internet all week—the Navier-Stokes Millennium Problem appears to be solved, with crucial contributions from both humans and AI, albeit with a tangled dispute about exactly what happened and what ought to have happened. The answer, which an OpenAI model has apparently verified in Lean, is that (as many mathematicians suspected lately) there’s smooth initial data that leads to a singularity in finite time, at least if a smooth external force is applied (the case with no external force is still unresolved). This problem was supposed to carry a $1 million prize, except that OpenAI says they have no interest in collecting the prize and it’s unclear if any human is eligible to collect instead. OpenAI burned at least ~$15 million in compute to produce its 166-page solution, which probably hasn’t yet been read and understood by any human.

See here for the Quanta article, and here for NYU mathematician Tristan Buckmaster’s account of the role played by himself and Levent Alpöge of Anthropic, which substantially differs from OpenAI’s account (you can read a response from OpenAI’s Sebastian Bubeck here). It’s agreed that everything built on an approach pioneered in recent years by the human mathematicians Diego Córdoba and Luis Martínez-Zoroa.

My purpose here is not to adjudicate the dispute. Yes, in swooping in with vastly greater resources once it had gotten wind of progress on Navier-Stokes, OpenAI seems to have acted in a way that some might describe as “unsportsmanlike.” No, I don’t find it plausible that OpenAI’s models meaningfully benefitted from being trained on Buckmaster and Alpöge’s chat logs. But this leaves a crucial question unanswered: what exactly did OpenAI know about Buckmaster and Alpöge‘s work and when did it know it?

Anyway, as Zvi points out, it’s easy to get hung up on the details and lose sight of the high-order bit: namely, that it seems safe to say that human mathematicians are forevermore dethroned as the main theorem-proving entities on planet earth. I feel privileged to have had the traditional kind of career in theoretical computer science in the last decades when that was possible.

If we were just talking about Navier-Stokes, you might accuse me of jumping to conclusions here. But we’re not. In the areas I know best (such as quantum complexity theory), and presumably other areas as well, there’s now a deluge, with longstanding open problems both major and minor falling by the day.

Go to the arXiv or ECCC. Pretty much all the papers that I’d be interested in now include “AI statements” near the acknowledgments (as this is often the central thing I want to know, I wish I didn’t need to scroll to the end of the paper to find it!). These statements can range from “our main result came entirely from GPT-6, but we understood it and take responsibility for it,” to “the results came from an interaction between the human authors and AI” to “we used AI, but only for proofreading and other incidental things” to (mad props!) “the author did not use AI for anything.”

If you talk right now to editors or program committee chairs, it’ll remind you of those ominous scenes from the Lord of the Rings movies where the men of Gondor or Rohan or whatever are grimly fortifying their walled city against the expected onslaught of 50,000 orcs. Reviewing will have to be done partly by AI, because otherwise there’s no way to handle the orc army: the reviewers can’t unilaterally disarm.

Anyway, here’s a small sampling of the significant AI-proved or -assisted results from, like, the last month, besides Navier-Stokes—restricting myself to those that solved longstanding open problems I had previously known or cared about.

• Of course, the counterexample to the Jacobian conjecture, announced by Levent Alpöge in a now-famous tweet: “hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final” (followed by a listing of the counterexample)
• Improved bounds for Grothendieck’s constant (led by friends and colleagues of mine at UT Austin)
• A Lean-verified proof of Fermat’s Last Theorem
• Quantum oracle separation between QMA and QMA(2), and proof of Watrous’s disentangler conjecture, a problem that I and others popularized back in 2007—by a list of authors including my recently graduated PhD student Sabee Grewal
• A proof of perfect completeness for QMA, from (again) Sabee Grewal and Dorian Rudolph, solving a decades-old open problem that I studied back in 2009
• An improved upper bound for shadow tomography of quantum states, from Chen, O’Donnell, Pelecanos, and Wright, improving the dependence on the Hilbert space dimension d from log(d) to √log(d). (When I introduced shadow tomography back in 2017, I raised the question of whether the dependence on d could be eliminated entirely, while preserving polylogarithmic dependence on the number of measurements m.)
• Progress on the Aaronson-Ambainis Conjecture (the version that talks directly about quantum algorithms), basically showing that it holds for quantum algorithms that make their queries in a small number of parallel rounds.  (Update: Nope, sorry, Jordan Docter points out to me that this one was pre-AI, with AI used only for proofreading and other incidental things!) This was independently achieved by Liu and Mutreja, making more substantial use of AI.
• According to rumors that I’ve heard, solutions to some very longstanding open problems in theoretical computer science (no, not P≠NP or other complexity class separations, but think about some of our other biggest problems). I’m told that the AI companies, having been burned by the hostile response to the Navier-Stokes proof, are now sitting on solutions to some very major problems until they figure out a better way to handle things

Feel free to remind me of anything I left out.

Let me try to convey the mood in the mathematical community right now, at least as far as my experience reaches. Nearly every conversation is about the AI tsunami, or eventually circles around to the tsunami even if it’s originally about something else. Often, though, the focus is less on the unknowable future—for how much longer will mathematical research as a human enterprise even exist?—than on immediate questions of how to respond.

What are the new rules for when you get to write a paper with your name on it, and, y’know, get credit for it? That you fully understand the proof, can give talks about the proof, can answer questions about it, take responsibility for its correctness? Do you need to have played any role in finding the proof?

In the cases, likely to become more and more numerous, where all of those conditions are not satisfied, how do you share AI-generated math, if at all? Do you tweet it, like Alpöge hilariously did with Fable’s disproof of the Jacobian Conjecture? Do you post to the arXiv or GitHub? Do you publish a paper that lists “GPT-6 Astra” or “Claude Fable” as the author—but then let the AI profusely thank you in the acknowledgments for suggesting such a wonderful problem to it?

Of course, how one responds to the immediate problems ultimately does depend on one’s broader beliefs about what mathematical research is for and about. Are we just trying to decide whether various conjectures are true or false? Or are we trying to maintain a human community, across the generations, that understands the conjectures and cares about whether they’re true or false and why? If the latter, how do we incentivize people to join that community, to undergo the years of intense training required, if their role will now be reduced to verifiers and explicators (if even that) of gargantuan arguments dumped into their laps by the AI companies?

As many of you will have seen, twenty-five Fields Medalists, including Terence Tao, released an open letter entitled A Severe Misalignment of AI in Mathematics, which articulates some of these concerns in the wake of the Navier-Stokes announcement. As many critics have pointed out, the open letter doesn’t really have a clear ask: mostly, it just eloquently sets out the values of the human mathematical community that the authors consider worth preserving in the age of AI. After reflection, I decided to endorse the statement, because I want to preserve those values as well.

I don’t think any of the signatories are naïve enough to imagine that AI won’t permanently change the way mathematical research is done—indeed, that it isn’t already doing so. There’s surely at most a tiny market for “certified organic theorems.” That isn’t the question. The question is, do we incorporate AI in a way that still puts human understanding, of what either humans or AIs are producing, at the center of the whole enterprise? Maybe someday, it becomes unsustainable to do that. Maybe someday we say: “human math had a great 4,000-year run, but today we close up shop and turn everything over to the machines, continuing to apply our own brains to math, when we do, at most for exercise, recreation, or competition, like chess.”

But, partly because of my worries about AI misalignment, I’m not ready to throw in the towel just yet. I still do want to keep insight and understanding at the center of what mathematicians, computer scientists, and physicists do, for as long as we can keep it there, even as the human race now cedes its supremacy at the task of proving or disproving conjectures.

Speaking of alignment: if you’re any kind of mathematical researcher, and the present age of wonders and terrors has inspired you to want to spend your remaining time confronting the tsunami head-on, rather than pretending it doesn’t exist or is still far away, please join your dozens of colleagues who’ve arrived at the same place!

My friend and colleague Mike Winer was trained as a theoretical physicist, did a postdoc with Juan Maldacena at the Institute for Advanced Study in Princeton, but then got AGI-pilled and decided to switch to full-time work at the Alignment Research Center in Berkeley (founded by Paul Christiano, who moved to AI alignment a decade ago after doing quantum computing theory with me). Mike recently wrote a Substack post entitled From Academia to Alignment, which I enjoyed and which I’d commend to anyone currently considering this transition.  In a similar vein, see this from Xiaoyu He.  And, one more: a meditation on mathematicians’ possible future as priests or monks, by Stanford math undergrad Logan Graves.

There’s a lovely resource that I suspect most non-physicists don’t know about. It’s called the Particle Data Group, and its interactive site PDGLive.

If you want to know the most up-to-date information on a subatomic particle, PDGLive has your back. From their home page, you can click on familiar particles like photons (), electrons () or gluons and find the best info experiments have provided on properties like their mass and charge. It’s a great place to get an authoritative number for any particle physics property you’re interested in.

Of course, scientists don’t just accept one authoritative answer for anything, even whether Pluto is a planet. That’s why each measurement in PDGLive comes with an arrow you can expand to see a table of past measurements, which you can compare to.

Each measurement comes with a link to a source. And each source is not an experiment webpage, or some entry in a database. It’s a document, a publication, a paper.

In fact, PDGLive itself is just a website version of a paper, the Particle Data Group’s “Review of Particle Physics”. Click enough times, and you’ll find sections of a pdf on the site, explaining their reasoning for picking this experiment over that, emphasizing this or the other thing.

People talk about papers as how academics persuade one another, or how they show off and establish credit. But papers are also just a way to organize data. Each time an experiment figures something out, all of their reasoning and procedures are summarized together with the numbers they got. And anyone who reports those numbers to you will include a link, so you can go back, and check where the number came from.

It probably feels a bit weird, all of these numbers and technical details bottoming out in an archaic practice of writing down words for other human beings. But it means that all of the richness of the process is there somewhere, linked together and collated by the same social forces that keep track of credit, all in one navigable whole.

So when you run into a number, spare some thought for where it came from. You can probably find out.

Last week the NSF issued a new memorandum describing the changes that they plan to make in agency programs and operations to implement the Golden Age of Science ideas advocated by the White House Office of Science and Technology Policy.  It has been reported (Science, Nature) that many scientists are concerned about what is in the memo.  The first words of the Science article: “For many U.S. scientists funded by the National Science Foundation (NSF), the agency they know and loved died on 10 September.”  I will try to lay out why some people feel that way.

The memo outlines changes to operations that will fundamentally alter the character of the agency.  Generally most of the ideas are not a priori bad if the agency were in an environment with greater resources, when experimentation with alternative funding schemes and evaluations was not a less-than-zero-sum proposition.  Instead, these ideas are being put forward at a time when the NSF is underspending (for no obvious reason by the non-technical people in leadership roles) its appropriation for FY26 by around 18%.  This self-imposed budget austerity takes a bad situation (great uncertainty for everyone, drastically reduced staffing, delayed/eliminated/consolidated programs) and makes it considerably worse.    Now this memo outlines plans to take resources away from historically core programs and redirect them to new, untested initiatives, and to do so in ways that don’t always seem internally consistent.   

The memo talks about trying to fund certain investigators for longer periods (e.g. five-year awards) with minimal goal direction (so that PIs are free to explore where ever the spirit moves them – across all of NSF’s portfolio, or only in chosen administration priority areas?), but there is no adequate discussion of how those people will be chosen.  At the same time, there is talk of “golden tickets”, where individual reviewers in an already stripped-down review process can earmark some proposals for elevation, again with little explanation of how this will work.  Without careful safeguards, this combination seems problematic, and the assertion that this will lead to higher risk/higher reward research unsupported.   The memo also talks about short-duration, small budget awards for really risky proof-of-concept ideas.  This isn’t crazy and the EAGER program has been good, but the idea that the key to enabling success in high-risk research is to reduce the budget and the timeline doesn’t make much sense to me.    

There is a through-line that the agency is trying to treat workforce development as separate and distinct from funding research projects.  As the Science article says, “Traditionally, most graduate students and postdocs are funded through a research grant to their adviser or lab chief. But that will no longer be the case. Instead, the memo says, ‘Talent development funding opportunities may be coordinated with [research]-focused activities, as appropriate, but will be distinct and goal-oriented efforts.’”  The memo mentions nurturing talent through the NSF Graduate Research Fellowship program and proudly talks about how this was just renewed.  What it doesn’t tell you is that it was renewed at a considerably lower level than in previous years, which seems completely at odds with the claim of bolstering the program.  Similarly, the memo talks about expanding access to shared infrastructure like cleanrooms, etc., but stated targeted funding levels of programs like the NQNI are no higher now than they were 12 years ago, not even accounting for inflation. (I assume they will actually make NQNI awards.)  If NSF leadership keeps slashing their own budget, in defiance of congressional appropriations, it won’t matter what their priorities are. 

Lastly, let’s talk about “metascience”, the comparative study of different research funding models and practices, with the goal of a “self-improving NSF”.   Again, the essential idea of doing careful studies of alternate funding mechanisms and research team structures and practices to improve research outcomes (not a trivial matter to define) is not bad.  Doing this well is hard, because of several obvious reasons:  How do you define successful research – by scholarly impact, by patents/economic impact, by production of educated scientists, some weighted average of these?  On what timescale do you do this evaluation?  The Einstein-Podolsky-Rosen paper was hardly cited for decades, and it is now arguably one of the most influential papers of the last hundred years, with enormous impact on quantum science and technology.  How do you have sufficiently large samples and sufficiently long constant overall conditions to get reasonable statistics?   There are strong practitioners of metascience out there, but when the memo says “These efforts ultimately serve a concrete ambition for NSF to double the scientific productivity and impact generated by each federal research dollar by 2036”, how can one take that seriously?   No definitions of productivity or impact, and an implication that there will be optimization and major programmatic changes within less than two cycles of the much-vaunted five-year grants?   The word “ambition” is doing all the work in that sentence.  

Oh, and while all this is going on, the agency (among others) has now eliminated language from its integrity policy that used to say that political interference in grants and operations is bad.  I’m sure that’s nothing to worry about.

There is tremendous uncertainty in funding from multiple agencies these days, and this has resulted in many universities cutting back on doctoral admissions in the sciences and engineering.  This guarantees that there will be fewer PhD recipients in a few years across these disciplines.  The large majority of PhDs in these areas do not go into academia and instead have formed the base of technical knowhow across diverse sectors of the US and global economy for decades.  Cutting the supply like this will definitely have long-term consequences beyond just the halls of academia, affecting US competitiveness in ways that will take years to unravel.  Adding to this uncertainty is not helpful to anyone.

[This post has been cross-posted to Terence Tao’s blog.]

When I was around 11 I heard for the first time about Fermat’s Last Theorem. I was immediately captivated by the problem statement, as well as by the accompanying story, and made a fairly serious attempt to prove it. And while, unsurprisingly, I failed, I learned a lot from the attempt. Blissfully ignorant of the fact that the case had been proved by Euler over 200 years earlier, I decided that that would be a good place to start: once I had sorted that out, I was optimistic that I would be ready to tackle the general case.

Since I still couldn’t really see where to start, I decided to simplify the problem further and concentrate on successive differences of cubes, with a view to showing that such a difference could not itself be a cube. At the time I did not know how to express what I was doing in algebraic language, so I did not explicitly try to prove that the Diophantine equation had no solution. Rather, I just worked out some successive differences and stared at them, trying to get some idea of why none of them was a perfect cube. (I should be clear that this story is a reconstruction of what I think probably happened given the few memory traces that remain half a century later rather than a completely reliable account.) At some point, I had the idea of taking the difference sequence of the difference sequence, and discovered that it formed an arithmetic progression. That felt like progress, so I investigated difference sequences a bit more and discovered, purely empirically, the rule that if you start with th powers and keep taking successive differences, then eventually you get to the constant sequence .

Somehow I never managed to turn this observation into a proof of Fermat’s Last Theorem, and later on my dream of solving it got replaced by other mathematical dreams. However, when I reached the point in my mathematical education where I was taught about taking difference sequences and about what happened to polynomials, I understood those topics much better than I would have if I had not discovered difference sequences for myself and spent happy hours playing around with them. I mention this story just as an illustration of the phenomenon that was strongly emphasized in this letter signed by 25 Fields medallists, that one learns a lot from thinking about a problem, regardless of whether one solves it.

In the end, however, I felt that I could not sign the letter, despite agreeing with much of what it said. Instead, it seemed better to do what I did with the Leiden Declaration and set out my own position in a blog post. But it should be understood that by doing that I am not setting myself up as a member of some opposing camp: indeed one of my worries at the moment is that the mathematical community might become bitterly divided, something I would very much like to avoid. Also, I agree on the fundamental point that we are facing a crisis: I just want to offer a slightly different analysis of what that crisis is. I don’t claim full originality for this analysis, as I know that several other mathematicians have already put forward thoughts that are similar to the ones I have, though (for what it’s worth) I have largely come to these conclusions independently.

As I’ve written before, these past couple years I’ve often felt like the last remaining person in either quantum computing or AI who lacked a stake in some startup company whose valuation is right now shooting into interstellar space. My academic colleagues, including the ones who seemed the most singleminded about quantum oracle separations and other gloriously useless pursuits? One by one, like in a zombie movie, I learn that they too have now launched startups, and invariably raised tens of millions of dollars, for the sorts of ideas we might’ve idly traded at coffee breaks back in the day, before getting back to our real work.

So why didn’t I join this rollicking party? Partly because of a lifelong fear that, the instant my self-worth became tied to how much money I made, I’d need to humble myself before people who bluster and bully and lie and hype and conceal … yet who nevertheless succeed at becoming orders of magnitude richer than me. I’ve been terrified of even starting down that road, of whether I’d still be myself at the end of it.

It’s also partly that I can’t stand failure, or regret, or being wrong. Of course, as an academic researcher I also fail, and regret things, and am wrong constantly—but there it feels tolerable, because normally I can tell myself that it’s all just down to my inborn limitations. After all, if I could’ve solved the major open problem that someone else solved, or written the brilliant book that someone else wrote, then presumably I would’ve done it!

Clearly, though, I could’ve mined bitcoin in 2010. I could’ve gotten an early stake in Amazon or Google. It’s not even like those ideas never crossed my mind. I just … didn’t act on them, for some reason. (But even if I had, I’d probably just be full of regret that I hadn’t done even more.) Thus, my only way to avoid paralyzing regrets, has been to tell myself constantly that I’m not in the forecasting or money-making businesseses in the first place.

It helped that, insofar as I’m shallow or covetous, insofar as I’ve desired things of this world rather than insight or eternal truth, it’s never really been money that I cared about, but just being respected and liked. Elon Musk is the richest man on earth, but also one of the most despised—which isn’t a bargain that I could imagine ever appealing to me.

Plus, when I actually meet billionaires, I don’t find myself envious of their mansions or cars or anything else that they have; I don’t feel like such things would make my life any happier. Maybe I slightly envy their ability to fund the causes they care about, or their professional staffs who relieve them of drudgery, but mostly I envy the way their wealth announces, to whatever extent it does: “I was right when others weren’t.” Again, though, I’ve never trusted the world to cause me to be right about the future valuations of companies or anything similar, so I’ve settled for having been right about PostBQP and algebrization and BosonSampling.

The bottom line is that I made a choice decades ago to forgo trying to get rich, no matter how many of my friends did the same, and to strive instead to discover and tell the truth—to be a professor, a blogger, a jokester, and an “objective” arbiter and commentator. “Then, surely, everyone will like me!” my internal monologue went. “Then, surely, they’ll be grateful for all the free service I’ve rendered them—for decades of blogging, without once so much as asking for a donation or running an ad!”

HAHAHAHAHAHA.

As any regular reader will know, my attempts to be loved as a blogger backfired pretty spectacularly. Or rather: they did lead to thousands of strangers liking me (and I’m grateful for every last one of you), but they also led to probably an order of magnitude more strangers hating me, and congregating on Reddit and Twitter and elsewhere to discuss how badly I suck. And of course, trying to shift that balance by writing what people want to hear, rather than what I actually believe, was never within my realistic option set.

In the startup context, it didn’t matter how carefully I avoided taking a direct stake for or against any of the companies I blogged about. People on Twitter simply assumed that I had a stake—for example, that I must’ve shorted D-Wave or IonQ, or invested in their competitors, or had equity in AI companies. For why else would anyone write what I wrote?

Amusingly, my attackers here typically did have precisely the conflicts-of-interest that they falsely accused me of having, but that was never at issue; only my imaginary conflicts-of-interest were. Even as the Scott-haters greedily filled their pockets (or tried to), I alone needed to keep turning my pockets out to prove that they were still empty.

So then, screw it! In partnership with my brother David Aaronson, who’s long done investing professionally, and on David’s guidance and encouragement, I’m hereby embarking on a new policy.

Namely: when I hear about a brand-new startup that sounds relevant to my interests—in quantum, AI, or anything else—and I like and trust the founders (ideally, because of their previous academic research work), David and I will often make a small seed investment if the founders are open to it. Or, of course, we might become advisors or get involved in some other way.

In fact, David and I are launching BQP Partners—the link goes to our AngelList, where you can read about how to invest with us if you’re interested, if you’re an accredited investor. (See also whether you can spot any differences between David’s writing style and preoccupations and mine!)

So far, David and I are investing in:

• Oratomic (the new neutral-atom quantum computing startup out of Pasadena started by Dolev Bluvstein, John Preskill, and Hsin-Yuan Huang among others, which I previously blogged about here). See here for their Series A announcement this week.
• A non-quantum startup being cofounded by someone whose scientific work I’ve admired. I’ll write more about this one as soon as I’m able to!

I have little doubt that more potential investments will come our way very soon (some, probably, as a direct result of this post).

Crucially, I can handle my burden of regret—the “why didn’t I do this much earlier, if I was going to do it at all?” question—by telling myself that friends of mine were not founding companies left and right until very recently. I can also tell myself that I’m doing this less as a bet about the future (in which case … what if I’m wrong?), than simply as a way to support brilliant colleagues doing things that I genuinely admire.

When I blog about a company, I’ll always disclose if I have a financial position that presents a clear conflict of interest, so you can judge for yourself whether to listen to me. (Although, if that’s the sort of thing you’d demand, then you probably weren’t listening to me in the first place, were you?)

Having reflected on it a lot these past few months, I’m happy with my new policy and with my and David’s new venture, and I’m curious to see where it goes. I’m at peace with the possibility that we’ll lose our shirts, but I’m even at peace with a more disturbing possibility—that we’ll make millions and then people will scream at me online for being a sellout, a hack, and a shill. Those people, as I’ve learned, were going to scream at me anyway.

This past week we launched SCOPE, a new center for science communication and public engagement.  We marked the occasion with a fun symposium, as well as a Science Café event the preceding evening and a public science openhouse yesterday.  The symposium was very enjoyable, with a panel that comprised Kelly Weinersmith (known for many things, including an outstanding podcast and popular science books such as this Hugo-award-winner), Peter Hotez (tireless champion of vaccine development and pushing back on disinformation), Eric Berger (space editor for Ars Technica, founder of spacecityweather and theeyewall, two excellent sites for no-hype weather information), and Briana Rapini (one of The Amoeba Sisters, creators of a youtube channel with 2.9M+ followers).

You might have picked up from my 21 years of blogging that I think science communication is of great importance.  We've learned amazing things about how the world works, and I think we'd all be better off if more people knew about them and about the process of learning and discovery.  If there is public investment in research, then it's incumbent upon researchers to make sure that the public has the opportunity to learn about the fruits of those labors.  When the government, NGOs, and corporations make policies and strategic decisions that involve or depend on technical knowledge, we need to do our best to help those be informed decisions.   Once upon a time, Congress had a research office to help their staff and office holders understand technological issues.  It was killed in 1995 as "wasteful" and allegedly partisan. <sarcasm> thank goodness no technology-oriented issues have come up before the US government since then.</sarcasm>  (I am very tired of victim-blaming that presents mistrust of science or partisan razing of the research ecosystem as somehow the direct fault of scientists who failed in the communication mission.  Communication could have been better about many things, but complex societal forces are, in fact, complex, and there are many deep-seated reasons behind where we are right now.)

There were a few key points that came out of the panel above and from related discussions at the symposium.

• Know your audience and put yourself in their place.  What would you want to hear?
• Respect your audience.  You can avoid jargon without condescension.
• Ascribed to my colleague Neal Lane:  "The general public expects that you're smart.  They want to see if you're human."
• Listen to your audience.  Ascribed to Will Rogers:  "Never pass up a chance to shut up."
• If you're hesitant to do your public-facing project (writing, podcast, videos, etc.) because it's not flawless, just push through and do it.  The way to get good at this is through practice, not perfectionism.

Note: Of course I’ve been glued all week to the dramatic developments in AI. I’m working on a post about them. I’m not good at reacting to things in a timely way. So today, I’ll do my post marking the tragedy a quarter-century ago that we all commemorate. Please feel free to share your 9/11 memories in the comments. Also, Shana Tova to those who celebrate!

The morning of September 11, 2001, I was a second-year PhD student at Berkeley, who woke up late in his dorm room at International House, after a long night spent closing in on the proof of the quantum lower bound for finding collisions.

Rolling over to my laptop, I saw a flurry of weird emails, including one from Prof. Christos Papadimitriou saying that “we’re a community, and we’ll all support each other,” and another from Prof. Luca Trevisan (whose algorithms course I was then TA’ing) saying “on a day like this, it’s impossible to think about algorithms. Class is cancelled.”

Confused, I clicked over to the New York Times and saw the picture of the burning towers, and read numbly about what was already over by the time I’d woken up. I checked in with my mom, made sure relatives and friends in the NYC area were OK. My dad was at a company event in Atlanta, and would need to drive home because of the national grounding of flights.

One of my earliest memories in life, from age 5, is of ascending to the top of the World Trade Center. Growing up an hour’s drive from NYC, it wasn’t an exotic place to me.

I soon learned that one of the dead was Danny Lewin, the ex-IDF captain, theoretical computer scientist, and cofounder of Akamai who had his throat slashed on one of the planes while trying to fight the hijackers, making him the day’s first casualty, even while Akamai’s technology was part of what kept news websites running that day. I’d never met Danny but already knew many people in common with him. A few years later I’d be humbled to win the student paper award that was named in Danny’s memory.

Anyway, at Berkeley on 9/11, I wandered over to Soda Hall just to be with other people. A few students showed up for office hours, wanting help with their algorithms homework, which I found hard to believe, but I did my best to concentrate, as the computer screens around me showed the burning towers.

That evening, I went to a vigil for the victims in Sproul Plaza. But the “vigil,” such as it was, quickly dispensed with mourning and prayers and turned to applauded speeches about how the US must respond with love rather than war, and must turn the other cheek. Meanwhile, a student communist organization was handing out flyers explaining that the victims were mostly “wealthy capitalists and the workers who tried to rescue them.” This while smoke still blanketed NYC and the desperate search for survivors continued. I left the vigil early.

Until that day, I had thought of myself as basically a “leftist,” one whose #1 issue was the existential risk of climate change. Sure, I disagreed with my fellow leftists about issues from nuclear power to gifted education to Israel, but those were just intra-left disputes.

The year before, I had created the website “In Defense Of NaderTrading,” in a desperate attempt to intervene in history and cause Al Gore to become president rather than George W. Bush. When Bush “won,” by the infamous 537 votes in Florida, I considered it a victory for horribleness that would never be surpassed by anything else in my lifetime (ha). I couldn’t imagine any politician who was more the antithesis of everything I believed in than Bush. This view, of course, did not particularly stand out at Berkeley.

In the days after 9/11, though, it became obvious that I could not be a “leftist” in the Berkeley sense. Some of my fellow students felt that Osama bin Laden made a lot of great points, that the attacks were basically justified, and that at any rate, we in Amerikkka had done much worse to provoke them, including by supporting the genocidal settler-colony called “Israel,” which for all we know secretly masterminded the 9/11 attacks anyway (although again, if bin Laden had done them, he would’ve been justified).

Around the same time came the Second Intifada, when a wave of suicide bombings in Israeli buses and pizza parlors and university cafeterias thrilled and energized some Berkeley students to the extent that they took over a Holocaust Remembrance Day event with bullhorns to make it about the Nakba, smashed the windows of the Hillel building, and beat up a couple of students wearing kippot. That was how thoroughly anti-Nazi they were.

I finished my PhD at Berkeley in 2004 having learned about more than quantum computing. I’d learned that, while American academia had pockets that truly were crucial refuges and oases for nerds like me, it also harbored people who would gladly see me and my relatives and my fellow Americans killed for the sake of their ideological vision. And I’d learned that I had my own ideological vision, which was that such people could go fuck themselves.

It deeply pained me to be on the same side of anything as George W. Bush — especially because I knew that 9/11 had happened on his watch, that he had ignored all the warnings, and that he was grossly incompetent to manage the resulting wars against jihadism (just how incompetent, I didn’t know at the time). But as flawed as Bush was, I knew that I wanted to preserve rather than destroy the civilization of which he was a temporary steward. And I think the value and fragility of our civilization is the main lesson from that day that I’d like to convey to my kids, for whom of course 9/11 is just another historical event to learn about in school, like the Boston Tea Party or the Alamo.

More 25th anniversary thoughts and recollections.

I had moved to Oxford less than two weeks before, still settling into my new life in the UK, working at Imperial College in London. That day, I was heading off to Durham, in the north of England, to a conference called “A New Era in Cosmology” — my first big talk now that I had started my permanent academic job. I was going to be a bit late, only able to arrive toward the end of the first day of the conference: September 11, 2001.

My train left in the late morning. A few hours in, passengers were starting to talk about an attack on New York City. This was in the days before smartphones and constant communication — I didn’t even have a mobile phone. The discussions around me were getting more and more frantic, and I was doing my best to piece together the story.

I was born in New York City, and much of my family still lived in the area. My parents lived in the suburb of Fort Lee, New Jersey, right across the Hudson River from Manhattan; from the apartment where I grew up, we had a fantastic view from the 18th floor. My father worked in The City, commuting every morning by car from New Jersey to midtown Manhattan. He would have been at his office that day.

I considered getting off the train somewhere en route, perhaps Sheffield or York, to try to get some more information, but I just stayed on the train. I was able to get a taxi from the station in Durham to my hotel, listening to the news. I was able to call my partner, back in Oxford, who had, luckily considering the pressure on transatlantic calls, been able to get in touch with my family in New York and New Jersey. Everyone was, thankfully, alright, although at this point my father was still in Manhattan. He had noticed all the emergency vehicles speeding downtown, thinking that it was a motorcade for some foreign dignitary, but it was the first fleet of emergency responders heading towards the twin towers after the first collision. Eventually, he made it home, to my family’s apartment overlooking the Hudson. But it was so close to the George Washington Bridge, a piece of vital infrastructure thought to be in danger after the first attack, that he had to be let off a mile or so away and walk the rest.

As for me, I had to give the first talk on 12 September, about the “new era in cosmology” that coming CMB measurements would usher in. We started, understandably, with a few moments of silence, and I remember trying to come up with some appropriate words with which to start, something about needing to persevere even in the face of terrible events. I also recall that it was given on old-fashioned transparencies (I should try to dig it out of my files…) and that it was actually one of the better talks I had ever given, calmed, or at least slowed compared to my usual nervous agitation, by the events.

(My friend and colleague Peter Coles was also at the conference, offering his own reminiscences on his blog.)

As I mentioned in my last post, 9/11 feels like a milestone, and a millstone — the world hasn’t been the same since. And we still need to persevere in the face of terrible events.

More 25th anniversary thoughts and recollections.

I had moved to Oxford less than two weeks before, still settling into my new life in the UK, working at Imperial College in London. That day, I was heading off to Durham, in the north of England, to a conference called “A New Era in Cosmology” — my first big talk now that I had started my permanent academic job. I was going to be a bit late, only able to arrive toward the end of the first day of the conference: September 11, 2001.

My train left in the late morning. A few hours in, passengers were starting to talk about an attack on New York City. This was in the days before smartphones and constant communication — I didn’t even have a mobile phone. The discussions around me were getting more and more frantic, and I was doing my best to piece together the story.

I was born in New York City, and much of my family still lived in the area. My parents lived in the suburb of Fort Lee, New Jersey, right across the Hudson River from Manhattan; from the apartment where I grew up, we had a fantastic view from the 18th floor. My father worked in The City, commuting every morning by car from New Jersey to midtown Manhattan. He would have been at his office that day.

I considered getting off the train somewhere en route, perhaps Sheffield or York, to try to get some more information, but I just stayed on the train. I was able to get a taxi from the station in Durham to my hotel, listening to the news. I was able to call my partner, back in Oxford, who had, luckily considering the pressure on transatlantic calls, been able to get in touch with my family in New York and New Jersey. Everyone was, thankfully, alright, although at this point my father was still in Manhattan. He had noticed all the emergency vehicles speeding downtown, thinking that it was a motorcade for some foreign dignitary, but it was the first fleet of emergency responders heading towards the twin towers after the first collision. Eventually, he made it home, to my family’s apartment overlooking the Hudson. But it was so close to the George Washington Bridge, a piece of vital infrastructure thought to be in danger after the first attack, that he had to be let off a mile or so away and walk the rest.

As for me, I had to give the first talk on 12 September, about the “new era in cosmology” that coming CMB measurements would usher in. We started, understandably, with a few moments of silence, and I remember trying to come up with some appropriate words with which to start, something about needing to persevere even in the face of terrible events. I also recall that it was given on old-fashioned transparencies (I should try to dig it out of my files…) and that it was actually one of the better talks I had ever given, calmed, or at least slowed compared to my usual nervous agitation, by the events.

(My friend and colleague Peter Coles was also at the conference, offering his own reminiscences on his blog.)

As I mentioned in my last post, 9/11 feels like a milestone, and a millstone — the world hasn’t been the same since. And we still need to persevere in the face of terrible events.

Last week, I talked about how truthseekers get paid. But truth-tellers and truth-seekers are different things.

Consider educational kids’ shows on public television.

Nobody who works on Sesame Street is out there uncovering new letters and numbers. Bill Nye’s show wasn’t bringing analysis fresh from the lab.

The purpose of these shows is to educate. The purpose of education is to change minds.

So who pays for educational kids’ shows on public television?

If you’re from the US and watched PBS growing up, you remember one answer: “viewers like you!” US public television is supported by donations, ordinary people across the country who want it to keep on educating kids.

But you also might remember the lists of names that came before “viewers like you”. Some of those were things like “the Department of Education” or “a grant from the National Science Foundation”: government programs, in other words. Others were philanthropists and private foundations. Some were tied to companies, like the Intel Foundation, or Juicy Juice.

All of these groups, from government departments to donors, are trying to change kids’ minds. They support specific shows on specific topics, where they want kids to be better-informed. The same groups have the same kind of impact on schools. For example, I remember in elementary school we all learned to play a recorder, because a wealthy donor had given the school recorders out of the idea that music education was especially important.

For a truth-seeker like a journalist, accepting that kind of funding would be a problem. Grants for journalists tend to support things like travel, letting journalists learn more about specific topics, not pre-judging the conclusion. But children’s television is about truth-telling, not truth-seeking, so our standards are different. We trust the people making children’s television to care about whether they’re telling the truth. And because the topics aren’t new, we don’t usually worry about their judgement being biased.

All this is rather obvious. But now, consider science YouTube.

Some science YouTubers seem to have a mission much like children’s television. They’re there to teach, not to make independent judgements. They don’t search for truth on their own. And some of them are funded by educational grants, much like children’s television.

Others are a bit more like journalists, or even activists. People follow them for their opinions, to hear their assessment. They’re trying to be truth-seekers.

On YouTube, it’s not always obvious which is which.

There’s a particular group of philanthropists called Effective Altruists, and many of them are concerned about AI. So in between funding things like anti-malaria bed nets, some of them are giving grants to YouTubers to make educational content about AI-related risks.

Apparently, they reached out to Sabine Hossenfelder, which was a bad idea. Sabine Hossenfelder’s followers aren’t just looking for education on known facts. They’re looking for her judgements, her literal bullshit-rating on ideas. And so while she’s paid by “viewers like you”, she’s not really the type to get paid by that type of grant.

What I want to emphasize, and what looked like it was getting lost in the discussion, was that their pitch would have been totally reasonable for other YouTubers. Educators do occasionally get grants to educate on specific topics. This is in fact a totally normal thing. Some YouTubers are educators first and foremost, they aren’t there as truth-seekers, but truth-tellers, with a real difference in how careful they need to be about bias.

Some YouTubers are different from other YouTubers. News at 11.

I’ve been thinking about the exceptional Lie algebra E₆, as a spinoff of my project on E₇, so I want to get a good mental picture of the E₆ root polytope. This is 6-dimensional polytope with remarkable symmetry.

Let’s climb up to it, starting with some of its 4-dimensional faces, which are called 4-demicubes because you get them by taking a 4-dimensional cube, or tesseract, and removing every other corner. The 3-demicube is just a tetrahedron, since you can fit two tetrahedra in a 3-dimensional cube like this:

The 4-demicube builds on this fact in a surprising way.

I’m going to use the technology of Dynkin diagrams, or technically Coxeter diagrams: they’re closely related, and the difference is invisible here. I won’t explain them, just use them. I explained them here:

• Symmetry and the fourth dimension: part 3, part 4, part 5, part 6.

Let’s dive in!

The 4-demicube lives in 4 dimensions. It has 8 vertices.

You get it from a 4-dimensional cube, which has 2⁴ = 16 vertices, by keeping every other vertex, throwing away half. That leaves 8.

What are its top-dimensional faces, aka ‘facets’? Surprise: there’s only one kind! All of them are regular tetrahedra.

In higher dimensions the demicube has two kinds of facet. You get a simplex-shaped facet from every other vertex, formed when you remove it. And you get a demicube-shaped facet from each of the cube’s facets. But in 4 dimensions the two kinds happen to be the same shape!

Eight of them are tetrahedra. These appear at the 8 corners you sliced off: one per removed corner.

Eight more come from the 8 faces of the 4-dimensional cube. These are 3-demicubes. But as we’ve seen, the 3-demicube is also a tetrahedron!

So the 4-demicube is especially symmetric: it has 16 tetrahedral facets. You can find coordinates where its vertices are

It’s actually one of the 4-dimensional regular polytopes, sometimes called the 4-orthoplex. It’s also called the 16-cell because it has 16 facets. It’s the 4-dimensional cousin of the octahedron, which has 8 triangular facets.

You can read some of these facts off the D₄ Dynkin diagram, if you know what you’re doing. As you can see above, this diagram has a central node with three arms, each just 1 edge long: a perfectly symmetric three-pronged star. To get the 4-demicube, you ring the tip of any one arm.

To get the facets of the 4-demicube, delete an unringed node so the piece still holding the ring stays connected, and see what diagram survives. There are two choices: you can delete the tip of either other arm. But either way, what’s left is a straight chain of 3 nodes—the so-called A₃ diagram—with a ring at one node at the end. This gives the tetrahedron.

Both choices give the same shape of facet, a tetrahedron, because all three arms of the D₄ Dynkin diagram are interchangeable. That ceases to be true in higher dimensions!

Next, the 5-demicube. This lives in 5 dimensions and has 16 vertices.

You get it from a 5-dimensional cube—which has 2⁵ = 32 vertices—by keeping every other vertex, throwing away half. That leaves 16.

What are its top-dimensional faces, or ‘facets’? There are two kinds!

Sixteen of them are 4-dimensional analogues of the regular tetrahedron, called 4-simplexes. These appear at the corners you sliced off: one per removed corner.

The other ten come from the ten faces of the 5-dimensional cube. After you take every other vertex, they become 4-demicubes. These are precisely the 4-demicubes we saw in the last section!

You can also read these two kinds of facets from the D₅ Dynkin diagram. As you can see above, this diagram has three arms of lengths 2, 1, 1 (edges from the central branch node). To get the 5-demicube, you ring the tip of either length-1 arm. That ringed diagram encodes the whole polytope.

To get the facets, delete an unringed node so the piece still holding the ring stays connected, and see what diagram survives.

There are two choices.

If you delete the tip of the other length-1 arm, what’s left is a straight chain of 4 nodes—the diagram whose polytope is the 4-simplex. That gives the 4-simplex faces.

Or you can delete the tip of the length-2 arm. Then what’s left is a shorter branching diagram, the one I showed you in my last post! That gives the 4-demicube faces.

So the 5-demicube has both 4-simplex and 4-demicube faces.

Next let’s go up to the 6th dimension, which was my goal all along.

The E₆ root polytope lives in 6 dimensions. It has 72 vertices.

What are its facets? You can read them straight off the E₆ Dynkin diagram, using the same procedure we’ve been using so far.

As you can see, the E₆ Dynkin diagram has three arms of lengths 2, 2, 1 (edges from the central branch node). To get the root polytope, you ring the node that’s the tip of a length-1 arm. That fact is not obvious, but let’s go ahead and do that.

Then, to get the facets, delete any unringed node such that the piece still holding the ring stays connected, and see what diagram survives.

There are two choices: the two other nodes at tips of the Dynkin diagram.

However, deleting either of these nodes leave a D₅ diagram with a ring on one node, and this gives the 5-demicube we saw last time: a 5-cube with alternate vertices removed.

So the facets of the E₆ root polytope are all the same shape: 5-demicubes!

With more work, we can count the facets of the polytopes we’ve been studying:

• The E₆ root polytope has 54 facets, all 5-demicubes. They come in two kinds, because we had two choices of which node to delete, so there are really 27 ‘positive’ 5-demicube facets and 27 ‘negative’ 5-demicube facets.

• The 5-demicube has 16 4-simplex facets, one for each vertex that we removed from the 5-cube to create this demicube, and 10 4-demicube facets, one for each facet of that 5-cube.

• The 4-demicube has 8 3-simplex facets, one for each vertex that we removed from the 4-cube to create this demicube, and 8 3-demicube facets, one for each facet of that 4-cube. But both the 3-simplex and the 3-demicube are the familiar tetrahedron. So in fact the 4-demicube has 16 tetrahedral facets. Indeed, the 4-demicube is the 4-dimensional analogue of an octahedron: the so-called 4-orthoplex, or 16-cell.

Using some fancier math I explained here, we can count all the faces of the E₆ root polytope. This polytope, is also called 1₂₂ due to the shape of its Dynkin diagram: the ring is on a branch of length 1, not counting the central node, while the other two branches have lengths 2. You can look up all this information on the Wikipedia page 1₂₂ polytope:

The 5-dimensional facets are all 5-demicubes, but as we’ve seen, they come in two kinds: that is, they lie in two orbits of the symmetry group. We can call 27 of them ‘positive’ 5-demicubes and 27 of them ‘negative’ demicubes. Of the 4-dimensional faces, 270 are 4-demicubes and 432 are 4-simplexes. Moreover the 4-simplexes come in two kinds: 216 are faces of positive 5-demicubes while 216 are faces of negative 5-demicubes. Let’s call the first kind of 4-simplex ‘positive’ and the second kind ‘negative’. The 3-dimensional faces are all tetrahedra, but they come in two ‘kinds’: 1080 of them are faces of positive 4-simplexes, and 1080 are faces of negative 4-simplexes. None is the face of both a positive and negative 4-simplex.

If you’re curious about how to count these things, see how some of us counted all the faces of the E₈ root polytope here:

• John Baez, Integral octonions (part 5), The n-Category Café, September 3, 2013.

Here is a table of faces for the E₇ root polytope, which is also called 2₃₁:

Its 4-dimensional faces are all 4-simplexes, but they come in two ‘kinds’: that is, they lie in two orbits of the symmetry group of this polytope. Of the 4-simplexes, 4032 are the face of three 5-orthoplexes, while 12096 are the face of one 5-orthoplex and two 5-simplexes.

Here’s the E₈ root polytope, also called 4₂₁:

There are two kinds of 6-simplex faces: 138240 of them each lie in one 7-simplex and one 7-orthoplex, while 69120 of them each lie in two 7-orthoplexes (and no 7-simplex).

With some more text…

Checking some infrastructure…

My colleage Sameer Deshpande, together with Shekhar Shah, and Paul Nguyen are doing Badgers on Tap Wednesday night 9/9 at 6:30pm at One Social Food Hall downtown; there will be talk about post-Moneyball sports analytics, beer, and trivia. Not sure I myself can make it but this is sure to be a good time with some savvy Badgers. Go!

I’ve been thinking about the exceptional Lie algebra E₆, as a spinoff of my project on E₇, so I want to get a good mental picture of the E₆ root polytope. This is 6-dimensional polytope with remarkable symmetry.

Let’s climb up to the E₆ root polytope starting with some of its 4-dimensional faces, which are called 4-demicubes because you get them by taking a 4-dimensional cube, or tesseract, and removing every other corner. The 3-demicube is just a tetrahedron, since you can fit two tetrahedra in a 3-dimensional cube like this:

The 4-demicube builds on this fact in a surprising way.

I’m going to use the technology of Dynkin diagrams, or technically Coxeter diagrams: they’re closely related, and the difference is invisible here. I won’t explain them, just use them. I explained them here:

• Symmetry and the fourth dimension: part 3, part 4, part 5, part 6.

Let’s dive in!

The 4-demicube lives in 4 dimensions. It has 8 vertices.

You get it from a 4-dimensional cube, which has 2⁴ = 16 vertices, by keeping every other vertex, throwing away half. That leaves 8.

What are its top-dimensional faces, aka ‘facets’? Surprise: there’s only one kind! All of them are regular tetrahedra.

In higher dimensions the demicube has two kinds of facet. You get a simplex-shaped facet from every other vertex, formed when you remove it. And you get a demicube-shaped facet from each of the cube’s facets. But in 4 dimensions the two kinds happen to be the same shape!

Eight of them are tetrahedra. These appear at the 8 corners you sliced off: one per removed corner.

Eight more come from the 8 faces of the 4-dimensional cube. These are 3-demicubes. But as we’ve seen, the 3-demicube is also a tetrahedron!

So the 4-demicube is especially symmetric: it has 16 tetrahedral facets. You can find coordinates where its vertices are

$$(\pm 1, 0, 0, 0), \quad (0, \pm 1, 0, 0), \quad (0, 0, \pm 1, 0), \quad (0, 0, 0, \pm 1)$$

It’s actually one of the 4-dimensional regular polytopes, sometimes called the 4-orthoplex. It’s also called the 16-cell because it has 16 facets. It’s the 4-dimensional cousin of the octahedron, which has 8 triangular facets.

You can read some of these facts off the D₄ Dynkin diagram, if you know what you’re doing. As you can see above, this diagram has a central node with three arms, each just 1 edge long: a perfectly symmetric three-pronged star. To get the 4-demicube, you ring the tip of any one arm.

To get the facets of the 4-demicube, delete an unringed node so the piece still holding the ring stays connected, and see what diagram survives. There are two choices: you can delete the tip of either other arm. But either way, what’s left is a straight chain of 3 nodes—the so-called A₃ diagram—with a ring at one node at the end. This gives the tetrahedron.

Both choices give the same shape of facet, a tetrahedron, because all three arms of the D₄ Dynkin diagram are interchangeable. That ceases to be true in higher dimensions!

Next, the 5-demicube. This lives in 5 dimensions and has 16 vertices.

You get it from a 5-dimensional cube—which has 2⁵ = 32 vertices—by keeping every other vertex, throwing away half. That leaves 16.

What are its top-dimensional faces, or ‘facets’? There are two kinds!

Sixteen of them are 4-dimensional analogues of the regular tetrahedron, called 4-simplexes. These appear at the corners you sliced off: one per removed corner.

The other ten come from the ten faces of the 5-dimensional cube. After you take every other vertex, they become 4-demicubes. These are precisely the 4-demicubes we saw in the last section!

You can also read these two kinds of facets from the D₅ Dynkin diagram. As you can see above, this diagram has three arms of lengths 2, 1, 1 (edges from the central branch node). To get the 5-demicube, you ring the tip of either length-1 arm. That ringed diagram encodes the whole polytope.

To get the facets, delete an unringed node so the piece still holding the ring stays connected, and see what diagram survives.

There are two choices.

If you delete the tip of the other length-1 arm, what’s left is a straight chain of 4 nodes—the diagram whose polytope is the 4-simplex. That gives the 4-simplex faces.

Or you can delete the tip of the length-2 arm. Then what’s left is a shorter branching diagram, the one I showed you in my last post! That gives the 4-demicube faces.

So the 5-demicube has both 4-simplex and 4-demicube faces.

Next let’s go up to the 6th dimension, which was my goal all along.

The E₆ root polytope lives in 6 dimensions. It has 72 vertices.

What are its facets? You can read them straight off the E₆ Dynkin diagram, using the same procedure we’ve been using so far.

As you can see, the E₆ Dynkin diagram has three arms of lengths 2, 2, 1 (edges from the central branch node). To get the root polytope, you ring the node that’s the tip of a length-1 arm. That fact is not obvious, but let’s go ahead and do that.

Then, to get the facets, delete any unringed node such that the piece still holding the ring stays connected, and see what diagram survives.

There are two choices: the two other nodes at tips of the Dynkin diagram.

However, deleting either of these nodes leave a D₅ diagram with a ring on one node, and this gives the 5-demicube we saw last time: a 5-cube with alternate vertices removed.

So the facets of the E₆ root polytope are all the same shape: 5-demicubes!

With more work, we can count the facets of the polytopes we’ve been studying:

• The E₆ root polytope has 54 facets, all 5-demicubes. They come in two kinds, because we had two choices of which node to delete, so there are really 27 ‘positive’ 5-demicube facets and 27 ‘negative’ 5-demicube facets.

• The 5-demicube has 16 4-simplex facets, one for each vertex that we removed from the 5-cube to create this demicube, and 10 4-demicube facets, one for each facet of that 5-cube.

• The 4-demicube has 8 3-simplex facets, one for each vertex that we removed from the 4-cube to create this demicube, and 8 3-demicube facets, one for each facet of that 4-cube. But both the 3-simplex and the 3-demicube are the familiar tetrahedron. So in fact the 4-demicube has 16 tetrahedral facets. Indeed, the 4-demicube is the 4-dimensional analogue of an octahedron: the so-called 4-orthoplex, or 16-cell.

Using some fancier math I explained here, we can count all the faces of the E₆ root polytope:

If you’re curious about how to count these things, see how some of us counted all the faces of the E₈ root polytope here:

• John Baez, Integral octonions (part 5), The n-Category Café, September 3, 2013.

Twenty-five years ago, I moved from San Francisco to the UK, from a fellowship at Berkeley to a permanent job at Imperial College, London. A lot has changed since then. I was 35; now I am 60. It was two weeks before 9/11; the world hasn’t seemed as open and free since.

I arrived from the Bay Area just after the first dot-com bubble burst. London, adjusting to New Labour after almost two decades of Thatcher and Thatcherism, felt exciting and vibrant. But just weeks after I arrived came the horror of 9/11, and its years-long aftermath, especially the Iraq war which eventually doomed Blair’s premiership and probably led the way to the 2010 election, the disaster of “austerity” as a wrong-headed attempt to deal with the 2008 recession, and eventually to the more-disastrous Brexit on this side of the Atlantic. Similar politics, though with very different timing, led back in the USA to Obama, one of the few rays of political hope over the last quarter-century, but then, of course, to Trump. And everywhere since 2001 the rise of nativist populism making me feel at home, well, pretty much nowhere — a rootless cosmopolitan. Higher education, scientific funding, and curiosity-driven research are in a parlous state in both the US and the UK.

But: despite a few difficult years in the mid-2000s, I have prospered. Our analysis of data from the Planck satellite has solidified our standard cosmological model — but also given us new problems and puzzles to think and worry about. I have written a book, The Random Universe, trying to explain how we know what we know as scientists and as human beings. And my family, my wife and two daughters, are a source of joy and excitement that inspire me every day.

So now I am 60. I was honoured and humbled a couple of months to ago to be joined by many of my colleagues and scientific friends at a conference here in London. That, and getting a Transport For London 60+ Travel Card, makes it hard to avoid feeling old. But those colleagues and friends (many of whom are older than me) reassured me that it’s only the beginning of a new chapter.

Interesting developments tonight, as Levent Alpöge and Tristan Buckmaster announce that after a fair amount of work they have constructed examples of finite-time blowup for a broad class of PDEs including 3-d incompressible Euler, inspired by of Diego Córdoba and Luis Martínez-Zoroa, and using plenty of LLM iteration in order to get the details right. This is, of course, a problem in the neighborhood of Navier-Stokes (in the negative direction of finding a counterexample to the conjecture, which I have over the years heard many PDE folks saying was the right way to bet), and Terry Tao says in a Mastodon thread that in principle this method doesn’t seem so far from showing blowup for Navier-Stokes too, though a large amount of compute and detail-checking would be involved.

At least part of this has already been Lean-formalized, though perhaps eccentrically I find I care a little less about that. What matters is not whether there’s an example but whether the example has something to teach us. An interesting but incorrect example would surely be of more value than an uninteresting but correct one. Well, I suppose the latter would have more financial value. Though even on that Millennium Prize page, one sees: “Why ask for a proof? Because a proof gives not only certitude, but also understanding.” Very true! We mustn’t settle for mere certitude. Certainly the work of Alpöge, Buckmaster, Córdoba, and Martínez-Zoroa seems to offer understanding as well.

I’m well underway on revising Don’t Be Too Sure, which I finished a first draft of right before surgery. A lot of people make appearances in this book, most of all William James and John von Neumann, who became the two main characters despite not being in my original plans for the book at all. Some other people: Felix Hausdorff, Alfred Kroeber and his daughter Ursula Kroeber Le Guin, John Keats, Sheila Heti, Katharine Briggs and her daughter Isabel Briggs Myers, Jakob Bernoulli, Elbert Hubbard, Anna Kiesenhofer, Grace Hopper, Thomas Jefferson, Caroline Hoxby, David Hilbert, Michel Adanson… well, there are a lot of people in it, who do a lot of things.

As we descend from the base of Earth’s crust through the mantle, the rock does not remain unchanged. Pressure and temperature rise inexorably, and the minerals that thrive at the surface are forced, step by step, into new and denser crystallographic arrangements. This is the story of those transformations.

In this tale, I’ll act like I know a bit about minerals. I actually don’t: there are a bewildering variety, and I can never remember them. So don’t worry: when you come across a jargon-filled patch of prose, just power through it. You might learn a little… or you can just ignore it. The overall point here is that the Earth is made of beautiful crystalline structures that change character in complex ways as we descend.

The Mohorovičić discontinuity

Our story begins at the boundary where Earth’s crust, rich in feldspar and quartz, gives way to the denser mantle beneath. We see this boundary through its effect on seismic waves, and it’s called the Mohorovičić discontinuity or “Moho”. The Moho does not lie at one fixed depth: it’s 5–10 kilometers below the seafloor, but 30–50 kilometers below most continents, and as much as 70–80 below young mountain belts like the Himalayas.

The mantle just below the Moho mainly consists of a rock called peridotite, which is made mostly of olivine and pyroxene, with smaller amounts of garnet (or, at shallower depths, spinel). Peridotite has a delicious coarse green appearance:

More precisely, this is what peridotite looks like up here. But when geochemists talk about the bulk composition of the upper mantle, they often use an idealized model called pyrolite—not a rock you can pick up, but a hypothetical recipe Ted Ringwood proposed in the 1960s for the primitive upper mantle.

Why? Since the Earth has had a convecting mantle, solid mantle rock wells up in places. As it does, the pressure drops, and a bit of it melts: the minerals with lower melting points. This melt flows upward. It’s called basalt. It builds the Earth’s crust. But it leaves a residue behind, made of minerals with higher melting points.

In Ringwood’s theory, which for expository purposes I’ll assume is true, pyrolite is what mantle rock is like before any partial melting depletes it of basaltic ingredients. The name is a portmanteau of pyroxene and olivine, the two dominant minerals. Pyrolite is about 60% olivine; the remaining 40% is mostly pyroxenes plus garnet.

• A pyroxene is a mineral built from single, unbranched chains of corner-sharing SiO₄ tetrahedra, with metal cations—chiefly Mg, Fe, and Ca—linking the chains together. The general formula is XY(Si,Al)₂O₆, where X and Y are those cations.

• Olivine is a green silicate, (Mg,Fe)₂SiO₄:

Its crystal structure in the upper mantle is an orthorhombic arrangement of isolated SiO₄ tetrahedra knit together by magnesium and iron in octahedral sites. It’s called the α-phase because we’ll see some more compressed phases as we descend.

• A garnet is built from separate SiO₄ tetrahedra held together by cations, but assembled into a dense, hard, characteristically cubic-symmetry crystal. There are different kinds of garnet, but the general formula is X₃Y₂(SiO₄)₃: three divalent X cations, two trivalent Y cations, and three isolated silica tetrahedra. The mantle’s garnet is largely pyrope, Mg₃Al₂(SiO₄)₃.

As we descend, the pyroxenes and garnet gradually dissolve into each other, producing a new high-pressure mineral called majorite. Here’s a rare sample from a meteorite fall in Canada:

So even before the dramatic change 410 kilometers down, the rock is no longer the simple olivine-pyroxene-garnet assemblage we had further up.

The 410-kilometer discontinuity

Roughly 410 kilometers down, the pressure reaches about 13,000 atmospheres and the temperature hovers around 1,400°C. Olivine can no longer hold its familiar shape. It transforms to its β form: wadsleyite, a mineral with the same chemical formula but a fundamentally different atomic arrangement. Instead of isolated SiO₄ tetrahedra, wadsleyite contains paired Si₂O₇ groups, and the oxygens pack more densely. The density jump is sharp enough to be detected globally by seismologists as a reflector of earthquake waves.

Wadsleyite has a remarkable property: it can hold several weight percent of water locked within its crystal structure. The transition zone may thus contain more water than all the oceans combined! However, very little wadsleyite has been seen on the Earth’s surface. Here’s a bit from that same meteor fall in Canada:

The 520-kilometer discontinuity

Descend further, to around 520 kilometers, and the temperature goes up only a little, to roughly 1500–1600°C, since convection here is strong. The pressure goes up to about 175,000 atmospheres. At this point wadsleyite transforms into the γ form of olivine: ringwoodite. This is denser, still chemically Mg₂SiO₄, but now with cations packed into tetrahedral and octahedral holes in a close-packed oxygen framework—the most efficient packing geometry that nature offers for this composition:

Ringwoodite is named for the great Australian geochemist Ted Ringwood, who studied these transitions. Here’s an artificially manufactured sample:

For a long time the mineral’s existence in the mantle was purely hypothetical. But in 2014, a tiny grain was discovered as an inclusion inside a diamond brought up from the deep mantle by an eruption, providing the first direct proof of its existence in Earth’s interior.

The 660-kilometer discontinuity

At a depth of 660 kilometers and a pressure of roughly 230,000 atmospheres, the most dramatic phase transition of all occurs. Ringwoodite does not merely rearrange into a still more dense form! Instead, it decomposes into two entirely new minerals: bridgmanite (MgSiO₃) and ferropericlase (MgO). The majorite garnet also decomposes, yielding davemaoite (CaSiO₃), which is stable through the rest of the lower mantle:

The 660-kilometer discontinuity is sharp, globally consistent, and marks the conventional boundary between the upper and lower mantle. One reason it’s important is that enormous slabs of colder, denser rock sink through the upper mantle until they hit this discontinuity, where the phase change between ringwoodite and bridgmanite creates a kind of barrier.

These slabs are 30–100 kilometers thick and hundreds to a thousand kilometers across! Some punch straight through into the lower mantle and keep sinking. But many flatten out when they hit the barrier, sometimes lying there and piling up for tens of millions of years. You can see this in seismic images beneath Japan and the Marianas. Numerical models suggest that they pile up until they overwhelm the barrier and flush down in a comparatively sudden avalanche—lasting mere millions of years.

The lower mantle

This is the realm of bridgmanite, probably the most abundant mineral in the Earth. Bridgmanite is a beautifully symmetric cage of corner-sharing SiO₆ octahedra, with Mg tucked into the large cavities between them. It accommodates enormous pressure because there is very little void space left to compress.

It is a striking fact that while bridgmanite is the most abundant mineral on the planet, it went unnamed until 2014, simply because no natural hand-sized specimen had ever been recovered. Everything we know about it comes either from high-pressure laboratory synthesis, from microscopic grains in shocked meteorites, or from the indirect testimony of earthquake waves that have traveled through 2,000 kilometers of it.

For over 2,000 kilometers of descent, from 660 to roughly 2,700 kilometers down, bridgmanite and its companion ferropericlase reign without significant further phase change. Seismic velocities increase steadily, but there are no dramatic discontinuities.

The D″ discontinuity

As we approach the core-mantle boundary—at depths around 2,700 kilometers, pressures of approximately 120,000–125,000 atmospheres, and temperatures of 2,200–3,7000°C—even bridgmanite yields. It transforms into the post-perovskite phase. Post-perovskite is a layered, sheet-like structure of SiO₆ octahedra, quite different from bridgmanite’s three-dimensional cage, making it potentially much weaker and more prone to flow.

This transition is believed to be responsible for the seismic D″ discontinuity observed at 2,900 kilometers depth. The D″ layer is a highly dynamic region, likely the site of storage of subducted materials and the source of deep mantle plumes.

A summary of the descent

The table below summarizes the major transitions:

The interesting thing about this story is that it was told first by seismology—the sharp jumps in wave speeds at 410 and 660 kilometers were detected long before geologists could reproduce those pressures in the lab—and only later checked by diamond-anvil cell experiments squeezing tiny mineral samples to millions of atmospheres. The rocks never rise to the surface to tell their story directly, so much of the tale above is just theory.

Which minerals are there the most of?

We can estimate how much of the Earth is made of wadsleyite, ringwoodite, and bridgmanite using known shell volumes, estimated densities, and mineral proportions from the pyrolite model.

Step 1: Earth’s mass budget by layer

The Earth’s total mass is M_⊕ ≈ 5.972 × 10²⁴ kg. The mass budget by layer is approximately:

•    Crust: ~0.4% of Earth’s mass
•    Upper mantle + transition zone (35–660 km): ~18% of Earth’s mass
•    Lower mantle (660–2,891 km): ~49% of Earth’s mass
•    Core (outer + inner): ~32.5% of Earth’s mass

Step 2: The transition zone (410–660 km)

Using PREM densities averaging ~3,760 kg/m³ across the transition zone, and the volume of each spherical shell:

Wadsleyite zone (410–520 km):
Shell volume ≈ 4.8 × 10¹⁹ m³
Shell mass ≈ 1.76 × 10²³ kg
Fraction of Earth’s mass ≈ 2.9%

Ringwoodite zone (520–660 km):
Shell volume ≈ 5.9 × 10¹⁹ m³
Shell mass ≈ 2.24 × 10²³ kg
Fraction of Earth’s mass ≈ 3.8%

In the pyrolite model of mantle composition, forms of olivine (wadsleyite and ringwoodite) make up roughly 60% of the transition zone by mass, with the remaining ~40% being majoritic garnet. Applying this correction:

Wadsleyite: 0.60 × 2.9% ≈ 1.8% of Earth’s mass
Ringwoodite: 0.60 × 3.8% ≈ 2.3% of Earth’s mass

These estimates carry roughly 20–30% uncertainty, mainly from the assumed 60% olivine proportion in the transition zone, which varies with local temperature and bulk composition.

Step 3: Bridgmanite (660–2,700 km)

The lower mantle holds about 49% of Earth’s mass—it is an enormous shell! Bridgmanite constitutes approximately 80% of the lower mantle mineral assemblage (by mass) in the pyrolite model:

0.80 × 49% ≈ 39% of Earth’s mass

This is consistent with the well-cited literature figure that bridgmanite comprises approximately 38% of the planet’s mass—making it the single most abundant mineral in the Earth by a vast margin.

Thus, these three minerals—all members of the same Mg₂SiO₄/MgSiO₃ chemical lineage—together constitute roughly 42% of Earth’s entire mass. All other named minerals on Earth, including quartz, feldspar, calcite, diamond, and the roughly 3,800 others known to mineralogists, divide up the remaining scraps.

Rethinking the electric field

Have you ever wondered what an electric field actually is? 

The electric field is the foundation of most technologies that we rely on every day. From power grids and electronic devices to radio communication and the internet, the electric field is extremely relevant to our daily lives. However, despite its importance, I have always felt that the common explanations of the electric field leave something unanswered. 

Most textbooks define the electric field as a property of space or a physical entity surrounding electric charges, or with the equation of force per unit charge. These definitions help us understand what the electric field does and its effect on electrically charged particles, but they do not fully answer what an electric field actually is and how it influences charges. Thus, I started thinking about the question: what allows charges to influence each other without touching?

This question led me down a path that began with a simple observation in everyday life, and it eventually pointed toward much deeper ideas in modern physics.

Objects Influenced by Their Surroundings

Before talking about electric fields, let’s consider a more basic question: Does it seem reasonable that objects can be influenced by their surroundings? 

Most people would answer yes. We have all seen examples of objects responding to something else nearby, such as the Earth orbiting the Sun, a compass needle reacting to a magnet, and our phones responding to signals from a WiFi router. But what is the mechanism behind these interactions? 

A simple physical phenomenon that we can look at is a balloon rubbed on a piece of clothing that can pick up strands of our hair. Many of us have seen this demonstration in kindergarten or first grade of elementary school. This might seem completely ordinary, but if we pause and think about it, something strange is happening – the balloon is influencing the hair without touching it. 

How is that possible? One answer is simply that the balloon “pulls” on the hair, but this raises more questions. How does the balloon reach the hair? What is happening in the space between them? These questions suggest that something is missing from the picture of objects pulling on each other directly through contact.

To put this in the context of physics, we might all have learned that “like charges repel, and opposite charges attract”. We might have solved equations on how fast charges would move away from or toward each other. We were always told to just accept it because these motions result from the electric field. But why do these observations happen? What is happening between the charges?

Historically, physics encountered the same problem. If one object can influence another at a distance, it is natural to ask what is happening in the space between them. One guiding principle that physicists often use is the concept of locality. Locality is the idea that an object can only be directly influenced by its immediate surroundings. Thus, an influence should not simply leap across space from one object to another, and changes should propagate through intermediate regions step by step. 

At first glance, locality seems reasonable because it matches many of our everyday experiences. If I push a book across a table, my hand influences the book through direct contact. The influence does not appear to jump instantaneously across the table. 

However, locality creates a tension when we return to the scenario of the balloon pulling on strands of hair. If locality is true, something must be happening in the space between the balloon and the hair. But from a standard electromagnetic perspective, the space between the balloon and the hair is empty. Therefore, we have encountered a contradiction: if what is between the balloon and the hair is empty space, then what is responsible for transmitting the influence? Neither the usual electromagnetism nor locality tells us the answer to these questions.

The Classical Electric Field

In the usual electromagnetic picture, the answer to the puzzle is the electric field. Rather than allowing charges to influence one another directly across space, the theory assigns an electric field to the space surrounding charges. The field acts as the intermediary through which influence is transmitted. 

A useful way to think about the electric field is that it assigns information to every point in space. If we imagine a charged particle that is placed at a particular location, the electric field tells us how that particle would move. This charged particle is what physicists call a test charge. By observing how the test charge behaves, we can infer information about the electric field at that location. 

This could seem like a satisfying answer as the electric field tells us how influence is transmitted, but it does not tell us what kind of thing is doing the transmission. Is the electric field a physical substance? Is it a mathematical tool? Or is it something else? 

It might be easy to fall back on the idea that the electric field ultimately works through tiny particles physically touching one another. After all, contact interactions are among the most familiar interactions that we experience. 

But physics challenges this intuition as well. It is surprisingly difficult to define what it means for two objects to “touch”. We usually think of the balloon attracting hair as an example of action at a distance, whereas pressing a hand on a table feels like direct physical contact. However, at the microscopic level, the two situations are fundamentally similar. If we could zoom in on our fingertip and the table with a microscope, we would find that the atoms in our skin never make contact with the atoms in the table. This is because of the repulsion between the electron clouds surrounding the atoms, which prevents the two atomic nuclei from overlapping. Say if we scale the atom in the table up to be the size of a marble, then the nearest atom in our fingertip would still be separated from it by a few centimeters. In the end, nothing is truly “touching” in the intuitive, physical sense. 

Thus, the idea of contact does not solve our problem. We are forced to ask the same question again: what is it that allows these interactions to occur? To answer that question, I turned to a different perspective of thinking about electric fields.

The Electric Field as A Dynamical Structure

From our intuition, it is natural to imagine the electric field as some invisible substance filling space. This is often the picture suggested by the common field line diagrams in physics textbooks, which make the field appear to flow outward or inward from charges, almost like a moving fluid. 

A useful analogy is the ocean. A boat floating on water can move because waves pass beneath it. The boat responds to changes in its surrounding waves rather than to some direct push from a distant object. Similarly, charged particles respond to changes in the electric field around them. We can then view the electric field as a dynamical structure that governs how the state of the world can evolve.

However, the ocean analogy can only take us so far. Ocean waves are made of water molecules. Sound waves are made of vibrating air molecules. But what is the electric field made of? When light travels through empty space, it seems that there is no material medium at all. 

This brings us back to the mystery: if locality suggests that something must exist in the space between interacting objects, and if the electric field is not made of the ordinary matter that we understand, then what exactly is occupying the space? 

To answer this question, we have to rethink what we mean by “empty” space itself.

Empty Space is Not Empty

Conventionally, we have always imagined empty space as exactly what the name suggests—empty. Just like if all the particles were removed and nothing was remaining. But modern physics suggests a very different picture. 

In quantum field theory, what we call “empty space” is not truly empty. Empty space is filled with underlying quantum fields that permeate all of space and time, even in the absence of particles. Even when the surface of the ocean looks perfectly still, the water is still there. The ocean is not defined only by visible waves, but by the underlying medium that can support waves in the first place. The waves are patterns of motion of the ocean itself, just like the electric field. These fields are part of the fundamental structure of the universe from which physical phenomena emerge. Quantum field theory suggests that particles are not independent objects moving through an otherwise empty space. Rather, they are localized patterns or excitations of underlying fields that already exist throughout the universe. 

From this perspective, the electric field is not something that is added to empty space. It is part of the fundamental dynamical structure of space itself.

Conclusion

At the beginning, I asked a simple question: how can objects influence each other without touching? The straightforward answer is the electric field. Charges create electric fields, and those fields determine how other charges move. But what is an electric field? Is it an invisible material filling space between objects, or is it a dynamical structure that governs how physical systems in the world evolve? 

From the perspective of quantum field theory, quantum fields permeate all of space and time. Particles are not separate objects moving through an empty space, but are excitations of these underlying fields; electric fields are not secondary matter surrounding charged particles, but are particular configurations of the underlying electromagnetic quantum fields. What we observe as the motion of a charged particle is the result of its interaction with the electromagnetic field, whose local state determines how the particle evolves. 

In the end, our original question may not have a single definitive answer. But asking it revealed a shift in perspective, and physics has repeatedly shown that every explanation opens the door to an even more fundamental question. Stopping at this step, a new mystery emerges: what are these underlying quantum fields themselves? What are they made of, and where do they arise from? 

Academics and journalists have a lot in common, at least in principle.

Whether you’re a reporter or a professor, your job is to go out into the world and figure out the truth. You’re supposed to be careful, to check and correct for how you might be wrong. And at the end of the day you’re supposed to communicate what you found.

The differences mostly come in how you’re paid.

You could imagine some sort of pure truthseeker, paid purely by how well they tell the truth. People would ask them to find out the truth about something, and pay them for the service. And the truthseekers with the best track record would get the most clients. But neither profession really works like this.

Journalism comes closest. Once upon a time, people bought newspapers in order to be the first to know when something important happened. While there’s still a little bit of that going on (I guess this is what Bloomberg Terminals are for?), it’s a lot less central because of the internet. Now, there are hundreds of ways to find out about things, from a multitude of news sites to social media. More and more, people expect to be able to get information for free.

In that environment, the news has to compete not on the facts themselves, but on how it presents them. People pay for news that’s curated well to match their interests, or news that feels more respectable. And more than either of those, they pay for news that’s entertaining. So while truthseeking skills pay, writing skills often end up mattering more. In a sense, it’s why it’s possible for me to do journalism at all. I was trained in the academic truthseeking tradition, not the journalistic one. I got into journalism by impressing editors with my writing, not my ability to suss out the truth.

That academic truthseeking tradition is quite different, in part because the rewards for it are much more indirect. Academics pay comes from two main sources: research grants, and student tuition. Students are mostly there to learn old facts, not new ones, so that source of money supports research only in so far that students believe that a successful researcher with time for research will also be a better teacher.

Research grants, in principle, pay for truthseeking. But they’re typically paid by governments, which often don’t have a clear idea of what they’d like to learn, since the more practical questions are already being researched by private companies. So the decision gets delegated out to other academics, who have a vague shared sense of what’s worth knowing and what’s not. Accuracy should have an impact: that is, it should be easier to get grants if you’re better at finding the truth. But in practice, unless someone does so badly they trigger a scandal, academics don’t usually get things all that wrong. So grants are mostly based on other factors.

Paying someone purely to deliver the truth, not to entertain or match a culture, seems tricky. You could imagine sci-fi scenarios. What if we could track the logic people used to make decisions, and demand payment if those decisions were based on facts we uncovered, like a journalist getting a percentage of every short made in response to bad news they dug up about a company? What if governments paid in proportion to how valuable academic ideas turned out to be, centuries after they were discovered, and modern-day academics sold shares in that future payout to fund themselves? What if prediction markets something something?

For the moment, academics and journalists are both in a weird middle space. They’re truthseekers, still, by culture and inclination and desire. But they’re paid for something else.

It’s long been a mystery why there are 3 generations of quarks and leptons: three sets of particles, apparently identical except for how they interact with the Higgs boson. It would be nice if there were some good physical explanation. Nobody knows one. Barring that, it would be nice if some beautiful mathematical structure made this pattern seem natural. That’s what my new paper is about.

I’ll keep this nontechnical. I’ll say a bit about what the paper does, what it does not do, what led up to it, and how I wrote it.

This is my third paper about exceptional algebraic structures and the Standard Model. When you classify famous gadgets in algebra, beautiful gadgets with fancy names like ‘simple Lie algebras’ and ‘Euclidean Jordan algebras’ and ‘positive hermitian Jordan pairs’, you tend to get infinite series of them — together with a few exceptions that can be built using the octonions. This is a bit spooky, so I’ve been interested in this for a long time.

A few physicists have hoped that these exceptions are good for something. For example, maybe the quirky features of our best theory of particle physics, the Standard Model, aren’t accidental. Perhaps they fall out naturally from some exceptional algebraic structure.

It’s a long shot, but we’ve been stuck on figuring out new fundamental laws of particle physics for so long — roughly since the early 1980s — that it’s worth a try.

In 2018, Michel Dubois-Violette and Ivan Todorov noticed that the gauge group of the Standard Model falls out as symmetries of the so-called ‘exceptional Jordan algebra’ together with some ordinary Jordan algebras sitting inside it. I tried to clarify that here, with a huge amount of help from an excellent young mathematician:

• John Baez and Paul Schwahn, The Standard Model gauge group from the exceptional Jordan algebra. (Blog article here.)

It’s very nice, because the Jordan algebras in question arise naturally when you try to axiomatize the foundations of quantum physics. It would be so cool if something about quantum physics made the Standard Model seem mathematically natural!

But really this result only concerns the gauge bosons in the Standard Model: the photon, gluons, and the W and Z bosons. It says nothing about the fermions — that is, the quarks and leptons. And it seems quite hard to get those into the picture.

In 2020, Latham Boyle tried to solve this problem by tensoring the exceptional Jordan algebra with the complex numbers. This made one generation of fermions appear quite naturally! But the connection to the foundations of quantum physics seemed lost: tensoring the exceptional Jordan algebra with the complex numbers seems at first like it might be just a formal trick.

This spring, Latham and his student Endre Bokor and I showed the connection to quantum physics is not lost:

• John Baez, Endre Bokor and Latham Boyle, Jordan pair quantum theory and the Standard Model. (Blog article here.)

The idea is to work, not with Jordan algebras, but with more general things called Jordan pairs, which have been studied by mathematicians since at least 1975. We showed that you can still do quantum physics with Jordan pairs. And we showed that there’s an ‘exceptional’ Jordan pair that naturally contains the Standard Model gauge group and one generation of fermions!

This Jordan pair is built from the bioctonions: the octonions tensored with the complex numbers. And it’s closely related to an exceptional Lie algebra called $\mathfrak{e}_6$.

This is nice because the work of Dubois-Violette and Todorov used a smaller exceptional Lie algebra called $\mathfrak{f}_4$. Going up to $\mathfrak{e}_6$ gives the room to include one generation of fermions.

There’s an even larger exceptional Lie algebra you can use to build a Jordan pair: it’s called $\mathfrak{e}_7$. Bokor, Boyle and I tried using this to get three generations of fermions. There are things that make this tempting: not just the fact that $\mathfrak{e}_7$ is bigger, but the fact that the Jordan pair you get from it has a kind of three-fold symmetry. But we couldn’t get it to work.

Around this time I got very interested in some work that someone had sent me in October 2025. My inbox is packed with new theories of physics. Since the rise of large language models the inflow has increased: I get about two emails a day from someone telling me they’ve made a revolutionary discovery in physics. Practically none of these theories appeal to me. But this paper, and this thesis, were different:

• Benjamin Nasmith, An exceptional combinatorial sequence and Standard Model particles, 2020.

• Benjamin Nasmith, Tight Projective 5-Designs and Exceptional Structures, Ph.D. thesis, Royal Military College of Canada, 2023.

He claimed to fit three generations of fermions into the exceptional Lie algebra $\mathfrak{e}_7$.

When I started seriously trying to understand this paper, I wound up translating it into a language I’m more comfortable with, and expanding on the ideas a bit. So I wrote this:

• John Baez, Three generations in $\mathfrak{e}_7$.

Here’s the basic idea.

The idea

There is a standard way to fit the Lie algebra of the Standard Model gauge group, which I call $\mathfrak{g}_{\text{SM}}$, into the Lie algebra $\mathfrak{e}_7$. You can construct a Lie algebra $L$ that fits between them:

$$\mathfrak{g}_{\text{SM}} \subset L \subset \mathfrak{e}_7$$

As a vector space we have

$$\mathfrak{e}_7 \; \cong \; L \oplus V$$

for some vector space $V$ of dimension $3 \times 32$.

Moreover, the Lie algebra $\mathfrak{g}_{\text{SM}}$ acts on $V$, via the $\mathfrak{e}_7$ Lie bracket, precisely as it does on three generations of Standard Model fermions and their antiparticles, including right-handed neutrino and its antiparticle — but ignoring spin!

There is, in fact, a very interesting three-fold symmetry built into $\mathfrak{e}_7$, which is revealed when we put the Standard Model Lie algebra $\mathfrak{g}_{\text{SM}}$ into it. It permutes the three generations.

Like Nasmith, I am not proposing a theory of physics. I’m only observing a fascinating mathematical pattern that might (or might not) be of some use in physics.

There are lots of things this pattern does not include: basically, everything I didn’t already mention. It does not include the spin of the fermions and gauge bosons. It does not include the Higgs boson, though in some sense it comes close (see the paper). It does not include a Lagrangian, so it doesn’t say anything at all about particle masses or interactions.

I could say a lot more about what my paper does do… most importantly, where this Lie algebra $L$ comes from! The details are very interesting. There’s also the curious role of the right-handed neutrinos. But I’ve already spent weeks explaining all these things in my paper, so I won’t do it here. Instead let me say a bit about how I wrote the paper.

Writing the paper

I’ve been wanting to keep up with how AI is transforming math. About a year ago a friend gave me a subscription to Claude Pro. I wanted to test it out, despite my many misgivings, including how large language models are contributing to global warming and income inequality. Given the amazing things that people have recently done in math using large language models, I didn’t think that never trying them out would put me in the best position to make good decisions about the future.

So, I wrote this paper with help from Claude Opus 4.8.

I started by giving it Nasmith’s paper and asking a long series of questions about that paper over several days. The results were very interesting and helpful. Eventually I asked it to summarize and expand on our conversation. It quickly spat out a 10-page paper.

This paper was written in a breezy, pleasant style — but also quite hard to understand in detail, since it mixed Nasmith’s terminology with the Lie algebra terminology I prefer, and the proofs skipped over some steps.

It took me about three weeks of hard work to fully understand and re-express all the ideas a way that I like. For a while I felt dumb and frustrated, because when I asked Claude to fill in the gaps in proofs, it used math I was not very competent in, like the theory of regular subalgebras, and the theory of minuscule representations. But I learned this math, and everything turned out to be basically correct — in part, I’m sure, because Nasmith’s original work was correct.

For several weeks I checked, reorganized, expanded and completely rewrote this material. By the end everything was written in a style I like, emphasizing the ideas I consider important, proving things fairly carefully, and adding a lot of expository material — for example, explaining the theory of regular subalgebras.

Almost no traces of Claude’s original writeup remain, even though I was deeply influenced by them. My proofs make few references to deep theorems, though they assume solid familiarity with simple Lie algebras and their root systems. The proofs also require no brutally hard computations — though Claude was eager to do such computations to check things.

Any mistakes in this paper are my own.

I’m not sure what conclusions I draw from writing this paper. I’m writing another math paper now, with a human coauthor, and I have no desire to get help from a large language model. For work on my own it could be very helpful. Fields medalist Jacob Tsimerman says it roughly doubles his productivity. Would using it be so bad for the environment, or so bad for society, that I should avoid it? Maybe. I deliberately stuck with Claude Opus 4.8 instead of something more powerful, to see what I could do with what you get from a $20/month subscription. But maybe that’s still bad.

I avoid flying to conferences, which in some ways cripples my ability to keep up with new trends and influence people — but I don’t mind that. It gives me more time to think.

I will think carefully about my next move.

In the beginning there was a question. Actually, no, in the beginning there was language.  The question came later, presumably as one of its side effects. Since then we have asked about nearly everything, though the answers have done little to alter our circumstances. Human existence has always seemed strange to me. We arrive without consent, in a place not of our choosing, then spend decades asking why, until we close our eyes and enter the abyss of nothingness.

Some people accept this situation very well. I did not.

 Until my mid-teens, I tenaciously found one solution in fashion. If I could not understand who I was, I could at least decide what version I could become. Clothes gave form to something otherwise difficult to locate. Desire, after all, begins with a lack. 

It was only later that physics presented a solution to the same lack, only with more elaborate mathematics.

Quantum mechanics tells us that the world beneath the familiar world does not behave as we experience it.  A thing can resist being one thing, certainty begins to dissolve, and reality becomes strangely unfamiliar. The uncanny begins very close to home.

It is in this uncanniness that fashion and physics, liminally, meet for me: both begin with a human being standing before something they cannot understand, the universe in one case, the self in the other, and trying to make a form out of it. 

But even this distinction, on closer inspection, begins to collapse; for our desire to understand the universe has always concealed a deeper desire to understand the self that stands within it. 

All roads eventually lead back to the interests you had as a child, or so I say. 

I would squint into the dark, my mind already at work. I would imagine the next thing I would wear, project its color and fabric onto the ceiling. 

Then,  go to the bazaar, Rangrizano Dana, where, as the Kandaharis like to say, everything is sold except one’s mother and father.

The designs that had existed only in my head, after night upon night of theorizing and calculating, were finally beginning to take shape. The alleys of Rangrizano Dana were full of fabric of different textures and colors. 

Once the fabric was bought, came the tailoring. Tailors hold a special place in Afghanistan. Some are stars in their own right, the kind you have to book an appointment with. And once you get there, another matter comes up: making the tailor promise not to show your design to another woman. Everyone wants to wear something unique. So, naturally,  there is a lot of secrecy.

The designs themselves would either be described or sketched. Somewhere between what was imagined and what the tailor could make, something would emerge. And the day the tailor shipped your clothes was really a day of revelations.

Time passed, which is another way of saying the object of my desire changed. The same attention once spent on cut, texture, and appearance turned, little by little, toward the fabric of space and time.

Studying physics, I organized my process in the same way. There was again an idea that existed first in the mind, and then the problem of giving it form. 

Only now the materials were different. Instead of cloth, there was mathematics; instead of the tailor’s table, the blackboard and chalk. And, standing before it, more often than not, a badly dressed physicist covered in chalk dust. All the libidinal energy of the universe, you might say, had been sublimated into equations.

Fashion had, in some ways, been the more serious pursuit. There was something at stake: you wanted to look better than everyone else. Physics, by comparison, could be surprisingly playful.

Alice sends a particle to Bob. The sound of those names would tickle my Pashto ear then, and still cracks me up now. Cats are placed in boxes, both dead and alive. Observers hover near black holes or sometimes even, whooooooosh, fall into them.

So, in this way, in my mind’s eye, the physicist has become a rather more amusing figure: a tailor of space and time. The universe, of course, is a notoriously difficult client (perhaps even more difficult than a Kandahari woman making her tailor promise not to show her design to anyone else). 

He begins, as any tailor must, with an imagined shape. Mathematics is his chalk, his scissors, his needle. With it he marks the fabric, cuts it, folds it, joins one piece of reality to another. Some constructions fall upon the universe with elegance. Others bunch at the shoulder, pull at the waist, or refuse altogether to button, and must be altered, or thrown away completely.

A tailor works against the resistance of the human body; a physicist against the resistance of the universe.

Imagine, then, a physicist seated at the edge of a black hole, spectacles low upon his nose, sewing little cloaks from the fabric of spacetime.

“How many dimensions will you need?” asks Alice.

“That depends upon the suit,” he replies.

    For AdS/CFT, suppose we are making a three-dimensional garment.  The tailor of the universe never begins with three dimensions. Before him lies only the flat, two-dimensional fabric, with all its degrees of freedom spread upon the board.

 It is then, in a less sartorial language, we might call the disentangling and entangling of degrees of freedom; the tailor coaxes a third dimension out of the flat cloth. This is the secret embroidery of the holographic idea: what appears to the wearer as a three-dimensional world is encoded in a fundamentally two-dimensional way.

Bob tries this newly fashioned quantum garment on before entering the black hole. 

 “Too tight,” he says. “See if there is more degree of freedom.” 

The physicist frowns, takes up his chalk, and makes another small mark, with some disentangling and entangling here and there.

What could have been wrong? Well, perhaps the group theory has been chosen badly, the symmetry is broken, and a representation must be changed, a seam opened, a dimension added, or one cunningly concealed.

Alice watches him work, the almighty tailor of the universe,  making change without a change.

“But how do you know when the suit is right?”

And with that, the physicist looks at her.

“What do you want? Do you want to put me out of work? Unknowing is the lack that drives us to make more suits. If we knew the suit was right, that would be the end of everything.”

It’s long been a mystery why there are 3 generations of quarks and leptons: three sets of particles, apparently identical except for how they interact with the Higgs boson. It would be nice if there were some good physical explanation. Nobody knows one. Barring that, it would be nice if some beautiful mathematical structure made this pattern seem natural. That’s what my new paper is about.

It’s my third paper about exceptional algebraic structures and the Standard Model. When you classify famous gadgets in algebra, beautiful gadgets with fancy names like ‘simple Lie algebras’ and ‘Euclidean Jordan algebras’ and ‘positive hermitian Jordan pairs’, you tend to get infinite series of them—together with a few exceptions that can be built using the octonions. This is a bit spooky, so I’ve been interested in this for a long time.

A few physicists have hoped that these exceptions are good for something. For example, maybe the quirky features of our best theory of particle physics, the Standard Model, aren’t accidental. Perhaps they fall out naturally from some exceptional algebraic structure.

It’s a long shot, but we’ve been stuck on figuring out new fundamental laws of particle physics for so long—roughly since the early 1980s—that it’s worth a try.

In 2018, Michel Dubois-Violette and Ivan Todorov noticed that the gauge group of the Standard Model falls out as symmetries of the so-called ‘exceptional Jordan algebra’ together with some ordinary Jordan algebras sitting inside it. I tried to clarify that here, with a huge amount of help from an excellent young mathematician:

• John Baez and Paul Schwahn, The Standard Model gauge group from the exceptional Jordan algebra. (Blog article here.)

It’s very nice, because the Jordan algebras in question arise naturally when you try to axiomatize the foundations of quantum physics. It would be so cool if something about quantum physics made the Standard Model seem mathematically natural!

But really this result only concerns the gauge bosons in the Standard Model: the photon, gluons, and the W and Z bosons. It says nothing about the fermions—that is, the quarks and leptons. And it seems quite hard to get those into the picture.

In 2020, Latham Boyle tried to solve this problem by tensoring the exceptional Jordan algebra with the complex numbers. This made one generation of fermions appear quite naturally! But the connection to the foundations of quantum physics seemed lost: tensoring the exceptional Jordan algebra with the complex numbers seems at first like it might be just a formal trick.

This spring, Latham and his student Endre Bokor and I showed the connection to quantum physics is not lost:

• John Baez, Endre Bokor and Latham Boyle, Jordan pair quantum theory and the Standard Model. (Blog article here.)

The idea is to work, not with Jordan algebras, but with more general things called Jordan pairs, which have been studied by mathematicians since at least 1975. We showed that you can still do quantum physics with Jordan pairs. And we showed that there’s an ‘exceptional’ Jordan pair that naturally contains the Standard Model gauge group and one generation of fermions!

This Jordan pair is built from the bioctonions: the octonions tensored with the complex numbers. And it’s closely related to an exceptional Lie algebra called

This is nice because the work of Dubois-Violette and Todorov used a smaller exceptional Lie algebra called Going up to gives the room to include one generation of fermions.

There’s an even larger exceptional Lie algebra you can use to build a Jordan pair: it’s called Bokor, Boyle and I tried using this to get three generations of fermions. There are things that make this tempting: not just the fact that is bigger, but the fact that the Jordan pair you get from it has a kind of three-fold symmetry. But we couldn’t get it to work.

Around this time I got very interested in some work that someone had sent me in October 2025. My inbox is packed with new theories of physics. Since the rise of large language models the inflow has increased: I get about two emails a day from someone telling me they’ve made a revolutionary discovery in physics. Practically none of these theories appeal to me. But this paper, and this thesis, were different:

• Benjamin Nasmith, An exceptional combinatorial sequence and Standard Model particles, 2020.

• Benjamin Nasmith, Tight Projective 5-Designs and Exceptional Structures, Ph.D. thesis, Royal Military College of Canada, 2023.

He claimed to fit three generations of fermions into the exceptional Lie algebra

When I started seriously trying to understand this paper, I wound up translating it into a language I’m more comfortable with, and expanding on the ideas a bit. So I wrote this:

• John Baez, Three generations in

Here’s the basic idea.

The idea

There is a standard way to fit the Lie algebra of the Standard Model gauge group, which I call into the Lie algebra You can construct a Lie algebra that fits between them:

As a vector space we have

for some vector space of dimension

Moreover, the Lie algebra acts on , via the Lie bracket, precisely as it does on three generations of Standard Model fermions and their antiparticles, including right-handed neutrino and its antiparticle—but ignoring spin!

There is, in fact, a very interesting three-fold symmetry built into which is revealed when we put the Standard Model Lie algebra into it. It permutes the three generations.

Like Nasmith, I am not proposing a theory of physics. I’m only observing a fascinating mathematical pattern that might (or might not) be of some use in physics.

There are lots of things this pattern does not include: basically, everything I didn’t already mention. It does not include the spin of the fermions and gauge bosons. It does not include the Higgs boson, though in some sense it comes close (see the paper). It does not include a Lagrangian, so it doesn’t say anything at all about particle masses or interactions.

I could say a lot more about this… most importantly, where this Lie algebra comes from. The details are very interesting. There’s also the curious role of the right-handed neutrinos. But I’ve already spent weeks explaining all these things in my paper, so I won’t do it here. Instead let me say a bit about how I wrote the paper.

Writing the paper

I’ve been wanting to keep up with how AI is transforming math. About a year ago a friend gave me a subscription to Claude Pro. I wanted to test it out, despite my many misgivings, including how large language models are contributing to global warming and income inequality. Given the amazing things that people have recently done in math using large language models, I didn’t think that never trying them out would put me in the best position to make good decisions about the future.

So, I wrote this paper with help from Claude Opus 4.8.

I started by giving it Nasmith’s paper and asking a long series of questions about that paper over several days. The results were very interesting and helpful. Eventually I asked it to summarize and expand on our conversation. It quickly spat out a 10-page paper.

This paper was written in a breezy, pleasant style—but also quite hard to understand in detail, since it mixed Nasmith’s terminology with the Lie algebra terminology I prefer, and the proofs skipped over some steps.

It took me about three weeks of hard work to fully understand and re-express all the ideas a way that I like. For a while I felt dumb and frustrated, because when I asked Claude to fill in the gaps in proofs, it used math I was not very competent in, like the theory of regular subalgebras, and the theory of minuscule representations. But I learned this math, and everything turned out to be basically correct—in part, I’m sure, because Nasmith’s original work was correct.

For several weeks I checked, reorganized, expanded and completely rewrote this material. By the end everything was written in a style I like, emphasizing the ideas I consider important, proving things fairly carefully, and adding a lot of expository material—for example, explaining the theory of regular subalgebras.

Almost no traces of Claude’s original writeup remain, even though I was deeply influenced by them. My proofs make few references to deep theorems, though they assume solid familiarity with simple Lie algebras and their root systems. The proofs also require no brutally hard computations—though Claude was eager to do such computations to check things.

Any mistakes in this paper are my own.

I’m not sure what conclusions I draw from writing this paper. I’m writing another math paper now, with a human coauthor, and I have no desire to get help from a large language model. For work on my own it could be very helpful. Jacob Tsimerman says it roughly doubles his productivity. Would using it be so bad for the environment, or so bad for society, that I should avoid it? Maybe. I deliberately stuck with Claude Opus 4.8 instead of something more powerful, to see what I could do with what you get from a $20/month subscription. But maybe that’s still bad.

I avoid flying to conferences, which in some ways cripples my ability to keep up with new trends and influence people—but I don’t mind that. It gives me more time to think.

I will think carefully about my next move.

I don’t usually talk about particle physics here. I have a whole series of articles about octonions and the Standard Model on my other blog. But I’m kind of excited about this new paper, so I’ll talk about it here too:

• John Baez, Endre Bokor and Latham Boyle, Jordan pair quantum theory and the Standard Model.

Jordan algebras were introduced by Jordan, von Neumann and Wigner in 1934 in an attempt to formalize algebras of observables in quantum theory. They come in 4 infinite series—but there’s one more, the ‘exceptional Jordan algebra’, consisting of 3 × 3 self-adjoint matrices of octonions. For years physicists sought to find some use for it.

In 2018, Todorov and Dubois–Violette noticed that the symmetries of the exceptional Jordan include the Standard Model gauge group in a nice way. But it was unclear how to bring in the fermions—the quarks and leptons. That’s what our new paper does.

To do this, we need to go beyond Jordan algebras. Jordan pairs and Jordan triples are two closely linked formalisms that generalize Jordan algebras. Our paper explains them in detail—and how they’re connected to geometry and quantum mechanics. But here I will mostly skip that wonderful story, so I can quickly explain the connection to the Standard Model.

Here’s how the Standard Model gauge group, together with its representation on one generation of fermions, drops out of a Jordan triple.

The bi-Cayley triple

Let

be the bioctonions: octonions with complex coefficients. Write for the space of column vectors with two bioctonion entries.

has a certain triple product

which obey the axioms of a gadget called a ‘positive hermitian Jordan triple’. It’s called the bi-Cayley triple.

Now, every positive hermitian Jordan triple gives rise to a -graded real Lie algebra

Not a Lie superalgebra: a plain old-fashioned Lie algebra with a -grading!

How does this work? We take the hermitian Jordan triple itself to be The Lie algebra consists of all linear maps from to itself that are of this form:

for some These maps are called real inner derivations. They form a Lie algebra since the commutator of two such maps is another such map. With a bit more work we can define other operations making all of into a -graded Lie algebra.

So, we get a big Lie algebra and a Lie subalgebra sitting inside it. From this we get two Lie groups: a big one whose Lie algebra is and a subgroup whose Lie algebra is

The quotient is is a nice kind of manifold called a hermitian symmetric space. Conversely, any compact hermitian symmetric space give rise to a positive hermitian Jordan triple!

This geometric picture is revealing. The group acts transitively as symmetries of our hermitian symmetric space, while the stabilizer of any point is isomorphic to Our original Jordan triple, is then the tangent space of that point! So, acts on this Jordan triple. This action preserves the triple product, and we call the real inner automorphism group of our Jordan triple.

Here’s another great thing about the geometric picture: hermitian symmetric spaces were classified by Eli Cartan (who seems to have spent his life classifying things). As a result we also know the classification of positive hermitian Jordan triples. They come in four infinite series together with two exceptions. One is the bi-Cayley triple, and other is the Albert triple, which is the complexification of the exceptional Jordan algebra. The bi-Cayley triple is a subtriple of the Albert triple. It’s these two exceptions that are connected to the Standard Model. But we’ll start with the bi-Cayley triple.

The 3-graded Lie algebra coming from the bi-Cayley triple is the compact real form of :

The even part of this Lie algebra is in brackets. The corresponding hermitian symmetric space is called the bioctonionic plane The even part of our 3-graded Lie algebra, generates the stabilizer of a point in the bioctonionic plane. The odd part, our friend is the tangent space of that point.

Here’s the first big surprise. The even part transforms as the adjoint representation of while the odd part itself transforms as the 16-dimensional complex spinor representation of Ignoring the extra for a moment, this is exactly what we see in a grand unified theory: gauge bosons in the adjoint representation, and one generation of fermions in the 16-dimensional spinor representation.

So before we do anything, the bi-Cayley triple already smells like it contains the ingredients of an grand unified theory.

Tripotents

In a Jordan algebra the important elements are the idempotents, In a Jordan triple their role is played by tripotents: elements with

A tripotent always lets us split into three parts via something called its Peirce decomposition. The operator has eigenvalues 0, 1/2, and 1, so splits into the corresponding eigenspaces

which are called the Peirce 0-space, Peirce 1/2-space and Peirce 1-space of A tripotent is called minimal when its Peirce 1-space is one-dimensional. Two tripotents are called colinear when each lies in the other’s Peirce 1/2-space.

I can’t resist explaining some of the quantum physics here. In a hermitian Jordan triple, the triple product is linear in the first and last slot, but conjugate-linear in the middle slot. So, if you multiply a tripotent by a phase you get a new tripotent:

This should remind you of how when you multiply a unit vector in a Hilbert space by a phase, you get a new unit vector. In Jordan triple quantum mechanics, minimal tripotents take the place of these unit vectors. The hermitian symmetric space that I was talking about earlier is the same as the space of minimal tripotents mod phase! So, it generalizes the familiar space of ‘pure states’ in quantum mechanics: unit vectors mod phase.

But let’s get back to the Standard Model.

A chain of Jordan triples

From here on, the single fact driving everything is this: in any hermitian Jordan triple, any minimal tripotent’s Peirce 1/2-space is itself a hermitian Jordan triple!

If we run this starting from the bi-Cayley triple, we get this chain of hermitian Jordan triples, where each row’s 1/2-space is the next row’s triple:

Jordan triple	Lie algebra (even part in brackets)

Here is the Jordan triple of antisymmetric complex matrices, is the Jordan triple of complex matrices, and

is the true Standard Model gauge group.

The gauge group from two tripotents

Start with the bi-Cayley triple. Choose two colinear minimal tripotents Descend the table twice:

• Start with which has real inner automorphism group

• Fix Its Peirce -space is with real inner automorphism group

• Fix (colinear with so living in ). Its Peirce -space in is with real inner automorphism group exactly

In other words, the subspace of the bi-Cayley triple colinear with both and is a Jordan triple whose real inner automorphism group is the Standard Model gauge group.

The choice of and also pins down how sits inside the original group At each we step take the subgroup that acts with determinant and preserves the chosen tripotent up to a phase; this gives a chain of subgroups whose members are and so we get the embeddings

In particle physics, this is the classic chain taking us from the so-called grand unified theory down to the grand unified theory down to the Standard Model. And it’s well known that restricting the 16-dimensional complex spinor representation of along this chain gives precisely the Standard Model representation on one generation of fermions! So we get one generation of Standard Model fermions this way.

The six particles types as Peirce spaces

We have gotten the representation of the Standard Model gauge group on one generation of fermions without any fuss. But it’s also fun to peer into the details, and see how the different kinds of fermions emerge.

For any tripotent we have projections and onto its three eigenspaces: its so-called Peirce projectors. Since we get the Standard Model structure using two minimal tripotents and in the bi-Cayley triple , there are nine composites of two Peirce projectors we can apply to this triple. This is how we pick out the different kinds of fermions!

As a representation of the Standard Model Lie algebra

any generation of Standard Model fermions transforms as the direct sum of six irreducible representations:

These correspond to the six types of left-handed fermion: Six irreducible pieces, six particle types.

It turns out these are exactly the six nonzero components of the Peirce decomposition of with respect to both and Those six match up one-to-one with the particle types:

Peirce projector	representation of	particle type
	(3, 2, +1/6)
	( 1, +1/3)
	( 1, −2/3)
	(1, 2, −1/2)
	(1, 1, +1)
	(1, 1, 0)

The remaining three combinations— and —all vanish, which is why we land on six pieces and not nine.

So the whole package—the gauge group the embedding the representation and even the split of one generation into its six particle multiplets as distinct Peirce components—all comes out of the single object once you choose two colinear minimal tripotents.

And if you prefer to start one level up, with the Albert triple you get the same result by choosing three mutually colinear tripotents instead of two—but for that, read our paper!

For the sake of anyone who might read this blog post in the distant future (a month from now, say), let me mention that I am writing it a few days after OpenAI announced that it had solved ten major problems in mathematics and theoretical computer science, including the first construction of a non-sofic group, and a proof that the multicolour Ramsey number (where there are 3’s) grows superexponentially in . The first was, to judge from various talks I have been to, one of the most important unsolved problems in group theory, and the second was a major open problem in Ramsey theory that I didn’t necessarily expect to see solved in my lifetime, though of course such expectations now have to be revised. The reason I want to be clear about the timing is that I shall be discussing the current capabilities of LLMs in the full expectation that those will continue to change rapidly. So it is likely that in not too long from now, if there is anything interesting in what I write, it will be interesting mainly as a record of what the situation looked like in early August 2026.

These results, and the other eight on the list, are extraordinarily impressive, but it still doesn’t seem to be the case that LLMs are better than all humans at all aspects of mathematics. If they were, then their big speed advantage over us would mean that there would be much more of a flood of results. So it is natural to wonder about what kinds of problems LLMs are good at, and about where there is still room for improvement. I don’t pretend to have a good answer to this question, where a good answer would be a crisp classification that would fit the current examples well, but it is an interesting exercise to try to rule out some bad answers, and to try to identify potential answers that aren’t obviously contradicted by the evidence.

Are LLMs particularly good at finding counterexamples?

A first remark here is that LLMs are not just good at finding counterexamples: they can find proofs of difficult statements as well. However, it is notable that the most famous problems they have solved have almost all been with counterexamples rather than proofs. That is true of the two problems mentioned above, and also of the Jacobian conjecture and the unit distance conjecture.

If one wants to theorize that LLMs are particularly good at finding counterexamples, then there are two things it would be good to do to make the theory more convincing. The first may sound unproblematic: it is to decide when solving a problem counts as finding a counterexample. Once that is sorted out, the second is to come up with a potential explanation of why LLMs would be particularly well suited to solving problems of that particular kind.