A recent tradition of the ICM is to have a non-technical popular article written for each of its medallists. (This is separate from the older tradition of having a laudatio for the winner, which is similar but aimed at a more advanced audience.) I was approached to write such an article for Hong Wang on the occasion of her Fields Medal. I have uploaded my initial version of this article, “Rotating needles in space: the road to the Kakeya conjecture, and why it matters“, to the arXiv. Chris Sogge gave the corresponding laudatio; both should eventually appear in the Proceedings of the ICM.

Mathematical Discourse is a new online, peer-reviewed mathematics journal, publishing videos of mathematics research talks of the highest quality.  Our inaugural scientific and editorial boards are listed at the end of this message.

Our goal is to promote and nourish a culture which values communication as an essential part of the research process. Giving a mathematical talk captures the human side of mathematics in a unique way, and supporting these human aspects of our discipline will be ever more important in the coming years. 

We are now accepting submissions for our first issue.  If you have seen (or given!) a truly excellent, mathematically stimulating research talk which was video recorded, we hope you will encourage the speaker (possibly yourself!) to submit it.  

Our current call for submissions is limited to ~1 hour research seminar or colloquia style talks in pure mathematics. A detailed description of the scope of the journal and submission requirements are available at https://www.mathematicaldiscourse.org/submit.

We hope you will spread word of this new initiative to your colleagues and collaborators!

Sincerely yours, 

Katie Mann, Akshay Venkatesh, Rachel Webb

Managing Editors of Mathematical Discourse

Scientific Board: Hugo Duminil-Copin, Larry Guth, Bryna Kra, Yair Minsky, Bjorn Poonen, Terence Tao, Ravi Vakil, Geordie Williamson

Editors: Matt Baker, Richard Bamler, Andrej Bauer, Alexei Borodin, Yaiza Canzani, Daniel Cristofaro-Gardner, Jordan Ellenberg, Hélène Eynard-Bontemps, Jessica Fintzen, Sergey Fomin, Zaher Hani, Peter Hintz, Shrawan Kumar, Justin Moore, Samuel Taylor, Richard Thomas, Isabel Vogt, Yilin Wang, Alex Wright, Zhiwei Yun

mathematicaldiscourse.org

Ben Krause, Hamed Mousavi, Joni Teräiväinen, and I have just uploaded to the arXiv our paper Quantitative bounds for sets lacking polynomial progressions with shifted prime difference. The purpose of this paper is to obtain quantitative versions of this theorem of Wooley and Ziegler:

Theorem 1 Let be polynomials of one variable with integer coefficients with zero constant term, and let be a set of integers of positive density. Then there exist infinitely many primes such that contains a progression of the form

for some integer .

This generalizes the famous theorem of Szemerédi in two ways: firstly, by considering “polynomial progressions” instead of arithmetic progressions, and secondly by requiring the shift parameter to be one less than a prime . The first extension of Szemerédi’s theorem is a theorem of Bergelson and Leibman; and the second extension is also obtainable by combining the arguments of Frantzikinakis, Host and Kra with the results of Green, Ziegler, and myself.

The proof of the above theorem uses ergodic theory, which makes it difficult to extract quantitative bounds from it; and standard methods of “finitizing” ergodic theory results, for instance by replacing Host–Kra seminorms by their Gowers uniformity norm counterparts, run into a number of technical difficulties here due to the need to work with multiple scales due to the presence of polynomials, as well as the fact that many of the conjectural uniformity properties of the prime numbers at small scales remain unproven.

Nevertheless, we are able to get reasonable quantitative results (with density bounds that are roughly single or doubly logarithmic in scale) in the following special cases:

• linear polynomials;
• polynomials of distinct degree; and
• multiples of a fixed polynomial.

The precise statements are slightly technical and not reproduced here.

Previous quantitative bounds in the linear case were obtained by Leng and by Teräväinen and myself, but our new method improves upon these bounds by roughly one iterated logarithm. On the other hand, in the case of two term progressions of spacing , there is a much stronger quantitative result (with polynomial dependence of constants) due to Green; our method do not recover that result.

Our methods use a variety of old and new methods in the subject. For instance, we use the (now quite standard) “-trick” to restrict the primes to a single congruence class to improve their uniformity properties (which, thanks to the recent work of Leng and Matthiesen–Teräväinen–Wang, are now quite strong quantitatively); and for good configurations of polynomials, one can also use a recent transference theorem of Altman and Sawhney to compare the polynomial averages with simpler linear ones, without having to pass to short scales. In order to get relatively strong bounds unconditionally, a “Siegel approximation” for the primes is used taking into account the potential influence of a Siegel zero. It will not be surprising to the experts that quantitative inverse Gowers theorems and nilsequence equidistribution theorems also play a major role.

A key technical difficulty is the presence of the modulus in the coefficients of the polynomial progressions after changes of variable, which forced us to make several of the existing estimates uniform over such coefficients (assuming they are not unreasonably large).This caused several complications that made a correct argument to more time-consuming to locate than initially planned.

AI usage in this work was fairly light, being restricted to proofreading and literature search only.

In recent months there has been a proliferation of AI-generated proofs of various old and new results, some of which have been formalized in the proof assistant language Lean. However, checking that a given Lean repository actually proves the claimed statement is somewhat non-trivial, especially for an audience which is not expert in the use of Lean: one has to first check that the claimed formal Lean statements have proofs that typecheck, that the proofs do not contain any “cheats” such as adding additional axioms, and that the formal statements also match (in a semantic sense) the informal description of the claimed results.

To help bring some clarity to this situation, I am happy to announce that Palomar registry of Lean verified mathematics, which is an initiative incubated by the Lean FRO and by ICARM, is now open for submissions. I am serving in several roles on this registry, including on the scientific advisory board, together with Jeremy Avigad, Matthew Ballard, Jaume de Dios, Nestor Guillen, Bryna Kra, Kim Morrison, Ravi Vakil, and Akshay Venkatesh.

A detailed motivation for Palomar can be found here, and further information about Palomar can be found here. A zeroth approximation of what Palomar intends to be is the analogue of a preprint server for Lean proofs. More precisely, Palomar (which is named after the astronomical observatory) is a registry of external Github repositories (or more precisely, “snapshots” of such repositories, as represented by a specific Github commit) containing Lean code adhering to the current best practices for such formalizations, in particular containing

• A “challenge file” containing a short, human readable description in Lean of the results claimed.
• A “solution module” containing an (arbitrarily long) proof of the results claimed in the challenge file.
• A “formalization.yaml” file describing the results in informal language, and also containing a number of other relevant metadata and disclosures.

(There are also some additional technical requirements for the repository which I will omit here.) If a snapshot of a repository is submitted to Palomar, it will check both (a) that the solution module typechecks and proves exactly the results claimed in the challenge file, and that (b) the informal description of the result in the formalization.yaml file appears to match the result claimed in the challenge file, and that the repository meets various minimal standards required for a registry entry. The first check (a) is purely mechanical, using the Lean tool Comparator; the second check (b) is non-deterministic, being performed by a large language model. If a repository passes both checks, it can be registered on Palomar. It is worth stressing that the checks in (a) and (b) fall well short of what a proper human peer review of a submission for novelty, interest, and accuracy would give; in particular, Palomar is not a peer-reviewed journal.

The submission process is thorough, but achievable: as a test, I successfully managed to submit my own recent formalization of the proof of Sendov’s conjecture to Palomar, and also plan to submit some older formalizations to the registry soon.

In any event, the registry is now open for formalizations of both old and new results. Submissions (whether human-generated, AI-generated, or some mixture of both) are welcome; please read the (somewhat detailed) instructions here before starting a submission. (I will however note that modern AI agents are quite helpful in assisting with the mechanical details of the submission, though a human review is still strongly recommended.)

Discussion and feedback on Palomar will occur on this Zulip channel.

This post concerns the following conjecture of Sendov, as well as its strengthening by Phelps–Rodriguez:

Conjecture 1 (Sendov’s conjecture) Let , and let be a degree polynomial with all zeroes in the unit disk. Then for every zero of , there exists a critical point of with .

Conjecture 2 (Phelps–Rodriguez conjecture) Let , and let be a degree polynomial with all zeroes in the unit disk. Then for every zero of , there exists a critical point of with , unless is on the unit circle and is a scalar multiple of .

By applying a rotation around the origin, we can normalize to be a real number with .

From the work of Rubinstein, both conjectures were already established in the case, so one can restrict to the case. Both of these conjectures then follow from

Conjecture 3 (Sendov’s conjecture in interior) Let . Let be a degree polynomial with all zeroes in the unit disk. Then if is a zero of , there exists a critical point of with .

All three of these conjectures were established for (in a sequence of papers culminating in this paper of Brown and Xiang) and for sufficiently large (in a paper of myself, which in turn built upon several partial results in this setting). This left the case of intermediate to be settled. My arguments used some qualitative ingredients (most notably analytic continuation) and as such did not easily lend themselves to quantifying the threshold of above which the argument was valid.

Recently, Lech Mazur was able to use an AI tool to resolve Sendov’s conjecture for all , with the proof verified in Lean. However, the AI-generated proof was not human-digested to be in the form of a publication-ready preprint; and it has taken me several days (with heavy AI assistance) to perform such a digestion, to place the proof in proper context with previous literature and to simplify and streamline the argument to highlight the main ideas. (Note: the above chat log only represents a portion of the digestion work: the rest was performed with pen and paper, or using some further AI agents.) The same arguments also give a new proof of Rubinstein’s theorem, which I also give below the fold.

One consequence of this digestion is that the argument in fact demonstrates Conjecture 3, and thus resolves both the Sendov conjecture and the Phelps–Rodriguez conjecture in full generality.

The proof ends up being remarkably elementary. No complex analysis is used other than the fundamental theorem of algebra (and very basic facts about Möbius transformations); and the deepest inequality used as input is the Maclaurin inequality (and we only need a special case of that inequality which can be derived from the arithmetic mean-harmonic mean inequality and an induction argument).

Using an AI agent, I have been able to formalize the entire argument in Lean, extended to by some minor modifications to the proof. This formalization is more streamlined than the original formalization (it has about 15,000 lines of code, compared with around 90,000 for the original proof).

We now prove Conjecture 3. The cases have long been known but need to be treated separately; a short proof using the machinery developed here is provided at the end of the post. Suppose now that we have a counterexample for some , thus one can find a degree polynomial with zeroes

for some and , in the closed unit disk, whose critical points all lie a distance at least from . We use notation here in the non-asymptotic sense, thus means that for some absolute constant (independent of ). We will also use the notation to denote a quantity that is bounded in magnitude by .

To capture the fact that the critical points lie at a distance at least from , we write these critical points as

for some (non-zero) in the closed unit disk.

Example 4 If and , then are the non-trivial roots of unity, while the are all equal to . Strictly speaking this is not actually a counterexample to Conjecture 3, because is not strictly less than one; nevertheless this is an important motivating near-counterexample for the arguments below.

Example 5 A generalization of the previous example was studied in Section 4 of my paper. Here one took

where was an asymptotic parameter going to infinity,

was a low-degree polynomial for some ,

and were constants. This polynomial has a zero at , critical points at , and additional critical points near . If all the critical points were at distance at least one from , one would have

and

while if all the zeroes were in the unit disk, the calculations in my paper showed that

Here denotes a quantity that goes to zero as . If one ignores the errors, one can show that these conditions are only simultaneously feasible if and all the vanish, but the argument was somewhat subtle (I had to proceed by inspecting the second Fourier coefficient of (1)). This illustrates the fact that the regime is particularly delicate.

We now have two sets of points in the closed unit disk: and . They “communicate” with each other through the polynomial and its first derivative , both of which can be expressed in terms of either set of points (as well as and ). Indeed, if we normalize to be monic, then we can factor in terms of the zeroes as

and thus upon differentiating

Here and in the sequel we adopt the convention of removing singularities when dealing with expressions that involve multiplication by both and , by cancelling such terms first in the event that .

In a similar vein, can be factored

and thus on integrating (and using )

It is convenient to rule out the easy case right away. In this case we see from (3), (4) that

which is absurd since the first product has magnitude at most one, and the second product has magnitude at least one. Thus we can assume henceforth that .

By inspecting or at various natural locations, we can thus obtain a number of identities relating the to the . We record the ones that we actually need here:

Lemma 6 (Communication identities) Let denote the function

• (i) (Centroid identity) We have
  That is to say, the centroid of the zeroes equals the centroid of the critical values.
• (ii) (Polar identity) We have

• (iii) (First origin identity) We have

• (iv) (Second origin identity) We have
  (Again, we are using the convention of removing singularities to deal with the case where some of the vanish.)

Proof: For (i), we inspect the behavior of as . From (2) we have

and thus on differentiating term by term

Meanwhile, from (4) we have

Comparing coefficients, we obtain the claim.

For (ii), we consider the expression . On the one hand, from (2), (3) one has

(Note from hypothesis that cannot be a critical point, so the denominator is non-zero.) On the other hand, from (4), (5) one has

Equating the two identities, we obtain (ii) after some algebra.

For (iii), we evaluate . From (2) we have

while from (5) we have

Equating the two identities, we obtain (iii) after some algebra using (6).

For (iv), we similarly evaluate . From (3) we have

while from (4) one has

Equating the two identities, we obtain (iv) after some algebra using (6).

Remarkably, the polynomial will play no further role in the argument: the identities in (i)-(iv), together with the hypotheses that and lie in the closed unit disk, will be sufficient by themselves to obtain a contradiction.

Example 7 Continuing the example in Example 4, in (i) both sides vanish. In (ii), both sides are equal to one. For (iii) and (iv), we have , with both sides of (iii) equal to one, and both sides of (iv) equal to zero.

Remark 8 The centroid identity is extremely classical, going back to this 1948 paper of Popoviciu. The comparison of the polynomial at a location and at the polar inversion of that location across the closed unit disk is a familiar trick in the literature; see, e.g., Lemma 5 and Theorem 8 of Dégot. The specific form of the polar identity is implicit in the first part of Section 5 of Mazur’s AI-generated proof, while the origin identities are extracted from equation (6.3) of that proof. The first origin identity is also very close to Theorem 6 of Dégot, while the second origin identity is similar to some identities appearing in the proof of Lemma 6 of Dégot, as well as the work of Mir–Nazir–Wani and (in the case) Rubinstein. The work of Meir–Sharma and Mir–Nazir–Wani also contain several further identities relating the to the ; see in particular Lemma 15 below. Variants of (5) also appear in Proposition 10 of Miller.

Remark 9 The first origin identity (9) is already strong enough to handle asymptotically all examples of the form in Example 5, except in the endpoint case where vanish and the are all . Indeed, as the are in the closed unit disk, (9) implies that

On the other hand, routine calculations (omitted here) show that

leading asymptotically to the constraint

But all terms here are non-negative (since ), so this forces a contradiction unless (and hence also ) and the all vanish.

As mentioned in Example 5, the most delicate regime occurs when . It is convenient to introduce the normalized version

of , thus , and the case corresponds to . Informally, measures how close is to (at the scale of ).

A key role in the argument will be played by the mean

of the , particularly the real part . As the all lie in the unit disk, the mean does also, so that

and

On the other hand, in the example in Example 4, is equal to the extremal value of , and . In Example 5, we have (and ).

It will be convenient to work with the quadratic polynomial

with a particular emphasis on the value at :

One should primarily think of as a measure of how close is to . Clearly we have

for all (note that is strictly less than ).

The arguments will revolve around the relationship between and . Specifically, we will establish the following two inequalities below the fold. The first inequality, which we call the “polar inequality”, comes in three forms:

Proposition 10 (Polar inequality)

• (i) (Raw polar inequality) We have

• (ii) (Polar inequality in , form) We have

• (iii) (Simplified polar inequality) We have
  In particular, since , one has

It will be the inequality (18) that we use in practice, but it will be derived from (17), which in turn is a consequence of (16), which will follow from the polar identity (8) together with the fact that the and lie in the unit disk. The bound (18) is only slightly weaker than (17); see the (Gemini-generated) image below.

I was not able to find an exact duplicate of the above polar inequalities in past literature, but the paper of Dégot contains several similar inequalities. The inequality (16) was extracted from (5.1) of Mazur’s AI-generated proof; the subsequent bounds (17), (18) arose from my attempts to simplify the arguments after that point.

The second inequality, which is more difficult, also will come in several forms:

Proposition 11 (Origin inequality) Let .

• (i) (Raw origin inequality) We have

• (ii) ( bound) We have

• (iii) (Origin inequality in , form) We have

Part (i) (which was extracted with some effort from Section 6 of the original AI-generated argument) will be deduced from the first and second origin identities (9), (10), as well as the centroid identity (7). Part (ii) will follow from (i) and the polar inequality (18), while part (iii) is an elementary consequence of (i).

As it turns out, the last three terms in (21) are asymptotically negligible as . Dropping those terms gives a competing feasibility region for and which is disjoint from the one coming from the polar inequality (17) (or (18)):

This already suggests that one can use this approach to recover my previous result on Sendov’s conjecture holding for all sufficiently large . In fact, even with the three error terms in (21) added, there is enough room between the two inequalities (18), (21) to obtain a contradiction for all (using the additional bound to control these errors), although showing this for medium-sized (such as ) requires a certain amount of computer assistance.

For fixed , the right-hand side of (21) is monotone increasing in (or equivalently, monotone decreasing in ). In view of (18), we can thus replace by in this inequality, so that is replaced by

and replaced by . The inequality (21) then becomes an inequality involving only and :

We also note that the bounds force the constraint

This prevents from getting too close to the upper limit (or getting too close to zero).

We can now eliminate all large degrees, e.g., , as follows. The quadratic attains its minimum at . For we have

while for (if this region is non-vacuous) we can bound the quadratic by its value at . Thus

Evaluating these expressions, we arrive at

Since , we have . Next, we claim that . As is monotone increasing in , it suffices to do this when . Here one can directly compute that

since the discriminant of the numerator is negative, we conclude that

as desired.

Dropping some and terms, we conclude that

Every term on the right-hand side can be seen to be decreasing in for . Thus the right-hand side can be bounded by

giving the desired contradiction.

The remaining range to handle is when

It turns out that (22) remains infeasible in this range. This can be illustrated numerically without much difficulty: see this applet. For instance, in the most delicate case , the right-hand side of (22) only gets as large as (and in particular stays below ) throughout the range :

I have also verified this bound in Lean.

— 1. The polar inequality —

We begin with a proof of Proposition 10.

As is well known, the Möbius transform maps the closed unit disk to itself. In particular, we have

for all of the zeroes . Inserting this into the polar identity (8) and using the triangle inequality, we conclude the lower bound

We now convert this bound to a bound involving the quantity in (12). From the arithmetic mean-geometric mean inequality we have

and from (12) we have

Since , we thus have

giving the raw polar inequality (16).

Bounding by and using the quantities from (11), (15), we observe that

Using the basic inequality , we thus have

with strict inequality for . From (16) we conclude (17). This also implies , since otherwise the integrand is always bounded by , which is absurd.

On evaluating the integral in (17), we obtain

and thus

so on taking logarithms we obtain

It remains to establish the bound

Here we use an AI-generated argument. One can directly calculate

where and . If we can show that

for all , then taking logarithms in (17) yields

from which (26) will follow by routine algebra.

Both sides of (27) vanish at . Taking derivatives, it suffices to show that

which rearranges to

To expand the left-hand side, we use the double angle formulae and

to rewrite it as

Collecting the coefficient of for and extracting a common factor of , one is left with

where . (The remaining coefficients, which also receive contributions from the polynomial terms, all vanish.) Thus the left-hand side has the Taylor expansion

in which every coefficient is non-negative, giving the claim.

Remark 12 As the image in the introduction suggests, the bound (18) is only slightly weaker than (17). For small , one can perform Taylor approximation on the latter bound to obtain

while the former bound is

Note that is slightly smaller than .

Relating to this, the constant in (27) cannot be improved.

— 2. The origin inequality —

Now we turn to the proof of Proposition 11, which is more difficult and revolves around an analysis of the function defined in (6). We begin with a heuristic analysis. Inserting the approximation for small into (6) and using (12), we are led to the approximation

at least when is small (which turns out to be the dominant regime in applications). This suggests a relation

between the two expressions involving in the origin identities in Lemma 6. Substituting in this approximation, we obtain some (slightly complicated) approximation for the sum in terms of , , , , and the product .

As lie in the closed unit disk, the product does also. However, past experience with the Sendov conjecture has taught us that the worst cases tend to be when lie very close to the boundary of the disk, so that is close to one. For instance, in Example 4 all the and lie on the unit circle, and . See Remark 3 of Dégot or Theorem 1.10(ii) of my own paper for other places where this heuristic is noted. To simplify the discussion, let us assume for now that is exactly one, so that all lie on the unit circle. This leads in particular to the inversion identities

The centroid identity in Lemma 6(i) relates the sum of the with the sum of the . Using (31), this gives a similar identity relating the sum of the with the sum of the . The latter sum is of course just . This combines well with the previous approximation, thus giving an approximate identity relating , to , , and . As it turns out, the roles of and are minor and can be quickly eliminated for the purposes of obtaining useful bounds, leading eventually to the relation in Proposition 11.

We turn to the details. To make the approximation (30) more precise, we note that , and hence by the fundamental theorem of calculus

The heuristic (29) predicts that , which would give (30). If we actually differentiate (6) carefully, we obtain the exact identity

Bounding , we write this

When faced with a similar expression in (24), we used the arithmetic mean-geometric mean inequality. Here, the analogous tool is Maclaurin’s inequality, which gives

and hence by Cauchy–Schwarz

Repeating the calculations used to show (25), we have

and so we obtain the bound

Integrating this, we obtain a rigorous analogue of (30),

and thus by the triangle inequality

From the first and second origin identities (9), (10) we have

The next step is thus to estimate . When , then all the were on the unit circle and we could use (31) (and the centroid identity) to proceed. Now, we are no longer assuming to equal , but we can still adapt the previous arguments with a loss proportional to . The key lemma is

Lemma 13 (Defect lemma) Let be some points in the closed unit disk. Then

Proof: By a limiting argument we may assume that none of the vanish. If we write for some , then we can calculate that

and

Thus the desired inequality reduces to the superadditivity property

But from the sinh addition formula we have

for all non-negative (this also follows from the convex nature of together with ), and the claim follows by induction.

We remark that the lemma can also be proven by direct induction, without an appeal to hyperbolic trigonometry.

From taking complex conjugates of the centroid identity (7) and performing some algebra, we have

Using the defect lemma (applied to the points ) and the triangle inequality we conclude that

where as before we are removing singularities when some of the vanish. Applying (12) and some algebraic manipulation, we arrive at

Substituting this back into (33), we conclude that

and hence after some algebra and the triangle inequality

Inserting this into (32), we obtain

We can simplify (34) by reducing to the case. Indeed, we shall show that

which implies that the right-hand side of (34) is non-decreasing in in the range . Thus we may replace by in (34) to conclude that

Let us now verify (35). Using and the triangle inequality, we can lower bound

Inserting this into (35) and clearing denominators, we reduce after some algebra to

But as a quadratic polynomial in , the left-hand side has discriminant , which one can check to be negative for sufficiently large (in fact suffices), giving the claim (35).

Next we eliminate the role of the imaginary term . Observe for any complex number with positive real part that

as can be seen by squaring both sides. The expression has real part

which lies between and (in particular, it is positive), and imaginary part of magnitude at most

by (13). We conclude that

The right-hand side can be rearranged using the quantity from (11) as

so the bound (36) gives (19).

— 2.1. Upper bound on —

Now we can prove (20). Suppose for contradiction that ; since , this implies that . Crudely discarding the term in (19) and bounding by , we have

The quadratic polynomial equals at and attains its minimum at with value . By convexity, we thus have

for and

for (this latter statement is vacuous if ). Since , we can therefore crudely bound

and hence

From (15) we have , thus by (18) one has

From another application of (18) one has

We conclude that

It is now convenient to introduce the quantity , thus with

and

Inserting these bounds and dividing by , we conclude

Since , we obtain

Since and , we conclude that

Routine calculus shows that has a maximum of at most , and that the right-hand side here is at most , giving the required contradiction. This proves (20).

— 2.2. A simplified estimate —

Now we show (21). Note from (11) that

while from (15) we have

and hence also

From (15) we have

By the mean value theorem (noting that is non-negative) we thus have

From the standard beta function identity

(and the fact that ) we can thus replace (19) by

From (15) we have

Thus by (37), (38), (39)

Dividing by the positive quantity gives the claim.

— 3. Rubinstein’s theorem —

We now adapt the arguments to give a proof of Rubinstein’s theorem that the Phelps–Rodriguez conjecture holds in the case, i.e.,

Theorem 14 (Rubinstein’s theorem) Let , and let be a degree polynomial with all zeroes in the unit disk. If , then there exists a critical point of with , unless is a scalar multiple of .

The argument here is essentially in Remark 5.1 of this paper of Tang and Zhang.

Taking contrapositives, we may assume that the critical points are of the form for some in the closed unit disk, and normalize to be monic; our task is to show that .

The polar identity (8), based on calculating degenerates to a triviality when , but we have the following usable substitute, valid for any choice of , first observed in equation (3.2) of Meir–Sharma:

Lemma 15 (Meir–Sharma identity) If and the critical points are of the form then all the zeroes are not equal to , and

Proof: By hypothesis, is not a critical point of , so and for all . Instead of computing , we instead consider the expression . On the one hand, from (4) we have

while from differentiating (4) we have

Meanwhile, from (3) we have

and from differentiating (3) we have

Using these identities to compute in two different ways gives the claim.

Now take . Since lie in the closed unit disk, has real part at least , while is at most . Thus, the only way that the above identity can hold is if for all , hence for all . Thus all critical points are at the origin, which forces for some . Since , we conclude that , giving the claim.

— 4. The cases —

We now prove the cases of Conjecture 3. The starting point is (23). Using the triangle inequality and , this implies that

(This also follows from (16) and .) From Hölder’s inequality and we conclude that

The right-hand side can be computed to equal

which is obviously less than for , giving the contradiction.

Remark 16 The same argument also works for , but breaks down for higher .

— 5. Further directions —

The Sendov and Phelps–Rodriguez conjectures are now resolved, but several related conjectures remain open. The following strengthening of Sendov’s conjecture, by Borcea, is open for any :

Conjecture 17 (Borcea conjecture) Let and , and let be a degree polynomial with zeroes satisfying . Then for every zero of , there exists a critical point of with .

Sendov’s conjecture is the limiting case of this conjecture. There has been relatively little progress on this conjecture: the cases were established by Khavinson, Pereira, Putinar, Saff, and Shimorin, and in this previous paper we reported the negative result that AlphaEvolve failed to find a counterexample to the conjecture. The proof methods here do not seem to extend easily; all the identities relating zeroes and critical points continue to hold, but now that the are only constrained to the unit disk in an averaged moment sense, all of the inequalities developed above now fail.

Another strengthening of Sendov’s conjecture that remains open is Schmeisser’s conjecture:

Conjecture 18 (Schmeisser’s conjecture) Let , and let be a degree polynomial with all zeroes in the closed unit disk. Then for any in the convex hull of the zeroes of , there exists a critical point of with .

Schmeisser proved several special cases of this conjecture, and AlphaEvolve again failed to find a counterexample, but there has not been much further progress. Here, the are now back in the closed unit disk, but we no longer have , again rendering most of the previous identities invalid. But perhaps some modification of the arguments here can make some progress on this conjecture.

A common generalization of the Borcea and Schmeisser conjectures was proposed in Conjecture 2.4 of this paper of Zhang. A slightly different strengthening was also proposed in Conjecture 1.10 of Tang and Zhang:

Conjecture 19 (Tang–Zhang conjecture) Let , and let be a degree polynomial with all zeroes in the closed unit disk and critical points . Then for any , one has .

Sendov’s conjecture is the limiting case . By Hölder’s inequality, the case is the strongest form of the conjecture.

Another well known variant of Sendov’s conjecture is Smale’s problem:

Conjecture 20 (Smale’s problem) Let , and let be a degree polynomial. Then for any zero of , there exists a critical point of with .

The constant is best possible, as can be seen by the example and . Using the Koebe one-quarter theorem, Smale proved this conjecture with replaced by . Some slight improvements of this bound have been obtained over the years; for instance for , the improved bound of was obtained by Crane. Again, AlphaEvolve failed to find a counterexample to this conjecture. This problem does not seem to have a direct relationship with Sendov’s conjecture, and there is no useful normalization of the zeroes and critical points that is confined to the unit disk. Nevertheless there may be some hope of making progress on this conjecture, perhaps working first in the asymptotic regime .

Needless to say, I did try some desultory attempts to use AI tools to attack these questions, but without much notable success.

One potential way forward is to find further proofs of Sendov’s conjecture that utilize other techniques that might be more broadly applicable to this larger family of problems. The proof here is remarkable in that the zeroes and critical points are treated almost as independent mathematical objects, communicating with each other only very narrowly through four identities in which one only inspects the underlying polynomial (and its derivative) at a small number of points. It could be that an approach focusing on more global features of the polynomial may lead to new proofs of Sendov’s conjecture, and perhaps also of its generalizations.

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.

What does it mean to find a counterexample?

Why am I suggesting that it is not completely obvious what it means to find a counterexample? Surely, one might suggest, all it means is that you have a statement of the form “Every object of such and such a type has such and such a property,” and you exhibit an object of the given type that does not have the given property.

However, this doesn’t always work. Consider a famous result of Vinogradov, which states that every sufficiently large positive integer is a sum of three primes. The negation of this statement is (or is equivalent to) the statement that for every positive integer there exists an integer such that is not a sum of three primes. In other words, it states that every positive integer has a certain property. Seen in this light, Vinogradov found an example of a positive integer that does not have the given property. Do we want to say that Vinogradov found a counterexample? Clearly not — the result should obviously be classified as a theorem and not a counterexample.

Thus, we cannot just naively say that LLMs are particularly good at negating universally quantified statements: there has to be something about the nature of the universal quantification. With the three-primes example, it is clear that Vinogradov did not think, “How am I going to find with this property?” Rather, what he thought would have been more like, “I’ve got an integer that is very large. How am I going to show that it is a sum of three primes?” In other words, all his focus would have been on the universally quantified , with the existentially quantified being a sort of afterthought once the details of the proof have been worked out.

In general, many interesting results, when they are stated formally, begin with an alternation of two or three (or more) quantifiers. The question then becomes to determine which is the first “interesting” quantified variable in some sense. Here’s another example to illustrate the point, from the theory of finite-dimensional normed spaces. I’ll give a few mathematical details for those curious, but if you don’t care about those, then you can skip the next three paragraphs and should get the gist of what I am saying about this example.

Let and be two -dimensional normed spaces and let be a linear map from to . We say that is a –isomorphism if there exists such that for every . By rescaling we can always take to be 1, in which case we have that for every . If , then this tells us that is an isometry. In general, the Banach-Mazur distance between and is defined to be the smallest such that there exists a -isomorphism from to . It is easy to see that the logarithm of the Banach-Mazur distance is a metric on the set of isometry classes of -dimensional normed spaces. A less easy fact, but still not too hard, is that the resulting metric space is compact: in fact, it is known as the Banach-Mazur compactum.

It is natural to wonder what the diameter of the Banach-Mazur compactum is, and here things get interesting. A result of Fritz John states that every -dimensional space has distance at most from . (The idea of the proof is as follows: pick inside the unit ball of an -dimensional ellipsoid of maximal volume; that is the unit ball of a normed space that is isometric to ; it can be shown that the identity map is a -isomorphism between and .) From Fritz John’s theorem and the (multiplicative) triangle inequality, it follows that for any two -dimensional normed spaces. That is, the diameter of the Banach-Mazur compactum is at most . But might it be substantially less than that?

An indication that the answer is not obvious comes from looking at the spaces and . The identity map between these two spaces is an -isomorphism, but one can do much better by mapping the standard basis vectors not to themselves but to vertices of the unit cube, with the vertices chosen to be as orthogonal as possible. In particular, if there exists an Hadamard matrix, then the corresponding linear map is a -isomorphism. One can push this observation and deduce that for any the Banach-Mazur distance between and is . It is also easy to show that , so -spaces hardly improve on the easy lower bound, and do not improve on it at all in dimensions for which an Hadamard matrix exists.

In 1981, Gluskin famously solved the problem by determining the correct asymptotics for the diameter of the Banach-Mazur compactum. Informally, what he showed was that the diameter is within a constant of the upper bound that follows immediately from Fritz John’s theorem. If we make the quantification explicit, then the statement we end up with is

,

where I have written for the set of all -dimensional normed spaces. (If you want to argue that it is not a set, then let me specify in addition that the underlying vector space is .) In words, there is a positive constant such that for every positive integer there are -dimensional normed spaces and such that the Banach-Mazur distance between and is at least .

I can’t continue without very briefly describing the beautiful and highly influential idea Gluskin had for solving this problem. He took and to be normed spaces whose unit balls were random symmetric convex sets defined as follows: take the standard basis vectors and a handful of other random unit vectors, as well as the negatives of all these vectors, and take the convex hull. Gluskin then showed that if two normed spaces are chosen from this distribution, then with high probability their Banach-Mazur distance is at least .

But back to the main point, which is that the logical form of the above statement is very similar to the logical form of Vinogradov’s theorem, which is

where I have written for the set of primes. And yet, Vinogradov’s result is unquestionably a theorem, while Gluskin’s result is unquestionably a counterexample, or at least an example.

What is the important difference between the two statements? It seems to be that in Vinogradov’s three-primes theorem the number plays a more essential role in the statement that is to be proved about the various quantified variables. In Vinogradov’s theorem, that statement is , whereas for Gluskin’s theorem the statement to be proved is

and ,

which we can write equivalently as

and .

In the case of Vinogradov’s theorem, the whole challenge is to get those three primes to add up to , whereas for Gluskin it is not remotely challenging to get the dimensions of and to equal : the challenge is to get and to be very far from each other, relative to their common dimension.

There is a further complication to bear in mind here, which is that via the process known as Skolemization, a universally quantified statement of the form can be converted into an existentially quantifed statement . (For this to be an equivalence one needs the axiom of choice, but it is certainly a sufficient condition.) This is not just a piece of logical trickery, but it often reflects quite accurately how we think about some problems. For instance, it is more natural to think of Gluskin’s example as a recipe for constructing (or at least proving the existence of) a pair of suitable normed spaces for any given dimension , or in other words to construct a suitable function from to pairs of normed spaces by giving its value at each , than it is to think of it as a statement that says that every positive integer has a certain complicated property.

Yet another complication is that some universally quantified statements follow naturally from existentially quantified statements, or may even be equivalent to them. For example, the theorem that a 2-dimensional torus is not homeomorphic to a 2-dimensional sphere is a universally quantified statement (every map from the torus to the sphere fails to be a homeomorphism), but the natural way to prove it is to prove the existential statement that there is an invariant that distinguishes the two spaces. For an example of where a universal statement is equivalent to an existential statement, consider a statement of the form that a vector does not belong to the convex hull of a certain compact set . The statement that no convex combination of elements of is equal to is equivalent to the existence of a linear functional and a such that and for every . In both these cases it feels natural to regard the result as a theorem that is proved via an existential statement, perhaps because it is the theorem that is ultimately what interests us. But using “what interests us” as a criterion to determine what counts as a counterexample seems a little vague, and is a difficult criterion to use if we want to explain convincingly why AI should be good at finding counterexamples.

A more general argument against the notion that there is something about existential statements that is particularly suited to AI is that the need to establish existential statements pervades almost all of mathematical research, regardless of the nature of the headline result being aimed for. For example, if I want to prove a statement by induction, I may well look for a strengthening of the statement that serves better as an inductive hypothesis. Or if I want to prove that every object of type with property also has property , then I may well look for a property that follows from and can be used to prove . These are more metamathematical existence problems, but the distinction can be somewhat blurred, and more importantly, when trying to prove a statement , it is often the case that the main question in our minds is less, “Why is true?” and more, “What could a proof of be like?” To give an example, I feel I understand pretty well why Goldbach’s conjecture is true — a highly plausible probabilistic model of the primes implies it and agrees closely with computational data — but if I were making a serious attempt to prove it, that understanding, which many mathematicians have had for a century or so, would be of limited help. Rather, my main task would be to try to find proof techniques that were powerful enough to make those heuristic ideas rigorous.

What is the difference between an example and a counterexample?

Logically, every statement of the form is a counterexample to the universally quantified statement . However, we do not describe all existential statements as counterexamples. For example, if I were to say, “The -spaces with are all separable, as is , but is not separable,” I would not describe the second part of that assertion as a counterexample to the claim that all Banach spaces are separable. Rather, I would present it as probably the most basic example of a non-separable space. The important point seems to be that there was no particular reason to think that all Banach spaces would be separable, and finding an example of a non-separable space is not very difficult.

I think the first point is more important here: we are more inclined to call an object a counterexample if the existence of that object disproves a statement that we had quite good reason to believe. It often happens that after repeated unsuccessful attempts to prove a statement, mathematicians begin to feel that it has no particular reason to be true, even if it seems to be hard to come up with a counterexample to it. In such a situation, if a counterexample is eventually found, it may have lost something of its “counter” feel. My impression is that the construction of a non-sofic group comes into this category. There have been several proposals in the literature for how one might construct such a group, and I don’t think there were many (or even any?) experts who strongly believed that all groups were sofic. So it feels more natural to say, “OpenAI came up with the first example of a non-sofic group” than to say, “OpenAI found a counterexample to the soficity conjecture” (despite the fact that that section of their paper is entitled “A counterexample to the soficity conjecture”).

Likewise, it seems to me that the new lower bound for multicolour Ramsey numbers is more of an example than a counterexample. I think quite a lot of people believed that the bound should be exponential, so for them it was a counterexample, but others, myself included, were more neutral about it. As a matter of fact, I have worked on the problem in the past (a long time ago) in an equivalent formulation, which asks how many triangle-free graphs on vertices you need if you want their union to be the complete graph . If you take bipartite graphs, then it’s easy to see that you need of them, but that bound can be improved if instead you observe that a complete 5-partite graph can be written as a union of two triangle-free subgraphs, and therefore it is possible to write the complete graph as a union of triangle-free graphs. It is then tempting to try to do better, with triangle-free graphs that are less dense but that make up for it with unbounded chromatic number — a necessary condition if one wishes to use a sublogarithmic number of graphs, which is equivalent to showing a superexponential lower bound for . All this is to say that when I worked on the problem, my efforts were concentrated on what turned out to be the right direction, so for me OpenAI found an example of what I (weakly) expected, rather than a counterexample.

Where does this leave us?

I would like to find a coherent explanation of the conjunction of the following facts.

1. The most notable mathematical results proved by LLMs have tended to be ones that we would classify as examples or counterexamples, where counterexamples are, broadly speaking, existence statements that disprove statements that we expected to be true.
2. Many statements can be formulated as existence statements when we would usually think of them as universal statements, and vice versa, so what we consider to be an example depends on the mathematical context of a statement as well as its logical form.
3. LLMs are pretty good at proving universal statements as well: it’s just that the strongest statements they have proved that we would think of as theorems have mainly not been at the level of the strongest statements that we would think of as counterexamples.

Given these facts, it seems likely that what LLMs are good at is something else, which happens to have as a consequence that they are good at the kind of existence problem that we would normally classify as asking to find a non-trivial example.

Let us consider two things that we can be confident that LLMs are good at. One of them is knowing a lot of mathematics: if a problem can be solved by means of a relatively standard argument, it is highly likely that an LLM will be able to find and use that argument. The other is the ability that an LLM has simply by virtue of being a computer: it can work at huge speed (compared with humans at least) and can therefore afford to make a large number of unsuccessful attempts at a problem before it finds a solution.

Without even looking at what LLMs have actually managed to solve, one might guess that these two features would lead to their having a somewhat different style from human mathematicians. Very roughly, LLMs would have the edge when there is more of a probabilistic element to the proof-finding process: they would be good at problems for which the best method is to try a lot of ideas, not necessarily particularly novel, until at some point you get lucky. Humans on the other hand would be better (for the moment) at finding more “surprising” and “conceptual” arguments, where the appropriate method is to dig deeper and deeper into a problem until the solution reveals itself. (It is hard to say exactly what this means, but I hope that any experienced researcher reading this will know what I am talking about.)

This raises two questions: does the guess above correspond at all to the reality that we are observing, and is there any reason to suppose that what I have tentatively described as the “LLM style” of doing mathematics would lead naturally to LLMs discovering several counterexamples (or just examples) to long-standing conjectures, even if that was by no means all they could do?

I don’t pretend to have a scientific answer to either question, but the reactions of experts to several of the remarkable solutions that ChatGPT has found do lend some support to the idea that LLMs work in more of a try-lots-of-things-till-you-get-lucky way. People often seem to react by saying something like, “Initially I was amazed that the problem had been solved, but on closer inspection I realized that the approach was actually not all that novel, and one that with the right small hint a suitably expert human could have found quite easily.”

For the second question — whether the LLM style is well suited to finding (counter)examples — I think matters are less clear, because there are many ways of searching for a counterexample, and some of them fit better than others the style I have described. Here are a few general methods. (I don’t claim that the list is exhaustive.)

1. Look for an off-the-shelf example. Here one has a stock of fairly standard examples and one simply tries them out one after another to see whether any of them fails to satisfy the given statement. For example, Ryan O’Donnell ends his wonderful book on the analysis of Boolean functions with some tips, one of which is, “If you have a conjecture about Boolean functions, test it on dictators, majority, parity, tribes (and maybe recursive majority of 3). If it’s true for these functions, it’s probably true.”
2. Build an example from basic examples and standard construction methods. For an algebraic problem, for instance, one might start with some standard examples, but then take products or quotients or limits.
3. Make heavy use of metavariables. The word “metavariable” comes from computer science, and in particular from automatic theorem proving, and refers to the practice that in mathematics would correspond to writing, “where is to be chosen later,” (in which case is the metavariable). In a paper we usually do this only in fairly simple situations such as when we need to choose a number that is small enough for later arguments to work. But when we search for an example of an object that satisfies some property (which may well be a conjunction of simpler properties ), it is often not a good strategy to specify completely and only then to check whether it satisfies . Instead, it can be more fruitful to do almost the opposite: we start by saying virtually nothing about and simply launch into proving that it satisfies . In the course of doing so, we find that we need to satisfy a property . If we are lucky we can describe in a nice way a very general class of objects that satisfy . For instance, we may be able to find a parametrized class: we identify some function and show that satisfies for every of a certain type. The problem is then reduced to finding such that $Q(f(y))$ holds, which is a more specific version of the original problem. There may be many iterations of this process, or a mixture of this process and other processes, before an example is eventually found.
4. Try to prove the opposite. If one wishes to find such that , it can be surprisingly helpful to start by attempting to prove the statement . The reason this can be helpful is that using our standard methods of attempting to prove something, we may end up identifying a key lemma that would suffice: that is, we may find an intermediate property that implies in a non-trivial way and thus reduce the problem to . Turning things round again, it may well then be that finding a counterexample to is easier than finding a counterexample to (that is, an example that satisfies ). Of course, there is no guarantee that a counterexample to will be an example of , but sometimes we are lucky and it is. More often, we can use the idea of the previous method, noting that it is at least a necessary condition of an example of that it should not be an example of , so one can try to describe a general class of objects that fail and in that way reduce the problem.
5. Successive approximation. Sometimes, when we are searching for an example of such that , we write down a moderately plausible guess not because we think it has a chance of working (if we did, then we would be using the first strategy), but because we hope that if does not satisfy , then we will be able to diagnose what went wrong and specify a new guess that does not have that defect. Again, this strategy can either be iterated or combined with one or more of the other strategies.
6. Just-do-it proofs. Sometimes we need to satisfy infinitely many properties , each of which is, individually, quite easy to satisfy. In such situations, we often “build” inductively bit by bit, ensuring at the th stage of the process that however the building process continues, will satisfy .
7. Pick a random example. Often it is very hard to give an explicit example of an that satisfies , but there is a natural probability distribution for which one can show that if one chooses randomly from that distribution, then with high probability (or at least non-zero probability) it will satisfy .
8. Pick a generic example. In more infinite contexts, it may again be quite hard to give an explicit example of an that satisfies , but one may be able to show that the set of that fail is or measure zero, or is a meagre set, or is small in some other way.

There is no particular reason to suppose that LLMs would be equally good at each of the methods above. So perhaps what we are observing is not quite that LLMs have a particular ability to find examples, but more that they are particularly good at finding examples (and proofs) in a certain way. Looking at the above techniques, one might imagine that they would be very well suited to checking off-the-shelf examples, finding just-do-it proofs (since that is a rather standard method with lots of instances in their training data), using the probabilistic method (unless, as often happens, significant new ideas are needed to show that the probabilities work out), and picking generic examples. The other three methods described above — use of metavariables, trying to prove the opposite, and using successive approximation — require more of an ability to judge whether the approach one is taking is likely to be fruitful. Here it seems at least possible that humans will sometimes have an advantage, but the conditions that a problem would need to satisfy are quite stringent. One would need an example to be one that lies at a leaf of a very large search tree — too large to be searched for by a combination of moderate mathematical ability and brute force — but that can be found by a mathematician with a sufficiently good nose for when they are making progress that they can prune the search tree very substantially.

Why wouldn’t LLMs also have that “nose”? I don’t rule out that “nose” is an emergent property of the way LLMs are trained, and that within a year or two they will have it to the same extent that we have it. But for now, in my interactions with ChatGPT, I do have a distinct impression that they haven’t got there quite yet. When I discuss an open problem with 5.6 Pro, I am often presented with approaches that sound promising until I think about them carefully, and then seem quite a lot less promising. And they will also often end a response by saying, “I have not managed to answer the question you asked, but have managed to reduce it to the following much narrower and more precise question,” which sounds very promising until it has happened five times without any obvious progress having been made. It isn’t completely obvious how they will get better at this, since their training data will not be full of examples of fruitful and less fruitful directions to pursue when trying to solve problems: all they will typically see is tidied up proofs that hide the thought processes of their discoverers. Of course, human mathematicians also don’t get to learn much about how to do research from the experience of other mathematicians, and yet we somehow manage to pick it up. But the situation is a little different for us, in that a lot of what we learn is by doing rather than emulating.

Another reason it is not obvious that “nose” is a property that emerges naturally when LLMs are scaled up is that if LLMs make heavy use of their broad knowledge and can afford to do a lot more brute-force search than humans can, then they will lack the incentive that humans have to prune the search tree ruthlessly. It could conceivably be that their successes so far are achieved using methods that for a human would be considered extremely inefficient, but that because of their superior speed and knowledge, the combinatorial explosion these methods will lead to has not yet become apparent.

It would be very interesting to try to test this experimentally, but it is also difficult, because if an LLM has what looks like the kind of idea that could only be the result of “deep thought” about a problem, we can never be sure that it has actually carried out that deep thought, as opposed to finding a model argument already in the literature, or in other words exploiting the deep thought of a human mathematician. It would probably be easier (but still not easy) to test it by using models that are less powerful than the latest ones and that have been to some extent shielded from the mathematical literature: one could give them a carefully designed suite of problems and see whether the ones that the LLMs solve have particular characteristics.

It may seem as though I am desperately clinging to the hope that humans will continue to be able to make meaningful contributions to mathematical discovery for a while yet, but while I do indeed hope that, I am not making any assertions of the form “LLMs will never be able to do X”. I think it is likely that they will, and given the pace of progress over the last three years it will probably happen quite soon. But I do think that there may be a hurdle for LLMs to clear and it seems at least possible that it won’t be cleared as straightforwardly as some of the previous hurdles.

In that connection, it would also be interesting to see whether a different reward structure leads to LLMs being able to solve different kinds of problems. For example, if during training an LLM (or machine-learning system of some other kind) is not just rewarded if it ends up with a solution, but also penalized if it explores too many dead ends or if it “cheats” by getting the answer from the literature, perhaps it would be incentivized to go about the research process in a more human way and thereby achieve better results for classes of problems where it is yet to make a big impact.

If the hurdle is cleared, either by pure scaling up or by some more thoughtful method, it will be quite difficult to know when that has happened, since, as just mentioned, an idea that seems very original and surprising may just be lurking somewhere in an LLM’s training data. But I would be confident that it had been cleared if an LLM were to come up with a proof that was as surprising to me as the solution of the cap-set problem was in 2016: the previous best known bounds were completely eclipsed, the method was utterly different from anything I had thought about trying, and afterwards there was a flurry of activity as people came to understand what this wonderful new technique was capable of.

Conclusion

I wasn’t quite sure where I would end up when I started this post, and now that I’ve got to the end, I feel that my main conclusions are not particularly new or surprising, but I hope that the route to them is of some interest. The main points I have made are the following.

1. “Finding an example” is in practice not the same thing as proving a statement that begins with an existential quantifier.
2. If it is true that current models are particularly good at finding examples, that is probably not because they have a particular affinity for existential statements, but more because the proof-discovery methods that are appropriate for finding certain kinds of examples play to the obvious strengths of LLMs: wide knowledge and the ability to explore many paths of the search tree that humans would judge to have a low probability of success.
3. It seems likely that LLMs will carry on improving very quickly. However, if, contrary to expectations (mine at least), there turns out to be some residual class of problems (or other mathematical activities) for which humans continue to have the edge for a while, it is likely that those will be problems for which the mysterious human ability to prune the proof-discovery search tree is particularly advantageous: that is to say, problems where the search tree is deep and has a large amount of branching, so that without rigorous pruning a search is not feasible even for a computer.
4. A good sign that LLMs have reached human level for a much wider class of problems will be if they start proving theorems using methods that, like much of the very best human mathematics, are new and surprising but that with hindsight come to seem beautiful and natural. They should also be methods that are difficult to stumble on by accident. It is hard to say precisely what would count as such a proof, but I think we’ll recognise it when we see it.

A function of one variable can be translated in space by a spatial shift to obtain a new function

and also modulated in frequency by a frequency shift to obtain a new function

One can compose these two operations to obtain a time-frequency shift:

As per the time-frequency uncertainty principle, the two shifts do not quite commute with each other. For instance, we have

As such, is not a representation of the abelian group , but rather a portion of the Weyl representation of the Heisenberg group, but we will not adopt a representation-theoretic perspective here.

Some functions obey finite linear relations between their time-frequency shifts. For instance, a sinusoid obeys the relation

However, the Heil-Ramanathan-Topiwala (HRT) conjecture states that once one imposes some reasonable decay condition on , no such relations exist:

Conjecture 1 (HRT conjecture) If is non-zero, then there is no relation of the form

for some distinct time-frequency shifts and some coefficients , not all zero.

A special case of the HRT conjecture, which was also open, makes the additional assumption that was Schwartz.

Many positive results towards this conjecture were known. I will mention only a few here. A simple case is when we only have frequency shifts rather than spatial shifts:

In this case, the operator is simply a physical space multiplier

with symbol

so that the relation (2) now takes the simple pointwise form

If the coefficients are not all zero, then is a non-zero analytic function and thus has isolated zeroes. It is thus not possible to solve this equation for any . Thus the HRT conjecture is true when all the time-frequency shifts lie on the vertical axis. Using the metaplectic representation, one can then handle the case when all the are collinear.

What about the non-collinear case? Suppose first that all the lie in the lattice , thus for some integers . Here, the phase shift in (1) disappears, and all the time-frequency shifts commute with each other. This suggests that it should be possible to diagonalize the situation with a suitable transform to convert (2) to a pointwise equation similar to (3). To find this diagonalization, observe that if one restricts the function to a coset of the integers, then just multiplies the function by the scalar , while shifts the function on this coset by . The latter translation operation can also be converted to pointwise multiplication by performing the Fourier transform on the integers. Thus, if one introduces the Zak transform

of then the equation

can be transformed after a brief calculation to the equation

where the symbol is now given by the formula

As before, if the coefficients are not all zero, then is a non-zero analytic function and thus non-zero almost everywhere. Thus has to vanish almost everywhere, which for can be used to show that also vanishes.

More generally, there is a result of Linnell that the conjecture is true if lie in a translate of a discrete subgroup of ; this (together with the argument handling the collinear case) establishes all cases where , and several partial results involving the cases are also known. The conjecture is also known if is decays at a suitably super-exponential rate, by work of Bownik and Speegle.

I was aware of this conjecture through various talks and conversations with colleagues, and even briefly tried my hand at it for a while, though not with particularly serious effort (or progress). It was thus a nice surprise to see that it has just been resolved by Faulhuber, Petersen, van Velthoven, and Voigtlaender, even in the Schwartz case:

Theorem 2 There exist complex numbers , not all zero, distinct points , and a non-zero Schwartz function such that

It is perhaps unsurprising that this result is AI-assisted. However, I think the authors have disclosed their AI use responsibly, with the final arguments written by hand with a readable overview of the argument, as well as proper discussion of methods, relation to past literature, and other independent numerical checks on the result.

The negative result lies only a little beyond the positive results: is now increased to , and all but one of the points lie in (a translate of) a discrete subgroup of (in fact the explicit subgroup is used). The functions constructed are smooth and rapidly decaying, but not analytic or super-exponentially decaying, which would start being in conflict with the known positive results.

In addition to AI being used to come up with the initial proof strategy, a more traditional numerical computation was used to verify one step of the argument.

I have not had the time to do a full digestion of the result, but (after reading the introduction, and using a little AI assistance of my own) I was able to understand the main ideas at a high level. The first few reductions are relatively standard. Setting and , one can view the problem as one of solving an eigenvalue problem

The time-frequency shifts are chosen to lie in a translate of the discrete subgroup by a certain irrational shift . As mentioned previously, if in the shifts of a standard lattice , it would be natural to work with the Zak transform of , but it turns out that the approach does not quite work when doing this for topological reasons (relating to the fact that scalar quasiperiodic functions of mean zero are forced to have zeroes), and so the authors used the slightly denser lattice instead , which relates to a vector-valued version of the Zak transform taking values in rather than . Here, the phase shift in (1) does not completely disappear, but becomes a sign change. This slight loss of abelianness means that we cannot hope to diagonalize the problem all the way to a scalar problem, but we can still hope to reduce it to a two-dimensional vector-valued problem. Indeed, by applying a suitable vector-valued version of the Zak transform, the eigenvalue problem can be transformed to a a “vector cocycle problem”

where is a non-zero smooth quasiperiodic vector-valued function, is an irrational shift , and is a certain explicit matrix-valued function depending on the choices of , , and .

How to solve this equation? The motivating scenario here is if the matrix function was replaced by a rank one function

for some smooth vector-valued function of unit magnitude. Then one could solve the equation by taking and . It is not possible to make the function exactly of this form, but through some numerical computation and clever AI-assisted guesswork, the authors were able to find a choice of and , and that made approximately equal to a rank one function of this form, in fact getting a uniform estimate

As it turns out, such an approximation is sufficient to run a contraction mapping argument to find a solution to a variant of (4), namely

for some smooth and . (Here it was important to get the operator norm bound below ; they are barely able to do this, with a numerically obtained bound of , though this bound might not be optimal.)

The main remaining obstacle is that the “eigenvalue function” is varying in the parameter rather than constant. (This issue was, by the way, anticipated to some extent in previous work of Demeter, who observed that eigenfunctions of the almost Matthieu discrete Schrödinger operator gave a near-miss counterexample to the HRT conjecture, but with an eigenvalue that depended on an auxiliary phase shift parameter rather than constant.) However, if one was able to solve the scalar cocycle equation

for some smooth , then one could solve the equation (4) by setting . The approach to solve (5) is standard: take logarithms, apply a Fourier transform, and then divide out by the multiplier associated to the shift. This can cause a well-known “small divisor” problem (which arises in various dynamical contexts, such as in the KAM theorem) if behaves too much like a rational vector, but the standard resolution to this is to select a shift that obeys good Diophantine approximation properties. For the purposes of numerics the authors selected an extremely concrete shift, namely

but I get the impression that the exact choice here was not crucial for the argument, and that many other irrational algebraic numbers could have worked here.

Last September I went to a workshop at the Lorentz Centre in Leiden to discuss mathematics and AI with historians, philosophers, computer scientists, AI researchers, and mathematicians of several different flavours (though there was a surprising preponderance of algebraic geometers). The whole event was extremely stimulating, with some talks but also a lot of time set aside for discussion. One of the concrete outcomes of the workshop was the Leiden Declaration, which has now been signed by over 3000 people. Given that I was part of the workshop, it might seem a bit strange that I am not one of the signatories of the resulting declaration. The reason is not so much that I disagree with it in any concrete way, but more that in several places it makes confident assertions and recommendations that I feel somewhat uncertain about. So instead I prefer to try to articulate my views about the issues raised by the declaration and put them in this blog post. Before I do that, I would like to make clear that I am very glad that the Leiden Declaration exists and I think that it has done a lot of good in focusing people’s minds on the issues that AI is forcing the mathematical community to grapple with, which are more acute now than they were last September.

Let me begin by quoting a passage from the declaration that sets out “what we take to be characteristic values of mathematical research that we have a joint interest in preserving”.

1. There are many reasons to pursue mathematical research, ranging from intellectual curiosity to a desire to solve practical and societal problems. Underlying much of mathematics is the activity of proof. Mathematical proofs are regarded as conferring the highest degree of certainty to their conclusions, as well as imparting understanding of why their conclusions are true. These characteristics of proof support the scientific integrity of mathematics.
2. Results are attributable to specific authors who take credit for their discovery and assume responsibility for their correctness. These principles ground the merit-based standards to which we aspire in mathematical research.
3. Mathematical arguments are regarded as transparent and subject to independent verification. They may be extremely long or difficult, but in principle no proprietary knowledge or equipment should be required to understand them.
4. Mathematicians share a concern for proper evaluation of mathematical work relative to shared standards of depth, difficulty, and significance.
5. Mathematics produces not only a body of results, but also understanding, clarity, and judgment among the communities of mathematicians who have shaped them, often in the context of their own autonomously guided research. This expert knowledge is essential, both to effectively use mathematics, and to continue to articulate new and significant research questions. A key source of strength of the discipline has long been the autonomous shaping of the direction of research and the methods used to pursue it.

The first thing I would say about these values is that they are undoubtedly values that are widely held by mathematicians, including, with some qualifications, me. The main qualification I have concerns point 4: I find the notion of “proper evaluation” somewhat problematic, given that different mathematicians can have very different judgments without either of them being clearly wrong, especially when it comes to the significance of a piece of mathematics. Also, these judgments are used for purposes such as the acceptance of papers in journals, hiring and promotion decisions, the awarding of prizes, and so on, that are part of a system that copiously rewards a few people — I myself have hugely benefited from it — but doesn’t necessarily adequately reward a lot of people who are doing less visible work that is essential to keeping the whole enterprise going.

But the more important point is whether these values are ones that we should fight for in the future, as the Leiden Declaration suggests. I find that clearer for some of them than others. For example, it seems to me that the importance of rigorous proof will be even greater in an AI age than it was before — if the output of AI is not underpinned by rigorous proof, then the kinds of difficulties one already hears about with certain areas of human mathematics (see for example many talks by Kevin Buzzard arguing for the value of formalization) would be hugely magnified. But what about the attribution of results to specific authors, who take both credit and responsibility for them? Suppose that at some point in the future AI becomes more autonomous, reading the literature and solving many problems that it finds. Suppose also that its solutions are autoformalized, so there is no serious doubt about their correctness. In such a situation, there would be nothing for a human to take credit for or responsibility for. Does that mean that we should declare such results undesirable and threatening to mathematical values?

Of course, something could well be missing in such a situation: perhaps the proofs would be badly written and hard to follow, which would mean that they lacked something we all very much value. So let me extend the thought experiment slightly. What if by that stage one could take one of these outputs and ask an LLM to explain the ideas, and what if LLMs did a very good job at that? That is not particularly hypothetical, since they are often pretty good at this job already, but I am imagining a world in which they are much better than they are now, as they will presumably become.

So now we would have a world in which a lot of problems had been solved, we were sure that the solutions were correct, and we had an LLM ready to explain those solutions in as much or as little detail as we wanted. Is that a future we should resist, and if so, why?

One obvious reason is that it would take a huge part of the fun out of the subject. It is extremely satisfying to struggle with a mathematical problem for months or even years and eventually solve it. But I worry about that argument, because it seems to be saying that we should resist doing mathematics the easy way because a tiny fraction of the world’s population gets huge pleasure from taking orders of magnitude longer to do it. That is not to say that I wouldn’t be sad that a way of life that has sustained me for the last forty years was not available any more — of course I would. I just find it hard to use it as a reason to argue that we should try to preserve the “ownership structure” of mathematical results. If we arrive at a world where mathematical theorems are no longer associated with mathematicians, maybe that won’t be any more problematic than the fact that stars aren’t named after astronomers and most aren’t named at all. I’m not necessarily in a hurry for that world to exist, but maybe once the transition had happened, people would be OK with it.

The third value I share in an uncomplicated way, and I have already discussed the fourth. The fifth value is one that I hold very strongly, though I’m not so keen on the idea of experts consciously “shaping the direction of research”, something that I see as happening more organically. Obviously there are some notable examples of mathematicians who have created wonderful programmes of research, but even there I would like to credit other mathematicians with understanding what is wonderful about those programmes and contributing to them enthusiastically as a result, rather than being told what direction to pursue and meekly doing so (which is probably not what the declaration is actually trying to suggest, but it has a slight flavour of that for me).

But that’s a minor quibble when set against my main worry about the effect of AI on mathematics, which is the possible destruction of mathematical culture. There is at the moment an extraordinary body of knowledge and expertise that exists not just in the mathematical literature but in the heads of mathematicians all round the world. Imagine if AI didn’t exist and a pandemic broke out that for some reason wiped out all mathematicians and nobody else. All the literature would still be there, but nobody would have the faintest idea what to do with it. To revive a mathematical tradition under those circumstances would be extremely difficult and take decades. Now imagine a slight variant of that, where AI does exist and because of it people are no longer motivated to put in the years of effort it takes to reach the level of expertise that a typical research mathematician has now. After a decade or two, we might arrive at a situation where the mathematical literature has, in some form, been vastly expanded, but there is no corresponding community of human experts who have a shared understanding of parts of it. Almost all of mathematics would be like the areas that we have more or less forgotten about today, areas that exist in papers written many decades ago that nobody reads any more. (I won’t name any such area because I don’t want accidentally to suggest an area that many people still love and work on.)

This, it seems to me, is a possibility that we should try very hard to resist, but I agree with many other commentators who say that in order to resist it, we will need to give less priority to some of our current values — and I would include ownership of mathematical results in that list — and more to others. For example, if Person A gets an LLM to one-shot a solution of an important open problem (which is formalized, possibly automatically, so there is no doubt about its correctness) but Person B makes the effort to digest the solution and explain it in a way that other mathematicians can understand and learn from, then I think we will want Person B to get the lion’s share of the credit. The credit would be of a slightly different from what it is now, which could be described as admiration for somebody’s talent, insight, speed (I mean here the purely factual statement that speed is often admired — I would prefer that to be less the case) and hard work. It would be more like the gratitude that one feels already for somebody who writes a beautiful textbook that makes a whole area of mathematics coherent and accessible.

Maybe that is what the “research mathematicians” of the future should do: make a selection from a vast sea of AI-generated mathematics and write a book about it in such a way that other mathematicians can read the book and feel the kind of enrichment that we feel when we get to grips with an area of mathematics.

At this point I have to admit that there’s a pessimistic side of me that asks the following general question whenever anyone says anything about what the role for humans might be in the future: why do you think that AI wouldn’t be able to do it? For example, with the suggestion I’ve just made, what reason is there to suppose that ChatGPT 8.2 wouldn’t be able to have a short interaction with you about your mathematical tastes and background and then write the ideal textbook just for you? Humans are likely to be better at this kind of curating for a little while yet, but is it a fundamentally human ability that AI could never hope to emulate?

In a world where AI wrote bespoke textbooks (or more likely, just taught people in some more direct way), something would be lost that feels important: mathematics as a collective endeavour. If we all just learnt cool bits of maths for our own private satisfaction, we would miss the considerable pleasure that comes from discussing mathematics with others, though even that could in principle be restored by a benign LLM that deliberately taught many people the same cool bits of the subject, though an LLM that could do that sort of social engineering would raise all sorts of safety issues.

Let me now turn to the section of the declaration about potential threats. I’ll put my comments on each one in square brackets.

1. Current automated techniques can produce plausible but unreliable (or even incorrect) arguments which are difficult to distinguish from correct mathematical proofs. This applies not only to informal arguments, but also to formalizations, where the difficulty lies in the translation between computer-encoded and human presentations of concepts. These fast-moving developments put our present system of review under increasing pressure, jeopardizing our ability to implement traditional standards for the correctness, transparency, and independent verifiability of proof. [This feels like less of a problem now than it did last September, partly because the best LLMs hallucinate a lot less than before, and partly because autoformalization is improving all the time — I have just used harmonic.fun’s Aristotle system to formalize a complicated paper in Lean and I didn’t need to know any Lean to do it.]
2. Technologies that draw extensively on the published mathematical commons undermine the traditional system of attribution. Models trained on published works frequently return outputs that do not properly cite the human works they synthesize. Many current models are also built on data obtained by systematically exploiting licenses and access arrangements that were not made with artificial intelligence in mind, or indeed by simply violating copyright protections. [This is a problem at the moment, when ownership of results is important, and I am very much in favour of people making an effort to give appropriate credit for mathematical ideas that AI may have used. However, in the longer term, as I have already discussed, I think this ownership structure will break down and the issue will become less important. It also seems possible that LLMs will become better at revealing their sources.]
3. Technologies which affect the way in which mathematics is practiced may disturb the current system of incentives. The use of artificial intelligence — and thus also the sort of problems which it can address — may become incentivized for its own sake, disrupting our mechanisms for hiring, funding, and recognition. This disadvantages researchers who do not have access to the technologies or decision-making related to them, or who are unwilling to use technologies controlled by organizations whose values they do not share. [These seem to me to be genuine problems. I think there is simply no point in hoping that our current system of incentives will not be disturbed — it obviously will. I am not necessarily too worried if our mechanisms for hiring, funding and recognition are disrupted, as I don’t find those mechanisms unproblematic as they are, but disadvantaging researchers who do not have access to good LLMs is something I certainly think we should worry about.]
4. Proper evaluation is endangered if results are communicated through informal channels such as press releases or blog posts, often without any research paper or other disclosure of information necessary for scientific evaluation. This practice seeks publicity for new results on market timelines before the accepted processes of community evaluation in mathematics can take place. In many cases this leads to simplifications in reporting, such as overemphasizing the significance of automated tools and undervaluing the prior human contributions which have made those tools possible. Such oversimplification risks influencing public opinion in a way that not only damages perceptions of mathematics, but also misleadingly uses specific mathematical tasks as metrics for the general reasoning capacities of commercial products. [I think this can be a problem, but I think it is not as serious a problem as some of the others, since when results get overhyped, there seems to be no shortage of people publicly (and rightly) pointing that out.]
5. These developments put the autonomy of mathematics under threat. The increasing involvement of technology companies in mathematical research raises the risk that research questions may come to be prioritized because of their amenability to automated mathematics, rather than expert judgment of their deeper significance. Indeed, broader understanding of the field may be permanently lost in the process of automation. With university budgets under pressure, this reshaping also changes professional incentives in a manner which encourages the collaboration of researchers with technology companies on asymmetric terms. If left unchecked, these trends go beyond threatening researchers’ autonomy, affecting the scope and depth of mathematical research itself. [I think this could be a problem, but it also seems to me that mathematicians have a lot of power here. For instance, if a technology company were to produce a lot of research that mathematicians did not find all that interesting or important, I don’t think they would be able to use their financial and other resources to persuade us to change our minds. Rather, what seems to happen is that mathematicians say, “Yes that does X but it doesn’t do Y,” and the tech companies then feel challenged to do Y.]

There follow eleven recommendations for individual mathematicians. I agree with almost all of them. The one that I’m not so sure about, for reasons I’ve basically already gone into, is this.

Affirm the humanity of authorship. Credit and responsibility continue to belong to humans within the mathematical community and should not be given to automated systems. Artificial intelligence may obscure, but does not replace, the collective human labor behind a result.

I’m not sure what that really means. For example, should we affirm the humanity of authorship in the case of the solution to the unit-distance problem? Some humans did a wonderful job of explaining the proof that OpenAI’s model came up with, and the model made use of some highly non-trivial mathematics produced by humans, but the solution itself has not been credited to any human, and nor should it be in my view.

Under recommendations for mathematical organizations and not-for-profit research funders I again agree with several of them but have my doubts about some. An interesting case is the following.

Protect the rights of authors. Automated mathematics presents new challenges to the rights of authors, and societies should be proactive in the development of sample licensing agreements to protect these rights. In particular, material should not be used as training data without consent, and publishing agreements should allow authors to opt-out [sic] of the use of their work in this way.

This recommendation seems to belong to a world in which journal articles are the main means of dissemination of mathematics. But that has long since ceased to be the case: almost all dissemination now takes place via arXiv preprints, with journals limited to providing a little extra mark of prestige. Once an article is on arXiv, it is on the internet and one can hardly ask for it not to be used as training data. So this recommendation, if it applies at all, will apply to a tiny fraction of articles that are published without first appearing on arXiv. More generally, what right of an author is being compromised when an article is used as training data? We don’t object if human mathematicians use our articles to help train themselves to become better mathematicians — indeed, we will typically be delighted that somebody else thought our articles worthy of their attention. So the objection to a machine doing the same would have to be that for some reason one did not want machines to get better at mathematics in a similar way. I can imagine grounds for such a wish: perhaps somebody is worried about the threat that LLMs pose to traditional mathematical practice, or perhaps they worry that mathematical ability of LLMs will transfer to much more dangerous reasoning ability. But there’s a more complicated discussion to be had here than one might think from reading the recommendation.

The next recommendation is this.

Insist on appropriate publication outlets. Demand that mathematical results continue to be published in peer-reviewed venues such as journals, proceedings, and books. Informal mechanisms such as press releases or blog posts can provide a valuable supporting role, but they cannot replace peer-review or community scrutiny.

For reasons that I’ve gone into many times, I am not too fond of the current publication system, so I can’t get behind this recommendation. Indeed, if the current system becomes unsustainable because of a flood of AI-generated and AI-aided content, I would regard that as a beneficial consequence of AI. However, that doesn’t mean that I would advocate a total free-for-all. I’ve already said that one of my worries is that if mathematical content is not sufficiently organized, then the traditions that we all value could die. I just think that what we will want to do to preserve those traditions is likely to be a lot more innovative than clinging on to the peer-reviewed journal system.

I have highlighted in this post the parts of the declaration that I have doubts about, either because I disagree with them or, more typically, because I sort of half agree with them but want to add many qualifications. That may make the post come across as rather negative, but that is not my intention. The parts I disagree with are in the minority, and I think it is important that a declaration such as this should be made. I should also make clear that my views are evolving all the time, largely because the speed of progress of LLMs has taken me by surprise, but also as a result of conversations I have had or opinions that other mathematicians have expressed online.

I’ll end with two further clarifications. The first is that it may seem as though I am taking it for granted that LLMs will soon be better than humans at all aspects of mathematical problem solving, and maybe also problem posing, theory building, formulation of definitions, etc. I do think all that will happen at some point, but whereas some people say that it will obviously happen within the next two to three years, I would say that it might happen as soon as that, but I don’t rule out that we’ll get lucky and find that we can do interesting AI-assisted maths for quite a bit longer than that before AI doesn’t need us any more.

The second is that I think I have acquired a reputation as somebody who celebrates what is going on. But if, for example, I post on Twitter saying that such-and-such an AI solution is a remarkable development, the word “remarkable” is meant to indicate no more nor less than that I found it very surprising. My feelings about the possibility of AI solving all sorts of problems that interest me are much more mixed. I’ve had the experience twice now of seeing GPT 5.6 Pro one-shot a solution to a problem that I very much liked and had thought about hard (in both cases with much younger collaborators, who, with my approval, were the ones who prompted the LLM). It felt very strange and not particularly pleasant to have the rug pulled out from under my feet like that. On the other hand, I was quite pleased to see the problems solved. It’s actually a similar feeling to the one I have had many times when a problem I am fond of and have thought about gets solved by another human mathematician.

Another factor for me is that I have invested a lot of thought into automatic theorem proving of a more traditional kind. One of my main motivations for that was the hope that the work I put into it would extend the state of the art, measured by which problems a computer can solve. That ship has sailed now, and that saddens me. I still think that there is value in the work that I and my group are doing, but it has become a tougher sell.

So I personally have already found AI quite disruptive, and this is just the beginning. I would have preferred the developments to happen at a slower pace. But I don’t see any practical way to slow them down, so the best we can do is probably to face up to the changes that are being thrust upon us and do what we can to maximize the benefits and minimize the damage. The Leiden Declaration may not be perfect, but it makes an important and positive contribution to that effort.

The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows.

Conjecture 1 (Jacobian Conjecture) Let be a polynomial map in complex variables, whose Jacobian is a non-zero constant. Then is invertible (with polynomial inverse).

The condition that the Jacobian is non-zero is equivalent to being locally invertible. (The implication of local invertibility from non-vanishing Jacobian follows from the inverse function theorem; the converse implication can be derived from the Weierstrass preparation theorem, but is omitted here; see also Lemma 5 of this previous blog post.) Also, from the fundamental theorem of algebra, once the Jacobian polynomial is non-zero, it must be constant. So the hypothesis “Jacobian is a non-zero constant” can be replaced with “ is locally invertible”. So the Jacobian conjecture can be viewed as an assertion that local invertibility implies global invertibility. The complex numbers can be easily replaced with other fields of characteristic zero by the Lefschetz principle, but I prefer to work in the concrete setting of the complex numbers.

It was recently shown (using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well):

Theorem 2 (Counterexample to conjecture) There exists a polynomial which has non-zero constant Jacobian, but is not invertible.

The conjecture remains open in two dimensions, and is easy to establish in one dimension.

The example can be stated completely explicitly: one can take

and one can verify by a brief calculation that

and

While this is an extremely quick verification, the construction presented in this fashion appears like a massive miracle. The polynomial has degree seven, so a priori the Jacobian ought to be a polynomial in three variables of degree as large as , so the fact that all non-constant coefficients of this polynomial vanish looks like a massive cancellation involving equations, which is much larger than the degrees of freedom for a generic degree seven polynomial map of three variables. So finding such a polynomial looks highly unlikely to be located by brute force.

The example has since been retroactively explained in more geometric terms. As a “digestion” exercise to myself, I sought to write this explanation with relatively little use of algebraic geometry, in a manner that minimizes the amount of “miracles” required, although there are still a few places where some remarkable phenomena occur.

It is convenient to use the local injectivity formulation, and to generalize the domain to an equivalent affine variety. Namely, we will show

Theorem 3 (Counterexample, reformulated) There exists an affine variety that is isomorphic to by polynomial changes of variable, and a polynomial map which is locally injective, but not globally injective.

Clearly one can get from Theorem 3 to Theorem 2 by composing with the isomorphism and using the previously mentioned fact that local injectivity implies non-zero constant Jacobian. Our objective is now to find data , that obeys three separate properties:

• (a) is locally injective on .
• (b) is not globally injective on .
• (c) is isomorphic to by polynomial changes of variable.

The advantage of splitting the problem in to these three components is that we can build towards each of them separately.

(A pedantic remark: strictly speaking, in the arguments below, we not only replace the domain of by an equivalent variety , but also replace the range of by an equivalent variety . But the equivalence between and is a boring linear isomorphism ( will just be a hyperplane in a four-dimensional vector space ), so we do not highlight this aspect of the construction.)

It turns out that and can be built out of the operation of multiplication of low degree polynomials. Namely, consider the following three simple affine spaces:

• The space of linear homogeneous polynomials of two complex variables .
• The space of quadratic homogeneous polynomials of two complex variables .
• The space of cubic homogeneous polynomials of two complex variables .

(The notation here refers to the symmetric power of a vector space .) Clearly these spaces are isomorphic to respectively. Furthermore, we have a multiplication map , mapping a pair of a linear polynomial and a quadratic polynomial to a cubic polynomial

(Right now, the domain and range of this map is larger dimensional than the target of three; we will cut the dimensions down to three as the argument progresses.)

The map , essentially a map from to , is clearly polynomial; it is given explicitly in coordinates as

The map also enjoys two basic (and commuting) symmetries:

• If one applies a scaling for some non-zero complex numbers , then the product is scaled by : .
• If one applies a change of variables for some invertible linear transformation , then the product is transformed by : .

So this map enjoys a huge amount of equivariance, basically with respect to an action of the five-dimensional group .

The five-dimensional domain is of course larger than the four-dimensional range , so the map clearly cannot be injective. This can already be seen from the scaling symmetry, as the specific scalings

for modify the linear and quadratic polynomials but not their product . But even if one quotients out by this symmetry (3) to cut the dimension of the domain down to four, the map is still not injective for the following basic reason. A generically chosen cubic polynomial will split into the product of three independent linear polynomials. Then there are three pairs

which all map to the same cubic polynomial

under the multiplication map , but are not related to each other by scaling symmetry (3). Thus, we see that even after quotienting out by the scaling symmetry (3), the multiplication map is generically non-injective in a three-to-one fashion. Thus we already have achieved something resembling goal (b)!

It will be convenient to “spend” the scaling symmetry to obtain a useful normalization. If is a linear polynomial and is a quadratic polynomial, the (homogeneous) resultant can be defined by the determinant

If we have a factorization

then the resultant can also be described as

Thus the resultant measures whether the linear polynomial and the quadratic polynomial share a common root. A fundamental fact about resultants is that they are -invariant: for any , we have

One way to see this is to check it first for translations (which translate the roots by while leaving unchanged) and for inversions (which map to while mapping to and respectively), and then noting that these transformations generate all of . They also interact very nicely with scaling:

In particular, the scaling symmetry (3) multiplies by :

Thus, we can (generically) normalize away this scaling symmetry by imposing the condition

We now have a restricted multiplication map (which by abuse of notation we will continue to call ) from the four-dimensional variety

to the four-dimensional space . This map is still not globally injective, as we can take the three pairs in (4) from before and apply the scaling (3) separately to each of the three pairs to obtain the normalization (7). So we have kept property (b). Furthermore, this map retains the -equivariance (and also one remaining scaling symmetry, though we will not make much further use of that symmetry).

But we now also have property (a)! Suppose we want to show the local injectivity of in the neighborhood of a pair with . As the resultant is non-vanishing, the root of (which exists in the Riemann sphere, or projective line if you prefer) is distinct from the two roots of (though the latter two roots could be equal to each other). Applying the action (which performs Möbius transforms on the roots), one can assume without loss of generality that is the point at infinity (or equivalently ), thus for some complex number and for some complex numbers , with the resultant condition (7) simplifies to (so in particular are also non-zero). It is then clear that if one perturbs and by a small amount (say, modifying each coefficient by ), then the root of will perturb to something large (), while the roots of stay bounded. Thus, just from knowledge of the product , one can reconstruct which of the three roots of this cubic polynomial will be the perturbed root of , and which two will be the perturbed roots of ; from this and (6), (7) we can also reconstruct the leading coefficient of , and this completely determines both and . This establishes the local injectivity property (a). (In fact it is étale, but we will not need the machinery of étale maps here.)

Unfortunately, (the four-dimensional analogue of) condition (c) fails: the quadric hypersurface (8) is not isomorphic to the affine space . But we can try to get around this by passing to a three-dimensional slice. Let be some three-dimensional affine plane of (which we will take to avoid the origin for technical reasons), then we can restrict as a map from the set

to . is clearly identifiable (by linear changes of coordinate) to . As was already locally invertible, it remains locally invertible under restriction; and because generic cubic polynomials had three preimages under in (8), this continues to be the case after restricting to (9) (unless was somehow so degenerate that it had no generic elements, but this turns out to be impossible). So we have retained properties (a) and (b). The miracle is that, with a good choice of , we can also obtain (c) and obtain the desired counterexample to the Jacobian conjecture: despite appearances, the variety (9) is in fact equivalent to the affine space by polynomial changes of variable!

Let’s see how. The affine hyperplanes in avoiding the origin are parameterized by the dual space of avoiding the origin, which one can think of as the non-zero third order homogeneous differential operators in two variables. Indeed, every such operator generates an affine hyperplane that avoids the origin, and conversely by duality every affine hyperplane avoiding the origin arises in this form uniquely. Just as the cubic polynomials in can be factored into three linear polynomials, the differential operators in the dual space can also be factored into three linear differential operators, e.g.,

in the case that is non-zero. The action moves the roots around the Riemann sphere by Möbius transformations. As these transformations are -transitive, the actual selection of such roots is not too important (and the scaling symmetry similarly makes the choice of leading coefficient unimportant); the only thing to keep track of is whether the roots repeat. Up to the symmetries, there are in fact just three different equivalence classes of differential operator (and thus of affine hyperplane ) to consider:

• Operators where the three roots are all distinct, thus for independent first-order operators .
• Operators where two roots coincide and one is distinct, thus for independent first-order operators .
• Operators where all three roots coincide, thus for some first-order operator .

It turns out that the affine miracle for (9) occurs precisely in the second case, when has two identical roots. I do not have a completely satisfactory geometric explanation for this miracle, but one can verify it by the following coordinate computation.

By applying the action, we can normalize so that , thus is now the affine hyperplane of cubic polynomials with . Using (2) and (5), the variety (9) can now be described explicitly in coordinates as

At first glance this seems to be a generic-looking variety cut out by a cubic equation and a quadratic equation – hardly a candidate to be affine! But observe that if is non-zero, then the second equation can be solved for ,

and the first equation can be solved for ,

Putting these two equations together, we see that as long as one removes the case , the quintuple is uniquely determined by by a change of variables which is Laurent in and polynomial in . Thus we have a nice birational equivalence

Thus we have already almost established property (c): the variety (9) becomes birationally equivalent to after cutting out the subvariety. In particular, for each fixed non-zero value of , the corresponding fiber

of (10) is equivalent to by polynomial changes of variable, since we can reconstruct from the coordinates by the polynomial formulae

So we just need to glue back in the fiber. Indeed, from (10) we see that the fiber at is just

Now we observe a key miracle: the cubic equation and quadratic equation have a unique affine solution (as opposed to the six possible solutions that Bezout’s theorem might suggest – the other five solutions live on the line at infinity). So the fiber here is also affine:

This is extremely encouraging for the purposes of establishing property (c), as it strongly suggests that the variety (10) has the structure of an -bundle over , which is already extremely close to being isomorphic to the affine space . The main remaining task is to make sure that nothing singular happens in the limit , and that a global polynomial coordinate chart for (10) that covers both the and fibers can be constructed.

The standard way to proceed here is to manipulate various tangent spaces using the modern machinery of algebraic geometry and commutative algebra, but given my own background, I prefer to adopt the language of analysis, and in particular big-O notation (in place of the ideals used in algebraic geometry), in order to investigate the limit by hand. On the variety (10), let us use to denote any multiple of by a polynomial expression in . Thus, for instance, the equation implies that

while the equation implies that

as well as the more refined estimate

In the case we could conclude that . Now we perturb this observation. Multiplying (13) by we have , which on substitution into (14) gives ; substituting this back into either (13) or (14) also gives .

We can get some more precise asymptotics by also taking advantage of (15). Substituting into (15), we obtain after some algebra

So if we write more explicitly as , then we have

and thus

Substituting this back into (11) gives an asymptotic for :

Finally, one can insert these estimates into (12), although one only gets a trivial bound in this case:

Expanding the error term in (16) as , and doing a little more algebra, we thus have a polynomial change of variables

which completely parameterizes the variety (10) by polynomial combinations of three coordinates . This already gives (c) and thus completes the proof of Theorem 3.

The previous computations, when expanded out, also gives polynomial inverse maps:

The map from to the coefficients of (dropping the coefficient which is constrained to equal ), we obtain a polynomial map

with

which theory predicts to have a constant Jacobian, and indeed one can calculate that the Jacobian is . This is essentially the original example up to trivial changes of variable; indeed, one can check that the map

is exactly the map given in (1).

AI disclosure: I used an AI chatbot to discuss various aspects of this problem and to confirm several of the calculations made here.

I believe that the creation of visualization apps to illustrate mathematical or scientific concepts is a particularly favorable use case for modern coding agents, as many of the downside risks attached to other LLM use cases are limited:

1. Not mission-critical. As such apps are not authorative sources of truth and only used for secondary purposes, a small positive error rate in the output can be acceptable.
2. Stand-alone. As the applets are not destined to be incorporated into a larger codebase or literature, the technical debt incurred by delegating all the coding to an LLM agent is bounded.
3. End product is deterministic (and sandboxed). As the applets run on a deterministic language (Javascript), are sandboxed against file or internet access, and do not make any LLM calls at run-time, security and privacy concerns are minimal, and the applet can be maintained without continued premium LLM access or resource-intensive compute.
4. Not replacing primary skills. While deskilling is the tradeoff one accepts when relying on these tools to accelerate output, I am perfectly willing to forego the opportunity to keep my Javascript skills at a high level, as this is a tertiary skill for me at best in my chosen profession. (I continue to manually program in Lean and in Python to keep in practice with programming in general.)
5. Not competing with humans. To my knowledge, there is no existing human effort that is being duplicated by these applets (the activity in this direction appears to have peaked two decades ago).

I would however caution against unrestricted LLM use when one or more of the above five favorable situations is not in effect.

With these points in mind, I have used such an agent to create two further apps. The first app illustrates the “zeta process” that was introduced in my recent paper with Alexeev, Barreto, Li, Lichtman, Price, Shah, and Tang, though it was first discovered by an AI. For each , the zeta distribution is a random natural number with distribution

It has long been known that this distribution has good number-theoretic properties: for instance, the number of times a given prime divides has a geometric distribution of mean . However, the new observation is that these random variables can be chained together into a single stochastic process, which we call the “zeta process”, which is an infinite divisibility chain. I used an agent to create an app to visualize this process:

The underlying process is generated by several exponential random variables at each prime: in the above instantiation of the process, two such variables are visible at the prime , and one variable at the primes . At a given choice of , is formed by collecting all the variables below this threshold (and for which all predecessors also lie below the threshold); in the above illustration, this amounts to one variable at each of the primes , leading to in this case. Additional visualizations in the app display the distribution of each , as well as the distribution of the hitting probability , which among other things can be used to give a quick solution to Erdős problem #1196.

The second app is rather different in nature, and is a somewhat whimsical attempt to display the motion of the heavens, both at “human” scales of space and time, and at more “astronomical” scales (in which the motion of the planets in particular are more apparent). It is very loosely inspired by the game “Katamari Damacy“, in which one absorbs both terrestrial and celestial objects of many different scales. Here is how the app typically looks at a human scale:

And here is how it looks when one’s perspective leaves the Earth’s atmosphere:

(As I did not want to render an entire explorable world in this app, the observer in the app is only limited to changing his or her size, from a human to a creature of comparable size to the Earth itself; they cannot move horizontally on the planet.) At the largest scales of space and time, the classic orrery diagram appears:

After lengthy conversations with the agent, I was able to implement many astronomical phenomena, including phases of the Moon, the effect of Earth’s rotation against the fixed stars (though one can also stabilize one’s view against those stars to see the Earth’s rotation more directly), and so forth.

One byproduct of learning how to use coding agents to create visualization apps is that it now becomes straightforward to convert any figure in one’s papers that had already been generated by code (e.g., in Python) into a more interactive, animated applet.

I can illustrate this with Figure 1 from my recent paper on the Gilbreath conjecture with Chase and Hunter, reproduced below:

This plot displays both exact and numerically simulated values of a certain poorly understood sequence relating to the Gilbreath conjecture, which I will call the “Gilbreath expectation sequence” here for lack of a better name. The definition of the sequence is as follows. Consider a “Gilbreath array” which is an inverted pyramid, where the top entries are independent exponential random variables of mean 1, and all the other entries are the absolute values of the differences of the two entries immediately above it. Thanks to the visualizer app, I can quickly give an example (with ):

The left diagonal entries are then random variables; the sequence are defined to be the expectation of these values. (The process is stationary, so in fact any entry on the row will have expectation .)

If one starts with the first normalized prime gaps (which have expectation about , and are conjecturally distributed asymptotically according to a geometric distribution), then standard conjectures (e.g., the prime tuples conjecture) predict that the row entries should decay like , at least for small . So the Gilbreath conjecture appears to be tied to how fast the sequence decays with .

One can in principle work out each value of as an explicit rational number by performing a certain complicated multivariate integral, but in the paper we only did this for (the orange line in the above figure); for the remaining we performed a Monte Carlo simulation with Gilbreath arrays to obtain a numerical approximation (in blue), which (as per the law of large numbers) agreed well with the theoretical values. A later calculation of Michael Ross extended the theoretical values to , maintaining the good fit:

The asymptotic behavior of the sequence remains mysterious. Clearly, it is not monotonic; in fact we cannot even prove it is bounded. The best we could do in our paper was establish an inequality which, roughly speaking, showed that cannot decay faster than .

In a recent preprint of Ross, these numerics were extended, and a rough empirical prediction

was proposed for some constants and (empirically ), where is the number of 1’s in the binary expansion of ; in particular, it is the fluctuation in this quantity that is intended to explain much of the non-monotonic behavior of . These are now all displayed in the following companion applet, which was a routine matter to generate in about an hour by the coding agent (which by this point has extensive experience with creating such apps, encoded via a “skill” markdown file that it maintains):

The appearance of the quantity may initially appear mysterious, but it is related to Lucas’s theorem, Kummer’s theorem, and the Sierpinski gasket. Consider for instance a Gilbreath array where all the entries are zero except for a single “spike”. Then the following Sierpinski pattern emerges:

Here is what an version of this picture looks like (with the spike positioned at the 32th entry):

The number of 1s in the row is then (if we index the rows starting from zero), which is at least of the same shape as the empirical prediction, albeit with different constants. (This sequence is also known as Gould’s sequence.)

Numerically, we seem to observe fragments of Sierpinski gaskets being generated before decaying (often due to “collisions” with other gaskets):

However, it is not clear to me at all what the asymptotic probabilistic model should be, even heuristically; it does not resemble any random shape model that I am familiar with. But perhaps there are readers more expert in probability theory or statistical physics who may be able to suggest such an asymptotic limit?

(I am writing here in my capacity as Director of Special Projects at IPAM.)

IPAM seeks program proposals from the mathematical, statistical, and scientific communities for long programs, workshops, and summer schools.  Most program proposals are reviewed at IPAM’s Science Advisory Board meeting, held in November each year.  Programs are selected on the basis of their scientific impact and contribution to IPAM’s goals. IPAM is committed to supporting a community where people of all backgrounds and points of view can engage, learn, and thrive.  If you would like to discuss your program ideas and prepare a proposal for IPAM’s consideration, you are encouraged to contact the IPAM Director. For more information visit: https://www.ipam.ucla.edu/propose-a-program/long-programs-2/

I am finding the newly revealed capability to code old applet ideas into reality to be very tempting to sink more time into, though I am certainly encountering the common “vibe coding” experience that the process can produce something that superficially resembles a finished product well before a satisfactory level of testing and review has been completed; indeed, it is the review process which is now the most time-consuming, to the point where I think any further advances in coding agent capability will have little impact on the new bottlenecks in the design process.

In any event, I spent a few hours working to realize a proposal I had made back in 2023 to automatically create diagrams to visually illustrate the logical flow of a given mathematical paper. At the time, Freddie Manners, extrapolating from the half-decent capability of the then-newly released ChatGPT 3.5 at this task, presciently predicted that “by the time a dedicated tool had been completed, the next general purpose engine would be better than it”.

With that in mind, I decided to focus not on the generation of the diagram – which now can be done at various levels of quality by any number of large language models – but on its presentation. The result is the following app, which can take a certain formatted JSON file of dependencies between theorem objects and produce an interactive graph which can be explored, edited and also exported (somewhat lossily) into other standard formats such as SVG, TikZ, quiver, or Mermaid. Here is a screenshot of a diagramming of the celebrated proof by Wang and Zahl of the three-dimensional Kakeya conjecture:

Using an LLM, I generated diagrams for eight papers for demonstration purposes, including for instance a diagram for Wiles’s proof of Fermat’s last theorem, or of Szemeredi’s proof of his famous theorem on arithmetic progressions (which sports a notoriously convoluted such diagram in the original paper), as well as a few papers of my own. If there are other requests to diagram particular papers, I can try to use an LLM to generate more examples; but my intention is for users of the app to create their own such diagrams, either by manually constructing them, or by directing their own AI tools to build the diagram in the required format (which is a JSON, with the precise specification given here).

I mentioned in the previous post that for these sorts of visualization apps, which work deterministically for a given set of inputs, the downside risk of LLM use to build the app is acceptably low. For this particular app, there is a complicating factor, which is that while the app does remain deterministic, the data I used to populate the app – namely, the above diagrams – are also LLM-generated. I have done spot checks comparing the diagrams against the source papers, and did not find any errors; however, they are not guaranteed to be 100% accurate, and should only be used as approximations to the logical structure of these papers rather than completely exact representations. (The latter might become deterministically extractable should the results of these papers become formalized in a proof assistant language, but this is not currently the case.) Still, I hope these sorts of diagrams can serve as a helpful initial guide when first trying to read and understand a complex paper.

With the advent of modern coding agents, many visualization projects that I had proposed in the past, but dropped due to the time and complexity of the coding portion of the task, have now become relatively feasible, in that a reasonable quality prototype (suitable for non-mission-critical tasks such as providing secondary visual aids, where it is not absolutely necessary that the product is 100% bug-free) can now be generated in a matter of hours using such tools. I don’t immediately plan to work on my entire backlog of such projects, but I did spend a few hours this weekend on one such project, namely the proposal from this 2016 blog post to visualize random variables as animated quantities, which can be either viewed numerically or displayed as a scatterplot. After some back and forth with the coding agent, I was able to come up with a working app. Here is a screenshot of the app displaying a visualization of Berkson’s paradox, which asserts that independent variables can become correlated to each other after applying a conditioning:

The app in fact is a “compiler” for a small, custom programming language (think of a simplified hybrid of Python and Excel) which allows for the introduction of random variables, manipulates them through operations such as arithmetic operations or conditioning, and then plots them as animations or as text. (The screenshot above is static, but when the app is live, it will update at the indicated speed.)

As always, I would be happy to receive feedback on the app, which I hope can be useful as a visual aid to understand basic probabilistic concepts such as independence or conditioning.

I have been interested in machine-assisted ways to do and teach mathematics from as far back as 1999, when I started coding several applets in Java 1.0, both for my complex analysis and linear algebra courses, as well as to visualize various mathematical objects I was interested in (such as honeycombs or Besicovitch sets). This was moderately successful; but the applets were time-consuming to program. Eventually, the standards for web pages stopped supporting this version of Java, and the applets became non-functional.

However, in the last few days I have begun the process of migrating much of my old web page and blog data to a more maintainable repository, using modern AI assistance. As an experiment, I asked the agent to port my old applets to a modern supported language (we landed on Javascript), and it managed to do so in a matter of hours, with all of my old applets now functional again, with even a few graphical upgrades (for instance, the Besicovitch set applet is now colorized, in contrast to my original monochrome version). I am particularly pleased to see the honeycomb applet that I wrote with Allen Knutson in 1999 come back to life, as this was a particularly tricky one to code by hand:

Notoriously, LLM-based coding agents can create various blatant or subtle bugs in their code; but in the porting of these two dozen or so applets, I could only find one minor bug (the handling of a drag event in one of the complex analysis applets had unwanted behavior when dragging outside of the main box), and in fact the agent identified two bugs in the original code that I was not aware of, so it ended up being a net wash as far as code quality was concerned. In any event, as these applets are meant to be secondary visual aids rather than critical components of a mathematical argument, the downside risk of such bugs is relatively low.

The process was painless enough that I decided to also try coding some new apps, in addition to porting the old ones. Back in 1999 I had an ambitious idea for a visualization tool for special relativity; this was before the release of the software tool Inkscape, but the idea I had in mind was basically “Inkscape, but in Minkowski space”. I had even started writing Java code for this app, but the code complexity became too much for me, and I abandoned the project. However, after a couple hours of “vibe coding” with an AI agent, I was finally able to generate an applet that matched the vision I had back in 1999, which can now be found here. A summary of the conversation I had with the agent to generate this code can be found here (it has been edited down to remove a large number of tedious technical implementation reports). While I have playtested the app somewhat, I would be interested in receiving further feedback on this “alpha” version of the applet, as I am sure (especially given the LLM-generated nature of the code) that there are still some bugs and rough edges to be ironed out.

After writing my blog post on the Gilbreath conjecture paper earlier today, I realized that I could similarly ask the agent to code a visualization tool for the Gilbreath conjecture to accompany the paper and blog post. After another few hours of conversation, this is now done; you can try out the visualization here. Again, the procedure was quite painless (see this transcript of the process), and I think I may add such interactive visualizations as supplements for future papers; as such supplements are not mission-critical to the core of the paper, I again feel that the downside risk of using guided interaction with LLM agents to generate such visualizations is acceptable.

Zachary Chase, Zach Hunter and I have uploaded to the arXiv our preprint Gilbreath’s conjecture: a Cramér random model and a deterministic analysis. This paper is motivated by a notorious conjecture of Gilbreath (also proposed eighty years prior by Proth), which one can state as follows: if one starts with the sequence of primes and repeatedly takes absolute differences of consecutive terms, then the first term of each subsequent row is always :

Coming from a PDE background, I like to think of this conjecture as a (discrete) nonlinear “wave equation” problem, where the primes are the “initial data”, the downward direction in the above pyramid is the arrow of “time”, and the “equation of motion” is that the value of the “scalar field” at any given point in “spacetime” is the absolute difference of the values of the two points directly above it. We will informally refer to solutions to such an “equation” as “Gilbreath arrays”.

Numerically, the conjecture has been verified for the first rows by Odlyzko. Asymptotically, the conjecture can be heuristically justified as follows. Firstly, because all primes other than are odd, it is easy to see that the first term of each row is odd, while all other terms are even. Next, if one starts with the first primes for some large and takes initial differences, then the prime number theorem tells us that the average size of the next row is about , and Cramér’s conjecture predicts that the maximum size should be . With each new row, the maximum size can only decrease (since for any natural numbers ), and so one would expect it likely on each row that the maximum size should drop by at least (unless it has already reached ). Since there are rows to go before one reaches the end, it seems extremely likely that the maximum size should drop down to at most by then, at which point the result is forced from parity reasons.

However, it seems well beyond current technology to try to make these heuristics rigorous; even the first step of proving Cramér’s conjecture is far out of reach. In our paper, we consider two more feasible directions:

• What is a realistic probabilistic model of the primes, and can one confirm the (asymptotic version of the) conjecture almost surely for such a model?
• Can one use deterministic arguments to reduce the (asymptotic) Gilbreath conjecture to more tractable looking (and heuristically plausible) statements about iterated differences of primes?

Let us first discuss the question of analyzing probabilistic models. One can strip away the first row and initialize using prime gaps rather than primes; it is convenient to also strip away the aforementioned parity structure, by eliminating the initial gap , and dividing all remaining gaps by , so that one now works with an initial sequence with no parity bias. The conjecture is now equivalent to the first row always being -valued:

The Cramér model suggests that the first normalized prime gaps should behave like geometric random variables of mean about . My co-author, Zachary Chase, established an analogue of the Gilbreath conjecture for a more slowly growing model. Here is a special case of his main theorem:

Theorem 1 Suppose the initial row entries of a Gilbreath array are drawn independently from a uniform distribution on for some . Then almost surely, all but finitely many of the rows have a -valued first entry.

The Cramér model morally corresponds to a value of comparable to , which is too large for the above theorem to apply. However, we were able to improve the argument, basically allowing to be anything of size . Furthermore, it was not necessary that the distribution be uniform: the important hypothesis was that the distribution not be concentrated in any -separated set, such as the even numbers, the odd numbers, or the multiples of . (See the paper for the precise formulation of “non-concentrated”.) Such a hypothesis is needed since if for instance all initial entries were divisible by , then this property would propagate down the array, and it would become extremely unlikely that the initial values would remain -valued. Our hypotheses are obeyed by the Cramér random model, and so we obtain a heuristic confirmation of the original Gilbreath conjecture for the primes.

One can informally explain our proof of the above result as follows. We consider the portion of the array generated by the first values for some large . Suppose that at some point deep in this portion of the array, a value that is larger than is attained. Then the two values above must satisfy the equation . So, either one of these values is at least , or one of them is and the other is . If one iterates this observation, one sees that is the base of an upside-down triangle of values, topped off by at least one location where the value is at least . If one iterates that observation in turn, we see that forms the base of a “tower” of upside-down triangles stacked atop each other, with the number of such triangles bounded by the maximum size of the initial data (in the “backwards light cone” of ). In the regime, it turns out that the number (or “entropy”) of such towers is subexponential in . So if we can show that each tower only can be created with an exponentially small probability, we can conclude by the standard techniques of the union bound and the Borel–Cantelli lemma.

At this point we use the following elementary observation. Suppose that some finite Gilbreath array coming from say initial data has been generated, and consider the effect of adding a new value to the initial data, which then triggers iterations of the absolute value difference operation for various values of until one reaches the new bottom vertex of the array. This difference operation has the property that the preimage of any -separated set is still -separated. Iterating this, we see that the set of values that make iterate to a -valued bottom vertex is also -separated. So as long as the distribution of avoids -separated sets, one can iterate this observation in to show that it is exponentially rare that large triangles of -valued vertices can be created.

We also consider an asymptotic continuous random model, in which the initial data are not natural numbers, but instead independently random non-negative real numbers with an exponential distribution, which we can normalize to have mean ; this heuristically is an approximate model for the Gilbreath array generated by the first normalized prime gaps, after dividing by the mean . In this normalized model, each entry of the row ends up having the same mean . The first few values of can be computed explicitly

However, the asymptotic behavior of remains unclear to us. We were able to show an inequality for any , indicating that cannot decay faster than , but we do not know whether this is the true decay rate. In any case a decay rate of (which is very weakly supported by numerical evidence) is consistent with the Gilbreath conjecture, as it would indicate that the Gilbreath array from the first prime gaps should end up being almost entirely -valued by merely steps, well before the steps needed to reach the bottom of the array.

Now we turn to deterministic analysis of Gilbreath arrays. Suppose we found some initial data that did not grow too quickly (e.g., one had a Cramér-type bound ), but still iterated to a final value that was not . What features of the initial data could generate such a failure of a Gilbreath-type conjecture? One way in which the conjecture could fail is if the Gilbreath iteration somehow produced a reasonably long consecutive string of zeroes (say, longer than ), as then the next few iterations would not act to decrease the magnitude of the non-zero entries bordering this string of zeroes. Such a scenario would be heuristically rate, as the parity of each element of the array can be worked out explicitly using the parity identity , and so constant-parity sequences of length say should be almost surely non-existent asymptotically by standard probabilistic heuristics.

Another bad scenario is if the Gilbreath iteration, after some medium number of iterations, produced an extremely long consecutive block (say of length ) which was entirely -valued for some . This block would then persist as a -block for a large number of iterations (equal to the length of the block), thus potentially delaying for a significant time the drop-down of the maximal value to below . For odd , one can use the parity analysis alluded to earlier to argue that the formation of such a block is extremely unlikely; but for even , we can only use such heuristics if we make strong assumptions of joint independence, as we did in the probabilistic analysis in our paper.

In any event, we were able to use purely elementary methods to establish an “inverse theorem” that states, roughly speaking, that the above two scenarios are the only ways in which a Gilbreath array can fail to have a -valued first entry. This basically arises from a more careful analysis of the towers of triangles alluded to earlier. (A previous argument involved considering ways to pack a large triangle by smaller triangles, leading to a MathOverflow question which was nicely answered by Fedja Nazarov and Anders Martinsson, but we later managed to optimize the argument to the point where the answer to this packing question was no longer needed.) So this in principle reduces the (deterministic) Gilbreath conjecture to several more tractable-looking (though complicated to state) assertions, though proving those latter statements seems well out of reach at the moment.

Suppose that one has a set of points in the plane, which we will think of as the complex plane . Let denote the number of unit distances determined by these points, i.e., pairs of points whose displacement obeys the equation

(It makes little difference for the asymptotics, but we will count the pair separately from here.)

The Erdös unit distance problem asks, for a given large number , what is the largest possible value of amongst all sets of cardinality ?

For instance, if one takes to be equally spaced collinear points with unit spacing, one can obtain a linear construction with . Erdös observed that one can improve this construction asymptotically:

Theorem 1 (Erdös construction) There exists point sets of arbitrarily large cardinality such that for some absolute constant .

In fact, in the construction one could take arbitrarily close to . Erdös famously asked whether had to be bounded above by ; and for decades there was significant effort expended on upper bounding , with the best known upper bound being , established by by Spencer, Szemerédi, and Trotter in 1984. We will note here that it seems extremely difficult to improve this upper bound. One reason for this is that if one replaces the equation (1) with the superficially similar equation

(i.e., replace the unit circle by a standard parabola), then the bound is best possible, as can be seen by taking to be a rectangle in the Gaussian integers of width and height . Hence any improvement of the bound would have to exploit some special property of the unit circle that is not shared by the parabola.

It came as some surprise recently when a team from OpenAI resolved the question of Erdös:

Theorem 2 (OpenAI construction) There exists point sets of arbitrarily large cardinality such that for some absolute constant .

The optimal value of is still unknown, but the best upper and lower bounds on are tracked at this page; currently we know that .

The construction in Theorem 2 is a heavily modified version of that in Theorem 1, and uses some non-trivial amount of algebraic number theory, in particular the device of Golod–Shafarevich towers of field extensions. However, it was later observed using the Mythos AI that one could get a weaker bound with less algebraic number theory, which after optimizing parameters yields the following intermediate result between Theorem 1 and Theorem 2:

Theorem 3 (Mythos construction) There exists point sets of arbitrarily large cardinality such that for some absolute constant .

Furthermore, by inserting Golod–Shafarevich towers back into the Mythos construction, one can recover the full strength of Theorem 2.

These results already have a number of expositions; see for instance this article of Alon et al., or this blog post of Bloom. As an exercise for myself, I recently spent some time trying to “digest” these constructions and place them on a common footing, with an emphasis on trying to find the minimal route to either heuristically or rigorously recovering these results relying on as little algebraic number theory as possible. The post here is a writeup of this exercise. (Disclosure: AI tools were useful for providing initial summaries of these arguments, as well as on explaining various fundamentals of algebraic number theory to me.)

The first (trivial) observation is that one can use rescaling to replace the unit distance by any other fixed distance. In particular, for any positive real , if we let denote the number of pairs whose displacement obeys the equation

then it is clear that any construction of a point set with a given value of can be rescaled to another point set of the same cardinality with the corresponding value of . It turns out to be convenient to work with values of that are asymptotically large, for instance the product of several large primes.

All the constructions of good point sets basically involve taking all the elements of a certain ring of algebraic integers up to some height. In the original construction of Erdös, was chosen to be the ring of Gaussian integers , but in fact any ring of integers in a non-trivial bounded degree field extension of would suffice to recover Theorem 1 (though always with the constant not exceeding ). To go beyond this, one has to start considering number fields of unbounded degree. As it turns out, the field extensions arising from Golod–Shafarevich towers are the most efficient for this purpose, and lead to Theorem 2; but one can work with the more elementary construction of number fields generated by many square roots of medium-sized primes, and this suffices for the intermediate result in Theorem 3.

The numerology can be explained as follows. Take to be a ring of integers in some number field of degree , and suppose for sake of argument that is the product of (rational) primes , which for simplicity we will assume to all have comparable magnitude, thus for some and all . Thus, is roughly of the size of . In practice one wants to impose some additional “splitting” conditions on these primes , but the prime number theorem, as well as variants such as the Chebotarev density theorem, suggest that we should be able to keep reasonably close to in size; for instance, if we select primes greedily then we can have . In particular we expect to have in practice.

By construction, splits into the product of rational primes. Moving up to the degree extension, one can optimistically hope that splits further into the product of primes in . Using conjugation symmetry, these primes might split into conjugate pairs . By selecting one element from each pair and multiplying, this generates solutions to (3) in . These solutions will of course have complex magnitude ; one can optimistically hope that they in fact have “height” in some sense.

To take advantage of this, take to be the set of points in of height . As has rank , we therefore expect the size of this set to be roughly

(For this heuristic discussion I will be deliberately vague about what the symbol means.) Meanwhile, using our solutions to (3), we expect to have

But this can be clarified by the heuristic (4). Taking logarithms, we expect to have

In the regime where the degree of the number field is held fixed, we thus expect to exhibit logarithmic type growth in , and on inserting this back into (5) we (heuristically) recover Theorem 1 (with the natural constant ). In fact it is not hard to turn the above heuristics into a rigorous argument, by setting equal the Gaussian integers and selecting all the primes to be , so that they split completely in by the Fermat two-square theorem.

But if one can permit the degree to grow in the construction, and in particular be superpolynomial in , then the above heuristics suggest that we can start improving upon Theorem 1 , and even get all the way to Theorem 2 if we can make the degree go to infinity while keeping the number of primes fixed.

If one naively tries this approach by forcing all the primes to split completely in a very high degree number field, one runs into significant technical difficulties, not least of which is the need to obtain good error terms in the Chebotarev density theorem, which touches upon such difficult questions as the Generalized Riemann Hypothesis and the existence of Siegel zeroes. From an algebraic number theory perspective, this is related to the breakdown of unique factorization in such number fields, as measured by the class group. The size of this group is in turn controlled by the discriminant of the field, as per the fundamental theorem of Minkowski in this subject.

But one can hope that the arguments are robust enough to tolerate a little bit of breakdown in unique factorization, so long as the class group is not too large. The most natural way to do this is to use all the standard machinery of algebraic number theory, such as the unique factorization of ideals. But there turns out to be a more elementary (though largely equivalent) approach, which is to weaken the target equation (3) to a congruence equation

This condition does not pin down the value of completely, but so long as we can keep the height of not too much larger than , it does restrict to a sufficiently small set of possible values that a simple application of the pigeonhole principle can allow one to conclude.

In order for this strategy to work well, one needs to locate high-degree number fields of controlled discriminant for which it is relatively easy to at least partially split one’s rational primes into ideals in this field . It turns out that requiring the field to have a tower structure and admit complex multiplication (which basically amounts to it including ) is already sufficient to get a satisfactory amount of splitting. To control discriminants, the most efficient choices are the Golod–Shafarevich towers, for which the (root) discriminant stays bounded; but a more naive choice of a tower of quadratic extensions also gives reasonable control on discriminants and is sufficient to establish Theorem 3.

The three constructions thus sit on a continuum, with the key differences being the selection of the key parameters (the number of primes multiplied together) and (the degree). The Erdös construction keeps the degree fixed and sends the number of primes to infinity. The OpenAI construction does the opposite, keeping the set of primes fixed but sending the degree to infinity. The Mythos construction is a compromise, in which the degree and the number of primes both go to infinity in a coupled fashion. In particular, one could easily imagine an alternate timeline of events in which the Mythos construction was the first to be discovered (by either humans or AI) after the Erdos construction as a reasonably natural modification of the latter, and then subsequently refined (again either by humans or AI) to the OpenAI construction once the significance of Golod–Shafarevich towers was realized.

In this recent paper of Pohoata, the terms “horizontal amplification” and “vertical amplification” were proposed for the technique of constructing large configurations by increasing and , thus the Erdös construction becomes a paradigm for horizontal amplification while the OpenAI construction becomes a paradigm for vertical amplification (and the Mythos construction utilizes both types of amplification). See also this paper of Bloom-Sawin-Schildkraut-Zhelezov for another recent application of vertical amplification.

— 1. Some more details —

Here we sketch how the quadratic extension approach can recover Theorem 3.

As indicated above, we will work with a product of (rational) primes of size . Our only requirements of these primes, beyond their size, will be that they are distinct and equal to mod , so that they split in the Gaussian integers. By the prime number theorem in arithmetic progressions, this allows us to take as small as , so in particular .

To construct , we start by taking distinct medium-sized (rational) primes of size for some medium-sized parameter (eventually, when we optimize parameters, we will take to be a small multiple of ). We will not need any further properties of these primes, so by the prime number theorem we can take as small as , so in particular . We will take to be larger than , so that the primes are distinct from the primes .

We will work in the number field
generated by and real square roots . For instance, if and , a typical element in this field would take the form

In this particular example, the ring of integers would consist of those elements (8) for which are rational integers. In general, the ring of integers can be slightly larger than this, but for our purposes we can just work with the “naive” ring of integers generated by and . A typical element of this ring then looks like

where are rational integers and we adopt the convention that . Let us define the (naive) height of such a ring element to be . Then the number of elements of of height at most is as long as is sufficiently large (again I will be vague about what means here).

Our main goal is to find a large number of solutions in to the congruence equation (7). As per the usual Minkowski embedding based on the various ways to embed into or , it is convenient to think of as a lattice in a -dimensional vector space, which (due to the presence of , which excludes purely real embeddings) is naturally thought of as the product of copies of . For instance, in the running example, one can identify a ring element with an element

of , and with this embedding becomes a lattice in . In general, the embedding of into is essentially the Walsh–Fourier transform, weighted by the various square roots of . (The appearance of the Walsh-Fourier transform reflects the fact that the specific number field we are working with is an abelian Galois extension with Galois group .) Because of this, one can readily compute the covolume of the lattice, which up to lower order terms is basically , which with our construction can be crudely bounded by . As long as we keep small compared to , this covolume will be small compared to and will end up being a lower order term.

The reason we care about the covolume (whichis essentially the square root of the discriminant) is because of Minkowski’s theorem, which we will use in this crude form: Minkowski’s theorem: any lattice in a -dimensional space of covolume will contain a non-zero lattice vector of length . Thus for instance will contain some vector of length , and any sublattice of of index will contain a vector of length , and thus also of height by inverting the Walsh-Fourier transform.

Anyway, suppose that we can find some sublattice (in fact, they will be ideals, but we will not need this) of for which we can obtain the inclusion

that is to say one has for all . Then clearly any element of will obey the congruence (7). An obvious choice of would be , but this is way too sparse: this lattice has index in , so the shortest vector one can hope to locate in it will have height , which is too large for our purposes. Instead, we would like to have index (which is the smallest it can be while still yielding the inclusion (9)); then will contain a non-zero element of height , which means that is equal to times an element of of height . The number of such elements is , which will end up being a lower order term that we can easily pigeonhole away.

We claim that we can find at least different sublattices of index that obey (9). To verify this claim, it is a straightforward matter to use the Chinese remainder theorem to work “prime by prime”. Indeed, it suffices to show for each of the primes dividing , that there are different sublattices of of index that obey the inclusion

We can descend now to the finite ring , which is a -dimensional vector space over the finite field . The complex conjugation operation descends to an involution on this vector space, and we are looking for subspaces of dimension with the property that

So now we just need to understand the structure of . The general Wedderburn–Artin theorem tells us that this ring is the product of finite fields, but we can be much more explicit here in this specific situation. Exactly as Minkowski embedding maps rings in number fields into product of copies of and associated to the real and complex embeddings of the ring, we can also embed into a product of copies of and , depending on how we assign square roots to or in or the quadratic extension . We can illustrate this with the running example. As mod , the square roots of in stay in . Suppose first that also splits into square roots in . Then we can embed into by mapping

By counting elements we see that this embedding is in fact an isomorphism. If instead does not split, so that now lie in , then we can embed into by mapping

thus we drop half of the previous embeddings as being conjugate to the half that we retain. Again, counting shows that this is an isomorphism.

In general, one can show that is isomorphic to either (if all the split in ) or (if at least one of the does not split). Furthermore, in the former case the copies of organize into conjugate pairs (corresponding to flipping to ), and in the latter case the copies of organize into conjugate pairs. By selecting one element from each conjugate pair, and taking to be the joint kernel of such elements, we can generate either or different subspaces of dimension with the desired property that . By the aforementioned Chinese remainder argument, this gives the claimed lattices of index .

Invoking Minkowski’s theorem, this now generates non-zero vectors of height obeying (7). There is a technical issue that some of these vectors could conceivably collide with each other; however there is a further Chinese remainder theorem argument (which I will omit here) that shows that any such can belong to at most such lattices. So the number of distinct generated by this argument is at least , and by pigeonholing one can now also get at least at least solutions to the equation

of height for some of height . As long as we select

then the type factors can be neglected, and we approximately have

up to lower order terms. If we choose to be a small multiple of , then we soon calculate that , and we recover Theorem 3.

I am happy to announce the third SAIR challenge, which is focused on obtaining numerical data for the infamous inverse Galois problem. This is a collaborative project with the L-functions and modular forms database (LMFDB), and is organized by John Jones, Jen Paulhus, David Roe, Andrew Sutherland, and myself. The challenge is somewhat similar to my own Equational Theories Project, in that one is trying to complete a large mathematical data set in a verified fashion, except that the target data set had an existing mathematical interest. Also, the verification will be done by MAGMA (as well as PARI/GP) rather than Lean.

Let me first quickly review the inverse Galois problem. Suppose one has an irreducible polynomial of one variable of some degree and integer coefficients; take for instance . Then will have distinct roots ; in this case the roots happen to be

The roots generate a splitting field over the rational numbers . Any automorphism of this splitting field must permute the roots , and thus generates a subgroup of the permutation group (defined up to relabeling of the roots), which we call the Galois group of . This is some subgroup of that acts transitively on the roots (because each root generates the field). Typically, it is all of ; but occasionally it is smaller. For example, the particular cubic polynomial above has the special property that each root individually generates the entire field , thanks to the identities

Because of this, the Galois group of is the cyclic group (or equivalently, the alternating group ), rather than the full symmetric group . (This is in contrast to, say, , whose roots , , cannot be expressed as rational polynomials of each other, and whose Galois group is all of .) In fact, in the cubic case, it turns out that the Galois group is when the discriminant is a perfect square, and otherwise.

More generally, we have

Problem 1 (Inverse Galois Problem) Let be a transitive permutation group on letters. Can be realized as the Galois group of some degree irreducible polynomial with integer coefficients (after identifying the roots of suitably with the letters)?

The answer to this problem is known to be positive for , with the single possible exception of the sporadic Mathieu group : there are transitive permutation groups on letters (cf. OEIS A002106), and for of them, a polynomial has been located with that Galois group; see this database of Klüners and Malle. The problem of locating a polynomial with Galois group is a notorious open problem, though this is likely to be quite a difficult problem, and not the objective of the SAIR challenge.

Instead, we will focus on “breadth” rather than “depth”, in order to leverage the power of crowdsourcing and modern AI technologies. It turns out that there are distinct transitive permutation groups on letters, which are conventionally labeled from (the cyclic group ) to (the permutation group ). The first stage of the challenge will be:

Problem 2 (First stage of SAIR challenge) For as many of the groups , , locate an integer polynomial with that Galois group (up to isomorphism). (Also of interest is to specify the number of real roots, and to keep the discriminant low; more on this later.)

The verification side of this problem is essentially solved: the MAGMA computer algebra system can take any candidate polynomial and locate its Galois group within seconds. The MAGMA team has kindly granted SAIR a limited license to provide an API for contestants to calculate a certain number of Galois groups per day without needing to purchase their own license, though of course they are free to use their other computational tools to also perform these calculations outside of the competition.

The LMFDB already has polynomials for 286 of the 25000 groups, so there is plenty of remaining polynomials to claim in the challenge.

For applications, it is of interest to track some other statistics of a polynomial besides its Galois group. One of these is the number of real roots, which is a number between and of the same parity as (and which has to be achievable as the number of fixed points of one of the permutations in the Galois group, namely the one corresponding to complex conjugation); in particular, this number must be even in the degree case. Combining the label of the Galois group with the number of roots turns out to generate pairs in degree , and the challenge is actually to attach polynomials to as many of these pairs as possible. (The LMFDB has already done so for just of these.)

Of course, there are infinitely many polynomials of degree , and any Galois group that is representable by one polynomial, will be representable by infinitely many others (e.g., one could simply translate the polynomial by an arbitrary integer shift). To avoid creating an unusable database filled with uninteresting polynomials, we will prioritize polynomials whose (absolute) discriminant is as small as possible. (There are some technical details as to how this discriminant is defined and computed; see this page for details). The way we have set things up, each pair will come with a leaderboard for the polynomials with the smallest discriminants that have been located so far by contestants, removing duplicates arising from trivial operations such as translating the polynomial. Contestant team will be awarded a score between and for each submitted polynomial based on how small their discriminant is compared to the best known discriminant, and how many other teams were also able to find a polynomial with that pair. Thus, pairs that are extremely easy to generate (such as those associated to the full permutation group ) will be worth only a negligible score (as every contestant will be able to submit a polynomial for that pair), while pairs which are difficult to locate a polynomial for will be worth more points.

For this competition, the unrestricted use of any sort of computational tool, including AI, to locate the polynomials, are expressly permitted; this first stage of the competition is a “black box” challenge where we are not directly interested in obtaining insights as to how the polynomials are located, but the sole objective is to resolve as much of the inverse Galois challenge as possible. As such, the notorious uninterpretability of modern AI is not a concern for this stage. However, we will encourage contestants to share techniques with each other in order to cover more ground, through the Zulip channel for this challenge.

This first stage of the competition will close on August 15. After this, we will launch a second stage (with details to be determined) to focus on some set of candidate Galois groups that could not be resolved by the first stage. Here we envisage a more collaborative, conceptual, and human-driven effort in which the role of AI tools may be more secondary, and with more of a focus on creating mathematically interesting results rather than simply trying to saturate a given benchmark. Stay tuned for more details!

The United States Office of Management and Budget (OMB) has proposed a vast and radical set of rule changes to how federal grants from all funding agencies are administered.  (A summary of the key changes, by a former Senior Program Officer at the National Institutes for Health, can be found here.) This is no mere tinkering at the edges of existing policy; many basic principles, such as the central role of peer review in grant-making decisions, are seriously compromised by the proposed rules, while the administrative burden of complying with grant rules are significantly increased, and hamstring the ability of funded scientists to react to new developments and forge new collaborations.

There is much to discuss in these proposals; see for instance this post by Karen Saxe (vice president for Government Relations at the American Mathematical Society), this news item on the response from the astronomy community, this op-ed from Ars Technica, this article from the New York Times, this article from Science, or this story from CNN. I will focus here on just one of the impacts, regarding the need to maintain agility and flexibility in a competitive and rapidly changing environment.  

Some types of research, particularly those closest to industrial or other real-world applications, can be planned in a predictable fashion, in which the timelines for hitting key milestones are clear, and schedules for events can be planned years in advance.  However, basic research — of which pure mathematics is a quintessential example — expects (almost by definition) to discover previously unknown directions and connections that cannot be predicted perfectly at the time a research project is proposed.  Many of the most striking breakthroughs in such subjects come from uncovering such expected developments and rapidly capitalizing on them – for instance, by quickly organizing seminars, workshops, or conferences on a suddenly “hot” topic.  

To give just one example of this sort of serendipitous discovery, a significant portion of the foundational theory of compressed sensing was initiated from a chance meeting in 2004 between myself, Emmanuel Candes (a statistician) and Justin Romberg (an electrical engineer) at a program at the Institute for Pure and Applied Mathematics (IPAM) on multiscale geometry.  This theory – has led to notable accelerations and other improvements to a range of technologies, from MRI scans to radio interferometry to electron microscopy.  The three of us, as well as the IPAM program we participated in, were all funded by grants from the National Science Foundation (NSF), but the extraordinarily fruitful collaboration was not fully anticipated in any of the proposals.  (Disclosure: I now serve as director of special projects at IPAM.)

This is the type of fortuitous interaction that would be severely impacted by the proposed rule changes. Consider for instance Section 200.432 of the Code of Federal Regulations, which concerns the use of grant funds to support conference costs:

A conference means an event whose primary purpose is to disseminate technical information beyond the recipient or subrecipient and is necessary and reasonable for successful performance under the Federal award. Allowable conference costs may include the rental of facilities, speakers’ fees, attendance fees, costs of meals and refreshments, local transportation, and other items incidental to such conferences unless further restricted by the terms and conditions of the Federal award. 

As just one of many significant rule changes proposed is the following addendum to the above text:

OMB proposes to expand § 200.432 to add a requirement that costs for attending conferences are allowable only if participation in the conference is expressly approved by the agency and included in the terms and conditions of the award. The revision would clarify that recipients are not authorized to attend conferences using Federal funds that do not serve to advance program outcomes. 

This rule change would limit conference activity support to pre-approved plans that followed the scheduled objectives in the original proposal, which is written some time before the research takes place.  However, it is the nature of novel research (particularly in fundamental sciences such as mathematics) to have serendipitous opportunities emerge that were not anticipated in the original grant proposal, such as an unexpected and exciting new connection between the problem one was initially studying, and another subfield of math or science that had previously been thought to be unrelated.  Being able to react quickly to such developments, either by attending or organizing an event around them, or by inviting key researchers to visit, is essential to keep up with such breakthroughs.  Requiring bureaucratic pre-approval in these circumstances would significantly hinder the ability for funded scientists to competitively take advantage of these opportunities.

An illustrative example would be the 2011 IPAM program on Navigating Chemical Compound Spaces. The premise sounded like a pie-in-the-sky idea: to develop computational tools to be able to somehow travel through the almost infinite space of all possible chemical compounds in the search for a compound we need — be it to create a novel drug, a better solar cell, or stronger glass. At the time, even with projected advances in computer power, accurate prediction of chemical properties of materials was seen as a distant dream. Simulating a simple protein for even a few milliseconds with existing methods would require weeks of time and an astronomical energy budget.  In addition to experts on computational mathematics and materials science, the program involved a group of people who worked in a then obscure subject called machine learning (whose practical applications at the time involved such feats as deciphering human-written zip codes).  Attending such a program might be regarded as out of scope for many material scientists. Yet the outcome of the program was the realization that machine learning methods could be used to learn and model the forces that govern electronic structure, molecular interactions, and ultimately determine chemical properties of materials through much faster and efficient computation. This idea was incredibly fruitful and literally changed the way electronic structure computations are done.  AlphaFold has become Nobel prize winning work, and AI is being used to discover new drugs. Now, 15 years later, scientists are building labs to literally navigate the chemical compound space, assisted by AI, a descendant of old machine-learning computations approaches.    

These examples also illustrate the time scales involved in fundamental research and in bringing it to the point where its application becomes an engineering endeavor.  Fundamental research means playing the long game, leveraging the richness and unpredictability of scientific discovery. It is not something a private company would fund, but it is the engine behind the continued technological transformation whose fruits we all enjoy.  It means taking risks, going in directions that are mere hunches and educated guesses, and going there only with the expectation to find new and surprising things. But it is necessary for technological progress.   

Importantly, it is unrealistic to expect that every conference attendance will result in a major and unexpected connection or breakthrough. At times, there is a slow accumulation of knowledge that suddenly produces unexpected results. It is important to understand that fundamental research operates on scales of years and decades.  The ultimate effect of attending a conference cannot always be known in advance, making the pre-approval process difficult to manage. This brings in a related point: the risk-averse nature of the proposed rules. We all know that making breakthroughs requires risk-taking; behind every successful project stand several that failed. Sometimes, communication of what failed is as useful (or more!) as communication of what succeeded, and this kind of information gets shared in informal settings at workshops and conferences. 

The willingness to take risks and move in unexpected directions has always been a particular strength of this country, both in science and elsewhere, as exemplified for instance by the Defence Advanced Research Projects Agency (DARPA)’s willingness to experiment with emerging technologies such as the internet, GPS systems, or high-energy lasers, long before they could be proven to be viable.  The additional regulatory burdens of these proposed rule changes would cripple this capability and set back the nation’s scientific competitiveness and leadership with the technologies of the future.  I encourage all stakeholders (whether individuals or organizations) to submit public comments on the proposal on the OMB site (the public comment period extends until July 13). You can also submit through the Stand Up for Science site.

(Thanks to Kevin Klowden and Dima Shylakhtenko for feedback on an initial version of this post.)

A couple months ago, Damek Davis and I launched the first mathematical challenge at the SAIR Foundation, aimed at “distilling” the ability to solve 22 million problems in universal algebra into a condensed form. Stage one of that challenge has now been completed, with several effective “cheat sheets” generated to guess the truth or falsity of these problems to reasonable accuracy; the leaderboard for that stage, with their winning cheatsheets can be found here. Stage two of that challenge, in which the competitors now have access to Python code as well as modest LLMs, and now need to generate Lean proofs or disproofs rather than just true-false answers, is currently underway.

With Alberto Alfarano, François Charton, Yongzheng Jia, Kristin Lauter, Cathy Li, and Emily Wenger, are launching a second challenge at SAIR, this time focused on seeing how efficiently neural networks can execute simple modular arithmetic operations. For this challenge we are focusing on the simple operation of modular multiplication: taking a prime modulus (up to about a thousand digits long) and two integers and between and , and computing the product . This is of course a solved problem using traditional computation, being a single line of code in any modern programming language. But it has been a fascinating toy problem in which to explore the basic capabilities of neural networks.

For instance, this problem has revealed the mysterious phenomenon of “grokking“. When one tries to train a neural network on this problem for small sizes of inputs , then initially one runs into the familiar problem of overfitting: the network learns to solve the problem for the training data too well, at the expense of performing well for held-out test data. However, if one continues training for sufficiently long periods of time, then the network can suddenly “grok” the problem and generalize surprisingly well to the test data. It appears that the neural network can suddenly “learn” powerful computational tricks, such as taking discrete logarithms, to find accurate and efficient ways to arrive at the correct answer.

This challenge is not about grokking, but instead about scaleability: we can create neural network models for modular multiplication that are extremely accurate for, say, 10-bit inputs, but they struggle at handling larger bit sizes. The competition is then simple: submit a neural network (with fixed weights) that can solve this task for larger input sizes with as high an accuracy as possible. Some pre-processing of the individual inputs , , is permitted (e.g., to convert these numbers into decimal or some other convenient representation), but other than that the main computation has to be neural in nature; one cannot simply run some Python code, for instance, to compute the multiplication. We are imposing limits on the size and allocated run time on the neural network, but otherwise we are deliberately being flexible in the architecture requirements, in order to encourage creative experimentation; in particular, we permit networks whose weights were arrived at by other means than the usual machine learning training process.

This is a relatively simple challenge to state, but we genuinely do not know what to expect from the competitor entries – is there a clever way to encode modular arithmetic for even quite large numbers into a medium size neural network, or is it going to be an exceptionally difficult task? Hopefully we will find out in a few months! Discussion of the ongoing challenge will take place on this Zulip.