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.

Theorem of the Day has 'new acquisitions': Theorems no. 247-248

Theorem of the Day has 'new acquisitions': Theorems no. 249-251

Theorem of the Day has 'new acquisitions': Theorem no. 228, 229, 230, by R.A. Fisher, Poncelet and Ore

Theorem of the Day has 'new acquisitions': Theorem no. 231, 232, 233, 234

Theorem of the Day has 'new acquisitions': Theorem no. 235, 236, 237

Theorem of the Day has 'new acquisitions': Theorem no. 238 and 239

Theorem of the Day has 'new acquisitions': Theorem no. 240 and 241

Theorem of the Day has a 'new acquisition': Theorem no. 242, the Polya-Redfield Enumeration Theorem

Theorem of the Day has 'new acquisitions': Theorems no. 243-246

Theorem of the Day has a 'new acquisitions': Theorem no. 227, Cauchy's Theorem in Group Theory

Theorem of the Day has a 'new acquisitions': Theorem no. 226, Wolstenholme's Theorem

Theorem of the Day has 'new acquisitions': Theorem no. 223, 224, 225, by Tutte, Green and Regiomontanus