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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The next recommendation is this.

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

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

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

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

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

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

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

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

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

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

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

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

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

The example can be stated completely explicitly: one can take

and one can verify by a brief calculation that

and

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

which all map to the same cubic polynomial

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

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

If we have a factorization

then the resultant can also be described as

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

and the first equation can be solved for ,

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

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

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

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

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

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

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

while the equation implies that

as well as the more refined estimate

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

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

So if we write more explicitly as , then we have

and thus

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

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

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

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

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

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

with

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

is exactly the map given in (1).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

— 1. Some more details —

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

More generally, we have

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

We are all having to keep revising upwards our assessments of the mathematical capabilities of large language models. I have just made a fairly large revision as a result of ChatGPT 5.5 Pro, to which I am fortunate to have been given access, producing a piece of PhD-level research in an hour or so, with no serious mathematical input from me.

The background is that, as has been widely reported, LLMs are now capable of solving research-level problems, and have managed to solve several of the Erdős problems listed on Thomas Bloom’s wonderful website. Initially it was possible to laugh this off: many of the “solutions” consisted in the LLM noticing that the problem had an answer sitting there in the literature already, or could be very easily deduced from known results. But little by little the laughter has become quieter. The message I am getting from what other mathematicians more involved in this enterprise have been saying is that LLMs have got to the point where if a problem has an easy argument that for one reason or another human mathematicians have missed (that reason sometimes, but not always, being that the problem has not received all that much attention), then there is a good chance that the LLMs will spot it. Conversely, for problems where one’s initial reaction is to be impressed that an LLM has come up with a clever argument, it often turns out on closer inspection that there are precedents for those arguments, so it is still just about possible to comfort oneself that LLMs are merely putting together existing knowledge rather than having truly original ideas. How much of a comfort that is I will not discuss here, other than to note that quite a lot of perfectly good human mathematics consists in putting together existing knowledge and proof techniques.

I decided to try something a little bit different. At least in combinatorics, there are quite a lot of papers that investigate some relatively new combinatorial parameter that leads naturally to several questions. Because of the sheer number of questions one can ask, the authors of such papers will not necessarily have the time to spend a week or two thinking about each one, so there is a decent probability that at least some of them will not be all that hard. This makes such papers very valuable as sources of problems for mathematicians who are doing research for the first time and who will be hugely encouraged by solving a problem that was officially open. Or rather, it used to make them valuable in that way, but it looks as though the bar has just been raised. It is no longer enough that somebody asks a problem: it needs to be hard enough for an LLM not to be able to solve it.

In any case, a little over a week ago I decided to see how ChatGPT 5.5 Pro would fare with a selection of problems asked by Mel Nathanson in a paper entitled Diversity, Equity and Inclusion for Problems in Additive Number Theory. Nathanson has a remarkable record of being interested in problems and theorems that have later become extremely fashionable, which has led him to write a series of extremely well timed and therefore highly influential textbooks. In this paper, he argues for the interest of several other problems, some of which I will now briefly describe.

If is a set of integers, then its sumset is defined to be . For a positive integer , the –fold sumset, denoted , is defined to be . Nathanson is interested in the possible sizes of given the size of . To that end one can define a set to be the set of all such that there exists a set with and .

An obvious first question to ask is simply “What is ?” When , the answer is the set of all integers between and . It is an easy exercise to show that if , then , so this result is saying that all sizes in between can be realized. However, it is not true in general that can take every size between its minimum and maximum possibilities, and we do not currently have a complete description of .

Another natural question one can ask, and this is where ChatGPT came in, is how large a diameter you need if you want a set with and having prescribed sizes. (Of course, the size of must belong to .) Nathanson showed that for every there is a subset of with and , and asked whether the bound could be improved. ChatGPT 5.5 Pro thought for 17 minutes and 5 seconds before providing a construction that yielded a quadratic upper bound, which is clearly best possible. It wrote up its argument in a slightly rambling LLM-ish style, so I asked if it could write the argument up as a LaTeX file in the style of a typical mathematical preprint. After two minutes and 23 seconds it gave me that, after which I spent some time convincing myself that the argument was correct.

The basic idea behind both Nathanson’s argument and ChatGPT’s was that in order to obtain a set of a given size with a sumset of a given size, it is useful to build it out of a Sidon set, which means a set with sumset of maximal size (that is not quite the usual definition but it is the simplest to use in this discussion), and an arithmetic progression. Also, for a bit of fine tuning one can take an additional point near the arithmetic progression. Then if one plays around with the various parameters, one finds that one can obtain sets of all the sizes one wants. Nathanson doesn’t express his argument this way (it is Theorem 5 of this paper), instead giving an inductive argument, but I think, without having checked too carefully, that if one unravels his argument, one finds that effectively that is what he ends up with, and the Sidon set in question consists of powers of 2. ChatGPT obtained its improvement by simply using a more efficient Sidon set — it is well known that one can find Sidon sets of quadratic diameter. (One might ask why Nathanson didn’t do that in the first place: I think it is because the obvious idea of using a more efficient Sidon set becomes obvious only after one has redescribed his inductive construction. Is that what ChatGPT did? It is very hard to say.)

Next, I asked ChatGPT to see whether it could do the same for a closely related question, where instead of looking at the size of the sumset, one looks at the size of the restricted sumset, which is defined to be . Unsurprisingly, it was able to do that with no trouble at all. I got it to write both results up in a single note, to avoid a certain amount of duplication. If you are curious, you can see the note here.

I then asked what it could do for general . I was much less optimistic that it would manage to do anything interesting, because the proof for makes fundamental use of the fact (due to Erdős and Szemerédi) that we know exactly which sizes we need to create. If we don’t know what the set is, then it seems that we are forced to start with a hypothetical set with and and build out of it a set of small diameter with the same property. As it happens, I still don’t know how to get round that difficulty (I’m mentioning that just to demonstrate that my mathematical input was zero, and I didn’t even do anything clever with the prompts), but Nathanson mentioned in his paper a remarkable paper of Isaac Rajagopal, a student at MIT, who must have got round the difficulty somehow, because he had managed to prove an exponential dependence of on for each fixed .

I’ll leave the previous paragraph there, but Isaac has subsequently explained to me that that isn’t really the difficulty. His argument gives a complete description of when is sufficiently large, and if one wants to prove a polynomial dependence for fixed , then assuming that is sufficiently large is clearly permitted. The real difficulty is that constructing the sets with given sumset sizes was significantly more complicated, and necessarily so because the degree of the polynomial grows with , and one therefore needs more and more parameters to define the sets.

In any case, the task faced by ChatGPT was not to solve the problem from scratch, but to see whether it was possible to tighten up Isaac Rajagopal’s argument. Here’s what happened.

1. After 16 minutes and 41 seconds, it came back with an argument that claimed to have improved the upper bound from exponential in to exponential in for any .
2. I asked it to write that in preprint form too, which took it a further 47 minutes and 39 seconds.
3. That preprint would have been hard for me to read, as that would have meant carefully reading Rajagopal’s paper first, but I sent it to Nathanson, who forwarded it to Rajagopal, who said he thought it looked correct.
4. Both ChatGPT and Rajagopal speculated a little on what might need to be done to push things further and get a polynomial bound, so I got greedy and asked ChatGPT to give that a go.
5. After 13 minutes and 33 seconds it told me it felt optimistic about the existence of such an argument but there were a couple of technical statements that needed checking.
6. I asked it to check them.
7. After 9 minutes and 12 seconds it got back to me with the check having been done, so I asked for this too to be written in preprint form.
8. After 31 minutes and 40 seconds the “preprint” was ready. Here it is.
9. Isaac Rajagopal looked at it and declared it to be almost certainly correct. It was clear that he meant this not just at a line-by-line level but at the level of ideas.

Isaac made some very interesting remarks about the nature of what the additional ideas were that ChatGPT contributed. Since, as I have already said, my mathematical input was zero, I invited him to write a guest section to this post. Just before we get to that, I want to raise a question (that will undoubtedly have been raised by others as well), which is simple: what should we do with this kind of content? Had the result been produced by a human mathematician, it would definitely have been publishable, so I think it would be wrong to describe it as AI slop. On the other hand, it seems pointless even to think about putting it in a journal, since it can be made freely available, and nobody needs “credit” for it (except that Isaac deserves plenty of credit for creating the framework on which ChatGPT could build). I understand that arXiv has a policy against accepting AI-written content, which makes good sense to me. So maybe there should be a different repository where AI-produced results can live. But various decisions would need to be made about how it was organized. I myself think that one would probably want to have some kind of moderation process, so that results would be included only if a human mathematician was prepared to certify that they were correct — or, better still, that they had been formalized by a proof assistant — and perhaps also that they answered a question that had been asked in a human-written paper. On the other hand, I wouldn’t want a moderation process that created vast amounts of work (unless the work was itself done by AI, but there are obvious dangers in going down that route). Anyway, until these questions are answered, this result is available from the link above, and perhaps, now that LLMs are so good at literature search, that will be enough to make it findable by anyone who wants to know whether Nathanson’s problem has been solved.

Isaac’s evaluation of what ChatGPT achieved

With just a few prompts, ChatGPT was able to improve the upper bound on (which I will define very soon) from exponential in to polynomial in . While its first improvement of the bound, from exponential in to exponential in , was a routine modification of my work, the improvement to polynomial in is quite impressive. To do this, ChatGPT came up with an idea which is original and clever. It is the sort of idea I would be very proud to come up with after a week or two of pondering, and it took ChatGPT less than an hour to find and prove, using similar methods to those in my own proof. My goal is to explain that idea, in a manner that will be digestible to my friends who are computer science majors as well as my math major friends.

The problem of bounding is closely related to a problem I worked on at the Duluth REU (Research Experience for Undergrads) program, of determining . In particular, is the set of possible -fold sumset sizes , where can be chosen to be any set of integers. is the minimal such that we can achieve all of the values of using -element sets . I spent last summer explicitly characterizing the set for large , by constructing sets such that achieves all sizes which I could not rule out as impossible. So, can be upper-bounded by optimizing my constructions.

I constructed these sets by combining smaller component sets which are simpler to analyze. Some of these components are the geometric series

for various values of and . Unfortunately, the elements of and are exponentially large in terms of . So, I asked ChatGPT (through Tim) whether there exist sets of elements which have similar sumset sizes to these geometric series, but contain only numbers of polynomial size in : I had no idea if this was possible, or how to begin constructing such sets. ChatGPT came back with an answer, constructing sets and which behave like “half a geometric series squeezed into a polynomial interval,” which is counterintuitive. Before I discuss the construction of and , I will explain the important properties of the sumset sizes of and which they recreate.

For , a set is called a set if the only solutions to

with in are the “trivial” solutions, by which I mean that one side of the equation is a reordering of the other side. If is a set of size , then elements of correspond exactly to choices of elements of , with repetition allowed. Using “stars and bars,” one can see that and this is the maximum possible value of among sets of size . So, another definition is that is a set if . Sidon sets, which Tim discussed, are exactly sets.

To make things more concrete, let us assume that in (1). Then, is a set, but it is not a set because of the relations

for any choice of in . In particular, , as these relations are the only ones preventing from being a set. lacks the relations in (2) because is not in . So, is a set, but it is not a set because of the relations

for any choices of in . This gives relations, and one can check that . To summarize, we have seen that

(a) is a set.

(b) is a linear function of .

(c) is a set.

(d) is a quadratic function of .

ChatGPT was able to find sets and of elements which satisfy (a)-(d), but whose elements all have polynomial size in . The construction of and uses -dissociated sets, which are sets where the only solutions to

with and in are the “trivial” solutions, i.e. and one side of the equation is a reordering of the other side. For , it is possible to construct an -dissociated set , where is approximately , and in particular polynomial in . Constructions of such a using finite fields date back to Singer (1938) and Bose–Chowla (1963) and are described in Appendix 1. Define

and

In hindsight, I have good intuition for the construction of and . All of the relations in (2) and (3) are formed by combining one or two relations of the form . There are approximately relations of the form in and , and approximately such relations in and . There are few other low-order relations in and , and similarly in and because is -dissociated. So, and manage to contain half as many -relations as their geometric series counterparts, while also containing few low-order relations.

We now see why (a)-(d) hold with and replaced by and , respectively. For concreteness, we assume that and , so contains no nontrivial relations as in (4) with . Then, is a set, but it is not a set because of the relations

for any choice of in . If we let , we can check that is linear in . In particular, (a) and (b) hold with replaced by , and the linear function replaced by . We can also see that is a set, but it is not a set because of the relations

for any in . If we let , we can check that is quadratic in . In a similar manner, (c) and (d) hold with replaced by , and the quadratic function replaced by .

Even though I can motivate it in retrospect, ChatGPT’s idea to use -dissociated sets to control relations of order at most feels quite ingenious. As far as I can tell, this idea is completely original.

ChatGPT’s proof that its construction produces the desired values of is very similar to my proof that the sets which I construct achieve all possible values of , after replacing and by and , respectively. Properties (a)-(d) capture many of the important properties of and (or and ) which are used in this proof. The final constructions involve combining the sets and (or and in my paper) for each value of between and with another set which is the union of an arithmetic progression and a point. Intuitively, and (or and ) have large sumsets, while arithmetic progressions have small sumsets, so it is plausible that one could get sets which achieve all the medium-sized sumsets by combining them. However, the proof of this is quite involved, and it occupies Section 4 of my paper and the entirety of the ChatGPT preprint. In Appendix 2, I work out the details of the ChatGPT construction to show that for sufficiently large,

For comparison, it is easy to see that is at least on the order of , and it is unknown what the real value is. In Appendix 3, I give details of the correspondence between my paper and the ChatGPT preprint, which will be helpful for those who want to read either.

Finally, I want to express my deep gratitude to Tim for allowing me to contribute to this blog. I am still stunned by the coincidence that the problem he chose to put into ChatGPT 5.5 Pro led him to my paper on the arXiv.

Tim on what this means for mathematical research

I would judge the level of the result that ChatGPT found in under two hours to be that of a perfectly reasonable chapter in a combinatorics PhD. It wouldn’t be considered an amazing result, since it leant very heavily on Isaac’s ideas, but it was definitely a non-trivial extension of those ideas, and for a PhD student to find that extension it would be necessary to invest quite a bit of time digesting Isaac’s paper, looking for places where it might not be optimal, familiarizing oneself with various algebraic techniques that he used, and so on.

It seems to me that training beginning PhD students to do research, which has always been hard (unless one is lucky enough, as I have often been, to have a student who just seems to get it and therefore doesn’t need in any sense to be trained), has just got harder, since one obvious way to help somebody get started is to give them a problem that looks as though it might be a relatively gentle one. If LLMs are at the point where they can solve “gentle problems”, then that is no longer an option. The lower bound for contributing to mathematics will now be to prove something that LLMs can’t prove, rather than simply to prove something that nobody has proved up to now and that at least somebody finds interesting.

I would qualify that statement in two ways though. First, there is the obvious point that a beginning PhD student has the option of using LLMs. So the task is potentially easier than proving something that LLMs can’t prove: it is proving something in collaboration with LLMs that LLMs cannot manage on their own. I have done quite a lot of such collaboration recently and found that LLMs have made useful contributions without (yet) having game-changing ideas.

A second point is that I don’t know how much of what I have said generalizes to other areas of mathematics. Combinatorics tends to be quite focused on problems: you start with a question and you reason back from the question or if you reason forwards you do so very much with the question in mind. In other areas there can be much more of an emphasis on forwards reasoning: you start with a circle of ideas and see where it leads. To do it successfully, you need to have some way of discriminating between interesting observations and uninteresting ones, and it isn’t obvious to me what LLMs would be like at that.

Of course, everything I am saying concerns LLMs as they are right now. But they are developing so fast that it seems almost certain that my comments will go out of date in a matter of months. It is also almost certain that these developments will have a profoundly disruptive effect on how we go about mathematical research, and especially on how we introduce newcomers to it. Somebody starting a PhD next academic year will be finishing it in 2029 at the earliest, and my guess is that by then what it means to undertake research in mathematics will have changed out of all recognition.

I sometimes get emails from people who are interested in doing mathematical research but are not sure whether that makes sense any more as an aspiration. I have a view on that question, but it may very well change in response to further developments. That view is that there is still a great deal of value in struggling with a mathematics problem, but that the era where you could enjoy the thrill of having your name forever associated with a particular theorem or definition may well be close to its end. So if your aim in doing mathematics is to achieve some kind of immortality, so to speak, then you should understand that that won’t necessarily be possible for much longer — not just for you, but for anybody. Here’s a thought experiment: suppose that a mathematician solved a major problem by having a long exchange with an LLM in which the mathematician played a useful guiding role but the LLM did all the technical work and had the main ideas. Would we regard that as a major achievement of the mathematician? I don’t think we would.

So what is the point of struggling with a difficult mathematics problem? One answer is that it can be very satisfying to solve a problem even if the answer is already known, but I don’t think that is a sufficient reason to spend several years of your life on this peculiar activity. A better answer is that by solving hard problems you get an insight into the problem-solving process itself, at least in your area of expertise, in a way that you simply don’t if all you do is read other people’s solutions. One consequence of this is that people who have themselves solved difficult problems are likely to be significantly better at using solving problems with the help of AI, just as very good coders are better at vibe coding than not such good coders, or people who have a solid grasp of how to do basic arithmetic are likely to be more skilled at using calculators (and especially at noticing when an answer feels off). Mathematics is a highly transferable skill, and that applies to research-level mathematics as well. By doing research in mathematics, you may not get the same rewards as your equivalents a generation ago, but there is a good chance that you will be equipping yourself very well for the world we are about to experience.

Appendix 1 (Isaac)

We will construct an -dissociated set , where is approximately . This construction is a very minor modification of Bose–Chowla (1963)’s construction of a set, which I learned about from this paper. For whatever reason, the GPT preprint (Lemma 3.1) uses a different, less efficient construction using moment curves.

Let be a prime, let , let be the finite field with elements and fix a generator of , so that is equal to . Define a set of elements

Then, each element corresponds to a unique value of , by taking . Now an additive relation of the form in (4) with can be reframed by taking powers of as

As is a degree- extension of and is a generator of as an -extension, this means that does not satisfy any nonzero polynomials in of degree . So, both sides of (6) are identical as polynomials in and thus the additive relation in (4) is trivial. So, is -dissociated, and of course one can prune a few elements to reduce to size .

Appendix 2 (Isaac)

Fix constants such that (in my paper I arbitrarily chose ). Let the two sets in (5) be called and . Let denote the set of integers satisfying . Similarly to my paper, the constructions of such that achieves the desired sizes will combine sets of the following four types:

• with choices of and .
• for each value of , with choices of .
• for each value of , with choices of .
• A set of the correct size so that .

One reason that this construction needs to be complicated is that we need to create at least many sets. To do this, we vary parameters and in the domain and parameters and in the domain . We can choose to be slightly bigger than , and then the above construction gives us different sets where can be made arbitrarily small. So, if we were to remove any of the above parameters from the construction, and not change the others, this construction would no longer create many sets. In comparison, Nathanson’s construction when only needs to create sets. He does this by combining a Sidon set, an arithmetic progression, and one extra value, and varying the size of the arithmetic progression and the extra value in ranges of size .

We want to combine sets , which are given by , for the values of , for the values of , and a set. By Appendix 1, for all , there exists a -dissociated set of diameter . By the constructions of and , we can take each , where . Let have basis vectors . To combine , we can define as

Similarly to my Lemma 4.9, this construction ensures that the generating function product holds, which is the identity that both my paper and the GPT preprint use (see either paper for a definition of these generating functions). By (the standard) Lemma 2.3 of the GPT preprint, is Freiman-isomorphic of order to a subset of . Therefore, for sufficiently large (the whole construction relies on this for the same reasons as in my paper),

Appendix 3 (Isaac)

In Section 4.2 of my paper, I use a different, simpler construction to construct sets achieving the values in which have , for some small . These sets are subsets of , meaning that all elements have polynomial size in . This is observed in Section 5 of the GPT preprint.

Section 4.3 of my paper carries out the construction which combines many components including and . This corresponds to Sections 2, 3, 4, and 6 of the GPT preprint. This section has a lot of moving parts; I give an outline in Section 4.3.1.

In Section 4.3.2, I describe how the different components will be combined, using a construction which I call the disjoint union, and introduce generating functions as a bookkeeping tool to keep track of the sumset sizes of a set . This corresponds to Section 2 and Section 4 of the GPT preprint.

In Section 4.3.3, I compute the generating function of each of the component sets, including (Lemma 4.15) and (Lemma 4.17). This corresponds to Section 3 and Section 6.1 of the GPT preprint. In particular, is computed in Lemma 3.3 and is computed in Lemma 3.4. Once these generating functions have been computed, the remainder of the proof is almost identical in my paper and in the GPT preprint.

In Section 4.3.4, I put all the pieces together to show that as we range over the sets which I have constructed, the values of will assume all of the elements of . The key idea is to show that the set of all values of forms an interval, and contains numbers both smaller than and equal to .

Boris Alexeev, Kevin Barreto, Yanyang Li, Jared Duker Lichtman, Liam Price, Jibran Iqbal Shah, Quanyu Tang, and I have just uploaded to the arXiv our paper Primitive sets and von Mangoldt chains: Erdős Problem #1196 and beyond. This paper (which is a work in progress) represents our efforts to digest and document the recent flurry of developments around the following problem of Erdős, Sárközy, and Szemerédi on primitive sets:

Conjecture 1 (Erdős problem #1196) Suppose that is a primitive set of integers, which means that no element of divides another. Then

as .

One can show that the upper bound of is best possible up to the error by taking to be the set of products of primes for some suitable parameter . This was one of the most well-known open problems in the study of primitive sets, and had attracted some number of partial results (for instance, Lichtman was able to show the upper bound of ). It was thus notable that this problem was first solved by an autonomous AI query (by the fifth author) a few weeks ago. This solution introduced a proof technique – based on Markov chains in the divisibility poset – which in retrospect is very natural for controlling primitive sets, but which had not been explicitly used in previous literature, though in retrospect many of the arguments in that literature involved a specific Markov chain which we call the downwards Mertens chain. The proof instead revolved around a different Markov chain, which we call the downwards von Mangoldt chain, which manages to neatly avoid the “” type losses in the previous Mertens-based arguments, and resolve Conjecture 1. In this paper we develop the Markov chain approach more systematically, and show that it settles several further conjectures concerning primitive sets, and also provides simpler proofs of some previous results in the literature. More precisely, in addition to Conjecture 1, we establish the following:

Theorem 2 (Erdős primitive set conjecture, #164) For any primitive consisting of numbers greater than ,

Theorem 3 (Odd Banks–Martin) Let and suppose is a primitive set consisting of odd numbers with at most prime factors. Then

where denotes the primes appearing as factors of elements of , and is the collection of products of primes from .

Theorem 4 ( is Erdős-strong) If is a primitive set consisting of even numbers, then

Theorem 5 (Ahlswede–Khachatrian–Sárközy) If is a primitive set, then

whenever .

Theorem 6 (Erdős–Sárközy–Szemerédi, #1217) Let be such that the upper doubly logarithmic density

is positive. Then there exists a strictly increasing infinite divisibility chain

in such that

Theorem 2 and Theorem 5 had been previously established by Lichtman and Ahlswede–Khachatrian–Sárközy respectively, but the Markov chain formalism gives shorter (and more unified) proofs of both. Theorems 3, 4, 6 were open conjectures that can now be settled by this method. These results were obtained with varying levels of AI involvement, ranging from completely autonomous AI queries to traditional pen-and-paper calculations, to various hybrid approaches (for instance, with humans suggesting key inequalities that could then be rapidly tested numerically or even proved by various AI tools).

— 1. Chain/antichain duality and Markov chains —

I’ll now discuss the basic method of proof and try to motivate the main ideas, which have become much clearer in retrospect. Primitive sets can be viewed as antichains in the divisibility poset , in which the partial ordering is given by the divisibility relation . So, one can pose the following more abstract question: given a general poset and a weight function , what is the maximal value of as ranges over all antichains in ?

One can attack this problem using the well known duality between antichains and chains (totally ordered subsets of ): every antichain and chain can meet in at most one point, thus one has

for any chain and any antichain . In particular, if one has a measure on the space of all chains (viewed as a compact subspace of the power set of , equipped with the product topology) with the property that

for all , then by integrating the previous inequality against and using Tonelli’s theorem one would obtain the upper bound

In fact this duality is completely tight:

Proposition 7 (Chain/antichain duality) Let be a poset, let be a weight function, and let . Then the following are equivalent:

• (i) for all antichains .
• (ii) There exists a measure of total mass at most on the space of chains, such that (1) holds for all .

Proof: We have already indicated how (ii) implies (i). Now we need to show that (i) implies (ii). A standard compactness argument allows us to reduce to the case when is finite. If (i) holds, then we also have

for all in the Stanley chain polytope, defined as the convex hull of the indicator functions of antichains. By a classic result of Stanley, this polytope can also be defined as the space of all obeying the inequalities

and

Applying linear programming duality (or the Farkas lemma), we conclude that the inequality (2) must be a non-negative linear combination of the inequalities (3), (4) (as well as the trivial inequality ). Equivalently, we can find non-negative weights for each chain such that

and

for all . The claim follows by viewing as a measure on the space of chains.

Thus, the “universal” problem of obtaining a uniform upper bound on for all antichains is replaced with the equivalent “existential” dual problem of exhibiting a single measure on chains of controlled mass, which hits each element of the poset with a mass of at least the original weight . Thus, such problems are now reduced to that of finding a sufficiently clever construction of such a measure . If the mass of was normalized to equal , this becomes a probability problem: find a random chain process in the poset that hits each element of the poset with a sufficiently high probability. (Though in our paper we found it more convenient for technical reasons to not normalize the measure, and allow the mass to take values other than .)

It turns out (in a manner that was not explicitly appreciated in past literature) that particularly good choices of random chain to use here can come from Markov chains. (Here, the term “chain” is now being used in two different ways, but fortunately the order-theoretic concept of a chain and the Markov process-theoretic concept of a chain will be quite compatible in this discussion.) There will be two types of Markov chains on the poset that will be relevant: downwards Markov chains and upwards Markov chains. Here is our notation for a downwards Markov chain:

Definition 8 (Downwards Markov chain) Let be a poset, and suppose we designate some subset of to be the “absorbing states” (in practice these will be the minimal elements of , although they do not have to be). A downwards Markov chain on with absorbing states is a collection of transition probabilities for obeying the following axioms:

• (i) , with equality unless and , or if and .
• (ii) For any , one has .

(Thus for instance one has for any absorbing state .)

Given such a downwards Markov chain and an initial state , one can generate a random decreasing sequence by having each transition to with probability after conditioning on the past history of the chain. This sequence will (almost surely) be strictly decreasing until it hits an absorbing state, in which case it stays there forever, although if the descending chain condition is not satisfied it is also possible for the sequence to be strictly decreasing indefinitely. We let denote the law of this decreasing sequence. This construction is already enough to recover the Lubell-Yamomoto-Meshalkin (LYM) inequality:

Theorem 9 (LYM inequality) If is the power set of with the inclusion partial ordering, and is an antichain in this poset (i.e., a Sperner family), then

Proof: We introduce the downwards Markov chain with absorbing state in which each non-empty subset of of some cardinality transitions to a -element subset chosen uniformly at random (i.e., with probability of transitioning to each). If we start the descending sequence from the maximal element of , then one can easily check that each is hit with probability . Applying Proposition 7 with , , and , we obtain the claim.

In the above argument we fixed the initial location of the Markov chain, but more generally one can start with any source mass and work with the measure for the purposes of applying Proposition 7.

One can also define upwards Markov chains in exact analogy with downwards Markov chains (reversing the order in the poset), which now generate random increasing sequences rather than decreasing sequences. There is a useful adjoint construction that can convert a downwards Markov chain into an upwards Markov chain: if we have a positive weight which is invariant under the chain in the sense that

for any , then we can define an adjoint upwards Markov chain (with no absorbing state) by the formula

for any in . More generally, if is merely sub-invariant in the sense that

for all , one can still construct an adjoint upwards chain as before, but now one must also add an additional absorbing maximal state to ensure that the transition probabilities still sum to one.

A downwards or upwards Markov chain, when equipped with an invariant or sub-invariant measure, also induces a flow network on the poset, in which an edge from to is assigned a flow capacity of

One can rewrite the Markov chain arguments in the paper in terms of such flow networks, in which case the arguments often boil down to an application of the discrete divergence theorem, giving very short proofs of many of the above results; see the paper for more discussion. However, we chose to focus more on the Markov chain approach in our presentation, as this formalism is also natural and could potentially be more flexible for further applications.

— 2. The Mertens and von Mangoldt chains —

For the purpose of analyzing primitive sets, there are two downward chains on the natural numbers (with absorbing state )that, in retrospect, are particularly natural to use:

• The downwards Mertens chain, in which each transitions deterministically to , where is the largest prime factor of ; and
• the downwards von Mangoldt chain, in which each transitions to with probability for each dividing , with the von Mangoldt function.

The von Mangoldt weight is a natural choice here thanks to the fundamental identity

which encodes the fundamental theorem of arithmetic. The two chains are similar in many way to each other: the von Mangoldt process favors the division by the largest prime factor, but does not require it.

The Mertens chain generates deterministic downward divisibility chains

starting from a product of primes , and as such this process was implicit in much of the previous literature on primitive sets. However, it does not quite interact well with the weight, which is not invariant or subinvariant with respect to this process. Intuitively, the process of dividing by tends to increasingly select for numbers for which is smaller than expected. Instead, the natural invariant measure for this chain is the Mertens weight

the verification that this is indeed an invariant weight is a nice exercise in telescoping series. Taking the adjoint of the downwards Mertens chain with respect to this weight and running that chain from gives the upwards Mertens divisibility chain

in which each transitions to for some prime with probability . A routine induction shows that each is hit by this chain with a probability of ; this for instance gives a weak version

of Theorem 1, and similarly for the other results discussed above.

The key innovation (which was uncovered by the AI-assisted proofs, though not quite in the notation and framework presented here) is to switch to the von Mangoldt chain, which removes the bias towards numbers whose largest prime factor is small. Indeed, the weight now turns out to be sub-invariant (after removing ) under this chain, and there is a modification

of this weight which turns out to be perfectly invariant. (We have an interpretation of this formula in terms of a zeta process that couples together various zeta distributions into a continuous divisibility chain; see the paper for further details.) Taking the adjoint with respect to this weight (or with the original weight ) can eliminate the loss in the previous argument, and give one of the proofs of Theorem 1 recorded in our paper (there are several other variants of this method that we also present).

One slight defect of the von Mangoldt chain, as compared to the Mertens chain, is that it can “jump over” primitive sets (such as the set of products of primes) due to the fact that it will sometimes multiply or divide by a power of a prime rather than a prime itself. This turns out to be a technical difficulty for many of our applications, resulting in a need to make various small ad hoc modifications to the von Mangoldt chain to eliminate this type of jump.

In order to establish some crucial sub-invariance properties, it turns out (after some standard manipulations) to be useful to obtain good bounds on the negative log-derivative

of the Riemann zeta function in the region . Here it turns out that there is a clean (and rather efficient) upper bound

which is equivalent to the non-decreasing nature of the Dirichlet eta function

in this region. There are many proofs of this fact in the literature, but I would like to record a particularly cute proof, that is in the spirit of other arguments in the paper: one can interpret probabilistically as

where is a gamma random variable of shape and scale . Because the sum of independent copies of and has the distribution of , one can couple together all the so that they are increasing in , at which point the claim follows.

— 3. Further directions —

Will Sawin and Ofir Gorodetsky have obtained analogues of several of the above results for function field models or permutation models respectively; we briefly discuss these in the paper, although we do not plan to cover these models in depth. We also note another recent use of this technique to solve a separate Erdős problem (#858) relating to antichains in a variant of the divisibility poset.

The zeta process that we have discovered hints at an emerging theory of the “developmental anatomy of integers”, which differs from the existing topic of anatomy of integers in that it views a large integer (and its prime factor “organs”) not as a static entity, but rather as an evolving process in which primes (or powers of primes, in the case of the von Mangoldt process) are added or removed to the integer over time. With this perspective, primitive sets can be viewed as singular moments in such a developmental process, which are only encountered at most once in the life cycle of a given integer. It seems of interest to study this developmental perspective further.

The paper is currently a work in progress; we have released an early version due to the public interest in this problem. We plan to explore some further applications, and also to formalize more of the above results in Lean (currently two of the six main theorems are formalized), before submitting the paper for publication. The situation here highlights a distinction I have recently made between three components of the problem solving process in mathematics, namely proof generation, proof verification, and proof digestion. In this particular case, the first two steps were extremely rapid due to modern AI tools; however, properly digesting the AI-generated proofs into a coherent exposition that places the arguments in context with both past literature and future directions remains a slower process that requires expert human attention.

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