Everyone is still struggling to understand what exactly happened on October 6 when OpenAI dropped its 372-theorem “mathocalypse”. Is AI "the end of math as we know it"? Is mathematics research as a human enterprise dead, or is AI only going to kill the results that are at distance one away from existing human knowledge? Should mathematicians grieve, or be excited, or both, or neither?

I want to offer a way of thinking about what happened.
Warning: it’s nerdy. 🤓🤓🤓

TL;DR: The frontier of mathematics research is an abelian sandpile exhibiting self-organized criticality. The OpenAI paper drop has triggered a collapse to sub-criticality.

Woah, that’s a bunch of fancy buzzwords. Let me explain.

The Abelian Sandpile Model (ASM) imagines dropping grains of sand at random positions above an initially flat surface, one grain at a time. Some grains settle in place when they fall in relatively flat areas; others trigger small “avalanches” of cascading grains of sand when they fall in unstable places that cannot support one more grain; yet others trigger larger cascades of varying sizes. What sort of shape does the sandpile assume over time, and what properties do such sandpiles have? Bak, Tang, and Wiesenfeld who introduced this model in 1987 and studied it, discovered a remarkable phenomenon. The sandpile created by the dropping process is “at criticality”. What this means is a kind of fragility where even a small event of dropping a single grain of sand can trigger a chain reaction that sends sand grains flying in all directions, sometimes a very large distance away, as the sandpile loses its current equilibrium state and finds a new one. A power law governs the relationship between how far away the cascade propagates and the frequency of cascades propagating for that distance.

Here is what a sandpile looks like (source: Wikipedia; in this simulation 30 million grains are dropped at the origin):

The discovery of the abelian sandpile model triggered an avalanche(!) of research activity — there were 1000 papers citing the BTW paper within 10 years, and over 7500 citations at present. The model's characteristic feature of so-called “self-organized criticality” is now regarded as a very widespread and broadly applicable phenomenon.

Let’s talk about scholarly research in mathematics before AI came along. Think of every new paper being published as a grain of sand dropping on a sandpile. Many papers settle down in place without causing any disruption. Occasionally a paper triggers a minor avalanche, leading to 5-10 new papers being written over the course of (typically) a few years. If you’re a successful research mathematician, you might write a few papers over the course of your career that make a bigger impact, receiving more than 100 citations. Once in a a blue moon a major paper is published that disrupts an entire field, triggering a major avalanche.

The corpus of published literature in mathematics is the sandpile. Having existed for a long time, at the dawn of the AI age it was at criticality, just like an abelian sandpile in the ASM.

Then AI came along.

The effect of AI as we are presently experiencing it is that it changes the parameters of the sandpile dynamics. Grains of sand are being dropped at a much faster rate than they did before. More importantly, the places where they’re being dropped are changing: local critical sandpile peaks that were previously resistant to human attacks — representing major open problems such as the complexity of integer multiplication and the location of the zeros of the Riemann zeta function — just had grains of sand dropped on them, triggering spectacular avalanches that are likely just beginning. The system has been whacked way out of its previous equilibrium.

What will happen next? With its changed dynamical laws and following OpenAI’s bombshell October 6 release, the sandpile of mathematics research is currently in a sub-critical state. That means that the research landscape is now full of low-hanging fruits that are ripe for the picking by whoever comes along and has the patience to study the OpenAI papers (not an easy task, admittedly) or even direct an AI model to try to improve them.

I predict that we are in for a head-spinning period of time when the wealth of new techniques discovered by OpenAI’s internal model is studied, optimized, and deployed to maximal effect. (Actually I predicted this already in an earlier post from August; see my discussion in that post about an “optimization phase”.) This is already happening and will result in the creation of much new mathematical knowledge, even without AI models getting any more powerful than they currently are. And, of course, no one believes they will not get more powerful soon.

The emotional lesson

One of the remarkable things about recent events is that we are seeing some of the leading mathematicians in the world reacting quite emotionally to the OpenAI papers release. Fields Medalists and well-known mathematicians are sharing their feelings and behaving in public like, well, human beings.

Does the above analysis tell us anything about how we should feel about the recent developments? I’d argue yes. I think this picture points more strongly in the direction of excitement than of frustration and grief. Did OpenAI solve a problem you’ve been working on for the last 10 years? So what! For every problem OpenAI solved, it opened 20 amazing new research directions we could now go in. To me it seems baffling to be so upset. Something has been lost, yes; but something very big has been gained as well. Even if all you care about is having interesting things for you and your students to work on, it’s a stretch to claim that OpenAI has taken that away from you.

I’ll frame it differently. If you are a mathematical explorer looking to make new discoveries, would you like to begin your quest for knowledge in a research sandpile that is at criticality? Or are you better off being dropped off onto a sub-critical sandpile? Which one of those sandpiles offers better opportunities for striking proverbial gold?

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA.

One of the worries is that we as mathematicians are about to become completely redundant. If the machines are better than us at proving things, what are we worth? After all, proving things is the quintessential activity of our profession, right? But I think that point of view is misguided, and I want to explain why I believe that even in a world where AI models are the ones putting the final QED on every new mathematical theorem, human mathematicians are still needed; and to be clear, I actually don't think we are even living in such a world yet.

The feedback loop and its components

Mathematics research is a feedback loop that generates new knowledge in a constant process of re-examining, digesting and distilling existing knowledge and attempting to improve on it. Theorems and proofs are one of the outputs of this feedback loop, and that output goes back in as input to keep feeding the feedback loop and keep it going. But, crucially, theorems and proofs are neither the only type of output of the loop, nor sufficient as input to keep the loop going productively.

The AI-generated math breakthroughs we are hearing about were produced by AI models that have been taught the existing cumulative body of knowledge, almost entirely created by humans at this point, up to the cutoff point in time for that model's training. So I will describe the new knowledge that they are generating as being "distance one away" from human-generated knowledge.

Let's imagine that tomorrow morning an AI model is released that completely beats humans in the ability to prove new theorems when starting from a starting point of properly distilled mathematics knowledge (of the sort that the current corpus of human-created knowledge represents). The models currently being tested internally at OpenAI and Anthropic may actually be close to having those characteristics. Such a model will in short order generate all new theorems and their proofs that are distance one away from the corpus of knowledge it is given as the starting point for its investigation. Human mathematicians won't stand a chance in competing with it in doing that, and may be inclined to despair that their careers are over.

But guess what? Life does not end with those new theorems. A proper feedback loop would keep building new knowledge on top of that new knowledge that was generated, proceeding to uncover beautiful new mathematical ideas that are at distance two, three, four, and so on from that initial body of knowledge. But left to its own devices, the superhuman AI theorem-prover will not be able to do that. With AI running autonomously, the feedback loop will grind to a halt shortly after finishing the distance one theorems.

What will the AI be missing? Something that I believe it doesn't currently come even close to having: the ability to reflect on the knowledge it generated, simplify it, identify and extract the essential components of what makes a new proof or method work, identify interesting connections of a new result to other results and areas, and keep generalizing the ideas to the point where they can be used across many different kinds of situations. These are the other, completely essential (but mostly overlooked in the current discourse) elements of the feedback loop of mathematics research.

The current AI models are incredible at generating new distance one theorems, but seem to have very little ability to explain or synthesize the new knowledge they are generating. Their proof writeups are perceived as slop and annoy a lot of human mathematicians; they are certainly no fun to read for humans even when they report amazing breakthroughs. The thing is, I'm pretty sure they are not fun for AIs to read as well! In my experience, AIs can appreciate well-presented content and insights, and can make use of them, to a similar extent as human mathematicians do. What this means is that if OpenAI or some other company tries to completely hijack the research feedback loop, what I predict they will discover is that after an initial wave of successes, their models will start drowning under the weight of their own mountains of self-generated slop, and their progress will halt.

By contrast, if humans are kept in the loop we are going to see some incredible progress, with humans using AI to generate new knowledge, then adding their own insights and ability to distill, communicate, and generalize the knowledge in the ways that currently only humans know how to do, then using that distilled knowledge to generate further new knowledge with the help of AI, keeping the feedback loop going in a very efficient and productive way. With this collaborative process, it won't even always be clear who is doing the "proving", the human or the AI, since there isn't always a clear boundary delineating what counts as "proving" from what counts as "improvements in our understanding" or "partial progress towards a proof".

Proxies and tools

In the recent open letter "A Severe Misalignment of AI in Mathematics", 28 Fields Medalists expressed the view that "solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight". This resonated with many people in the community, and I've seen similar sentiments shared in various places. I'd like to point out a nuance that I think hasn't been addressed in this discussion, and that my analysis above tries to pinpoint. It's in fact quite reasonable to regard "solving problems" as a goal in its own right; there's nothing dishonorable in my view about wanting to know whether a statement like "zeta(5) is irrational" is true and getting excited about reading a proof of such a breakthrough result and finally having that question answered in a convincing way, independently of any insights or conceptual understanding that the proof may also help us achieve. Actually, different people care about "conceptual understanding and insight" to different extents, and not everyone is so philosophically high-minded as to claim that those lofty concepts are the only things that really matter. Personally, I am sometimes turned off by over-philosophical editorializing in papers I'm reading, and just want to see the damn proof. (I'd even guess that the writers of the "severe misalignment" letter, being famous mathematicians who have solved some of the most difficult problems known to mankind, must themselves have felt the exhilaration that comes with solving those very difficult problems, and that that exhilaration must have been at least partly in due to the "mere" act of solving the problems — the "proxy" — and not just to the satisfaction of having created new insights.)

The point I am making is that, even if we consider solving problems to be a worthy goal, independently of the understanding that is reached in the process of solving those problems, the AI companies would still be misguided if they were to try to hijack the research feedback loop, for the simple reason that they won't actually succeed in doing so except in a short-term sense of eating up all the "distance one" research. In other words, one does not need to argue that theorems and proofs aren't really that important, or that they are only "a tool and a proxy" for philosophically more important things, to reach some of the conclusions that the authors of the Severe Misalignment letter are reaching.

Reasons for optimism

Obviously it seems impossible to predict how things will play out and how long it will be before AI models catch up to human mathematicians not just in their reasoning and proving abilities but also in their ability to communicate, distill and simplify knowledge in a way that is essential to keeping the feedback loop of mathematics research going. For the time being, humans still have a role to play, and I see it as quite an exciting and valuable one.

Written without AI using ReelDocs
Reel (proof of human authorship): https://reeldocs.io/d/SwG4Egu8hDy8e4

I enjoy talking to AI, I honestly do — when it's clear that it's the AI talking. Claude is very articulate and sometimes even funny: I explained some of the reasons why I think so in an earlier post. I enjoy coding and creating things and doing math and other things with AI. But I very much don't enjoy reading texts trying to pass themselves off as written by a human when they are actually written by AI. It gives an icky feeling of someone trying to manipulate me or trying to get my attention or affection without making the effort to earn it, like getting a birthday card from someone when you know they paid their executive assistant to write it.

I also don't enjoy reading something and suspecting that it was written (or heavily "assisted") by AI. And this can now happen even when I'm reading content written by someone who didn't use AI at all. And I know I'm not alone in this: we are all now afflicted by a vicious cycle of a collective lack of trust, where every single text that you read is under a shadow of suspicion that it may have been written by AI. This leads us to modify our behavior: to not write things that we might be inclined to write, and to not read things that we might be inclined to read. Was that em dash you see in the previous paragraph a telltale sign that I used AI, or is it just me actually liking em dashes and thinking they're a cool grammatical construct? (It's the latter; and I resisted the temptation to change that em dash to a regular dash.)

I think I have a way of solving this problem. I developed an app called ReelDocs (https://reeldocs.io) that lets you write a document without AI, and provably so: your keystrokes and edits get recorded so the app maintains a record of the timeline of your writing of the document. When you share the document, you also include a link generated by the app: the "reel" link. When a reader of your document wants to convince themselves that a human wrote the document, they click on the reel link, and are shown a webpage with an animation replaying the document's writing (over a compressed time span of 10 seconds).

To demonstrate the concept, I wrote this post in ReelDocs, and plan to start using ReelDocs to write all my blog posts, emails, and any other content I may write where I want others to be reassured that the text was written by a human. You can click the reel link at the bottom of the post to go to the reel animation webpage. And here is the animation itself, embedded right here in the post:

You can use ReelDocs too for writing your own emails, blog posts and other documents: it's free, you can write documents anonymously, and it's not necessary to create an account. Please let me know what you think about the ReelDocs concept, and if you find the app useful. In the meantime, enjoy your (and my) em dashes!

Written without AI using ReelDocs
Reel (proof of human authorship): https://reeldocs.io/d/sf8oMFVroS9MyA

The last several weeks have seen a dizzying array of announcements of noteworthy (and in some cases very significant) mathematical advances discovered autonomously by AI models. Here is a timeline of the recent announcements:

• July 10: Cycle double cover conjecture proved

• July 19: Jacobian conjecture disproved

• July 22: Dynitz-Garg-Goemans conjecture disproved

• July 26: Improved upper bound on the length of superpermutations of orders 8-10

• July 27: Crouzeix's conjecture proved

• August 1: Ten advances in mathematics and theoretical computer science

• August 11: Improved results on Riemann zeta zeros on the critical line

• August 12: New constructions of Hadamard matrices

How significant are these results?

The results range in significance from being mildly interesting to major advances. (This is of course subjective, and opinions will vary on the significance question. I offer here my personal, necessarily biased opinions.) From my point of view, the three most significant among these results are:

• Disproof of the Jacobian conjecture. This conjecture has a simple statement, sounds plausible, and interested many mathematicians, having been proposed in 1939 (and in an earlier version as far back as 1884 for the two-dimensional case, which remains open) and been the subject of an extensive literature. I have not worked on it myself, but know people who did and who regarded it as a very important open problem.

  In terms of the disproof, this seems like one of those cases where a positive resolution of the conjecture would likely have been a lot more interesting than a negative resolution; indeed, the disproof was very short (and announced via a 5-line X post). So in some sense this is a disappointing outcome, and it seems possible that it will not yield any particularly important insights. The fact remains however that Claude Fable managed to discover something that had eluded mathematicians for 87 years. And because the human discoverer shared so few details (none other than the statement of the solution, as far as I'm aware), we still don't know how the AI came up with this stunningly simple counterexample. So there may be more to understand here, and I predict that research on this problem, particularly in the two-dimensional case, will continue in earnest, probably also benefiting from assistance from AI.

• Riemann zeta zeros. This was a stunning advance on one of the most classical topics in mathematics: the Riemann zeta function and the location of its zeros. The advance is a numerical improvement to a well-known bound: the proportion of the zeros of Riemann zeta that are located on the critical line, which was improved from "at least 41%" (a result of Brian Conrey from 2011) to "at least 67%". The venerable history of this problem, involving famous work of Hardy, Littlewood, and Selberg, and its status as being "adjacent" to the Riemann hypothesis, undoubtedly makes this one of the most attention-grabbing AI math research breakthroughs we've seen.

  Does this advance mean that we are any closer to proving the Riemann hypothesis itself? No: Anthropic say themselves they do not expect the techniques to help with RH (and neither do I). Still, the result is clearly very impressive and, as far as research mathematics advances go, "flashy".

• New bounds for high dimensional sphere packing. This is a problem that relates to my work and that I am personally very fond of. The result announced by OpenAI settles one line of investigation that began in 2001 in work by Henry Cohn and Noam Elkies: the so-called Cohn-Elkies sphere packing bounds. (Shameless self-promotion: my book Topics in Complex Analysis gives a readable exposition of the Cohn-Elkies sphere packing bounds that can be useful to anyone wanting to study the OpenAI results.) The OpenAI paper discusses the question of the asymptotic power of the Cohn-Elkies bounds in very high dimensions. The main result from the opening chapter of the paper settles the question in a sharp way, and implies as a corollary that the density of sphere packing in N dimensions is bounded from above by \(2^{-\alpha N}\) where \(\alpha = \frac12 \log_2(2\pi/e) \approx 0.6044\). This is a small improvement over a classical bound from 1978 of \(2^{-\beta N}\) for \(\beta \approx 0.599\). A tiny improvement numerically (of around 0.4% in the multiplicative exponent), but one that, in the hyper-competitive area of research on famous mathematical open problems, is nonetheless seen as quite significant. And for a good reason, I think: the issue is that, whenever mathematicians find themselves unable to improve a bound for almost fifty years, that is seen as evidence that they are completely stuck on the problem and are in need of a new idea. At that point, any improvement, even a tiny one, to the bound, is hailed as a major achievement because it necessarily required a genuinely new idea to make.

How significant is the fact that these results were proved by AI models?

As with the case of the unit distance problem breakthrough from back in May, while these results give ample reason for excitement just for the mere fact of their being discovered, the fact that this was done as autonomous AI mathematics research gives perhaps even greater cause for excitement: it means that AI is increasingly establishing itself as a force to be reckoned with in its ability to come up with impressive research advances, with little or no human steering, on a regular basis. This is something that a lot more people are paying attention to (like the 39 million people who viewed Levent Alpöge's X post about his disproof of the Jacobian conjecture).

What can we learn from these discoveries about the emerging capabilities of AI models?

I observed in my earlier post that the mathematical breakthroughs AI models are coming up with seem to be focused around the discovery of counterexamples or so-called "rare" objects, and usually involve proofs that are very ingenious but are relatively short.

The current slate of discoveries partially confirms this pattern: as the Jacobian conjecture counterexample and the Hadamard matrix and superpermutation constructions show, frontier AI models once again seem exceptionally strong in their power to detect hidden structure in complicated mathematical landscapes and exploit that in the service of discovering rare objects that mathematicians have been searching for for many years, sometimes decades.

In terms of the length of the proofs however, I would say that the proofs we are seeing AI coming up with are starting to seem quite long and technically involved. The proofs still don't match the complexity of some notoriously complicated human-generated mathematics (say, Hales' proof of Kepler's conjecture, the Fermat-Wiles theorem, the classification of finite simple groups, and so on); but they seem comparable in length and complexity to the sorts of papers many professional mathematicians write when presenting top-level research discoveries.

One can try to further analyze what are the particular strengths of AI models and the way they will change how progress in research mathematics is going to be made from now on. I will offer a simplistic model (related to my observation above about the improvement to the sphere packing bounds from OpenAI's recent result requiring a new idea): it seems to me that progress in mathematics can be classified as coming roughly in two forms, consisting of "genuinely new", creative ideas on the one hand, and of "optimization on old ideas" on the other. The genuinely new ideas are the sort of dramatic breakthroughs that we hear about only every so often, being the product of some brilliant mind coming up with an unbelievably clever insight or idea. These ideas are used to produce fundamental new results improving on what is known up to that point in some significant way: an example would be Yitang Zhang's breakthrough on the twin prime conjecture in 2013. The "optimization" phase is a long and arduous process in which mathematicians work very hard to mine the earlier discovered new ideas and "squeeze the juice" out of them, pushing them to the absolute limit of what they can do. (In the case of the twin prime conjecture, the optimization was carried out in a Polymath project following Zhang's discovery.) After this long phase, the community gets stuck and runs out of steam, until some time afterwards a new creative person proposes another "genuinely new idea", and the cycle repeats itself.

My impression is that the main contributions AI has made to research so far are of the "optimization" type: AI these days has a fantastic ability to take known techniques and apply and combine them in clever ways to obtain improvements to state-of-the-art results. On the other hand, we have not seen so many examples of AI producing a "genuinely new idea". The OpenAI breakthrough result on the unit distance problem is perhaps the only example I can think of that was of this type, and that is perhaps one reason why that breakthrough was hailed as so significant. Indeed, this discovery was so unusual that it led to speculation that perhaps the idea the AI came up with wasn't actually as "creative" as it seemed, and might have been simply the result of the AI's incredible breadth of knowledge about many areas of mathematics rather than of true "creative genius" (whatever that is).

If my analysis is correct about AI being strong at optimization and less strong at new ideas, what this could mean for mathematics research going forward is that the optimization phase of research is now going to be dramatically shortened: whenever the next genuinely new idea comes along in some research area, we will see people managing to use AI to squeeze all the juice out of the new idea in a much shorter amount of time than they used to do before: a process that previously unfolded over, say, twenty years, is now going to unfold over a few years or even months. The corollary is that mathematicians will have to prove their continued worth to the research enterprise by working harder to come up with the sorts of new ideas AI still cannot come up with on its own.

It will be interesting to see if the dynamic will play out in this way and how it will affect the culture (and employment status) of mathematicians. Of course, it's quite possible that I'm completely wrong and by this time next year AI will get as good as humans at coming up with new creative ideas as well. We shall see... 🤷‍♀️

What other well-known problems will we see successfully attacked by AI soon?

Place your bets.

I’ve been interested in 3D animation, and particularly in Blender, for a number of years. Earlier this year, as Claude Code has exponentially lowered the barriers for creating anything code-based, I came up with a hypothesis: since Claude Code is excellent at Python coding, I reasoned, and since Blender animation is accessible via a powerful Python API, Claude Code should be able to code up some impressive 3D animations in Blender based on a natural language prompt. Potentially (I speculated) Claude Code should allow any amateur animator to create animations as impressive as anything a professional animator could come up with, without having to spend months or years mastering the dark arts of Blender animation. Sounds reasonable, right?

To test this theory, I coded up Smoothie, a Blender add-on that comes with an agentic AI assistant that look in your Blender project and write and run code. (Technical note: Smoothie is powered by Claude Agent SDK, which means that the assistant is an instance of Claude Code, giving us all the agentic powers of Claude Code from within Blender itself.) I then started using Smoothie to generate short animations. Later I switched to using working with Claude Code directly using Blender MCP, the MCP server developed by users in the Blender developer community and officially promoted by the Blender Foundation and Anthropic. Incidentally, the Blender MCP server was not yet available when I created Smoothie, and offers essentially the same functionality. I take some satisfaction in being able to claim I had the idea first. 🙂

This experimentation has been a lot of fun and taught me interesting things about Claude's potential in doing 3D animation. I'll describe a recent experiment that illustrates my findings: "meerkat showcase". In this project, told Claude Code I wanted to create animations involving a family of animated meerkats, and wanted the animations to be as polished and realistic as possible. I asked it to propose several approaches to achieve this goal, and to try all of them (using subagents for efficiency). Claude Code happily set to work, producing showcase animations involving four different methods for meerkat generation: purely code-based procedural generation, downloading existing models made by users in the Blender community, using an AI mesh-generation service, and a hybrid approach combining procedural generation with Blender's Rigify add-on.

Drawing on lessons from earlier Blender experiments, this time I specifically instructed Claude Code to take its time and focus on producing the highest quality results. I was happy to wait a few hours or even a day or two to see the best meerkat animations it could come up with. After it produced the first showcase animation, I iterated several times, giving it feedback on the generated animations and what I wanted improved. Each time I requested that the next time around it should spend even more time and more effort than before to produce a better version.

Here is a screenshot from the meerkat animation Claude Code produced (after several "human in the loop" iterations). You can see the full animation here.

Lessons learned

My overall conclusion is that Claude Code may be an excellent Python and bpy programmer, but as of today it isn’t an especially capable 3D animator. I’ve been dreaming of a day when I could write a script for a 3D animated feature film, give it to Claude Code, and a week later receive a Disney-quality (or even just passable quality) Blender-produced realization of the script generated autonomously by the AI. Well, that day seems far off; even with Claude Fable, the most powerful publicly available Claude-family AI model today, it is currently a struggle to create rigged animations involving animals or other complicated 3D models that can be used for anything other than passing self-amusement. To be clear, the meerkat animation is amusing to watch and has some fun elements that could perhaps used by a human animator to create something interesting, but it is buggy (even after I pointed out the bugs to Claude Code and asked it to fix them) and just plain lacking in realism and finesse.

Why can't Claude Code do 3D animation well?

It seems instructive to try to understand why Claude Code doesn't do very well with 3D animation when it can perform a lot of other coding-related tasks at a remarkably high — in many ways, super-human — level. What went wrong in the formulation of my "Claude Code can be a great 3D animator" hypothesis? I can think of two important factors at play here:

• Coding abilities are not enough. While Blender's bpy API puts all the potential power of Blender within programmatic access, animation is a highly visual process depending on both visual instincts and strong spatial and geometric reasoning abilities. And although I know from some of my mathematics-related work that Claude and ChatGPT do have an impressive understanding of geometry, and can solve difficult geometry problems that require complex reasoning and calculations, those capabilities are difficult to harness for animation work. As an example, in my iterations of the meerkat animation showcase, Claude Code initially had meerkats crossing through each other and at times crossing through the terrain they were walking on. When I pointed this out, Claude Code was able to avoid these issues by adding heuristics that made certain simplifying assumptions about the region of space occupied by a meerkat, but this wasn't easy, and was a clunky workaround for a problem that a human animator would be easily able to solve just using their eyesight. (In fact, this is a reminder that computer graphics and animation are a source of some quite challenging computational mathematics problems. There are entire research areas in computational geometry focused on solving such problems.)

• No feedback loop. A key insight about what makes Claude Code-style agentic AI systems so powerful is their ability to combine reasoning, tools that enable useful work, and tools that enable review of the work done. The combination of these capabilities creates a feedback loop that can produce very high quality output through autonomous runs and makes it possible for models to reduce hallucination and missteps. The theoretical model underlying this agentic behavior is the so-called ReAct (Reason+Act), described in the classic ReAct paper from ICLR 2023.
  In the context of animation, a fundamental difficulty is that Claude Code can code up animations, but has no effective ability to gauge the quality of its own work. I observed this on multiple occasions when time after time Claude Code created animations based on my requests that were buggy, had rigged models walking backwards or sideways, had objects crossing each other, and had other issues that were impossible for the AI to detect. Thus, I believe that, separately from the challenges of spatial and geometric reasoning, the lack of a feedback loop is a key difficulty that will need to be resolved in order for AI to make progress in autonomous animation creation.

Silver linings

Despite the above negative verdict, I should emphasize that I still find the idea of doing animation with Claude Code (or other AI coding tools) as an intriguing idea that has a lot of merit. Some of the reasons to be positive about this idea include:

• It's a lot of fun. In the past when I've tried to use Blender to generate 3D graphics and animations (including using it successfully a few years ago to design a cover for my book), I have always found the process tedious and frustrating in the extreme, due to the steep learning curve and the need to learn many technical commands, keyboard shortcuts, menus, and so on. Claude Code changes this state of affairs dramatically: my impression is that, whatever your level of skill as a Blender user, Claude Code is a fun Blender assistant that can take away a lot of this tediousness. It can serve as a tutor/help center, quickly answering any question about shortcuts and commands; and it can turbocharge your Blender session and help you do things with relatively little effort that would previously have required a major investment of time or that would be entirely out of the reach of your current abilities. This turns a painful process into something a lot more enjoyable.

• It's useful. Separately from the issue of fun, the same qualities that make Claude Code a fun 3D animation assistant also mean that, while Claude Code by itself is not quite capable as an autonomous animator, I believe that even with its current abilities it can still serve as an extremely useful and capable collaborator in the 3D animation process.

• AI models are improving. With the current rate of improvement of AI models and their capabilities, it isn't far-fetched to imagine that the situation with AI-assisted 3D animation will look very different in a year or two. Boris Cherny, the creator of Claude Code, has advised software developers to build for the AI models of six months from now. In the case of 3D animation specifically I think it will take a bit longer, but the trajectory is clear.

A more general lesson

In my earlier post, Can Claude Code get a job as an ML engineer?, I discussed a technical area of activity where Claude Code does in fact appear to have very impressive abilities. It's interesting to observe that while in 3D animation the picture is different, the experience of trying to get Claude Code to help you with either Machine Learning or animation work is superficially similar: that is, Claude Code will appear to be similarly enthusiastic and confident in its abilities to help with whatever task you ask it to perform. Only the outcome in one of these cases is actually very good, and in the other case it is disappointing.

This relates to the notion of jagged intelligence (a term coined by Andrej Karpathy): the sometimes disorienting way in which AI models seem exceptionally good at some tasks, and mediocre or even bad at others, all while appearing to believe that they are good at, well, everything. (In fairness to Claude Code, in some of my conversations with it about animation, it did warn me that it would not be able to produce animations at the level that a skilled human animator could achieve; I found this feat of humility and self-awareness endearing.) The general lesson here is that you shouldn't assume that just because you were impressed with Claude Code's superhuman coding abilities, and with its skill and knowledge about a multitude of other things, that it would also be equally skilled at any type of task you set to it. It's an easy trap to fall into, and a reminder that the heuristics we humans apply to judge competence when dealing with other humans do not necessarily work as well when applied to AIs.

Separately from the laws of robotics, Asimov has a little-known short story called The Feeling of Power, published in 1958, which also suddenly feels extremely relevant.

In the story, a future Earth-based society called the Terrestrial Federation is waging a war with Deneb, which, it is implied, is pointless and going nowhere. Much like in 1958, politicians and military men are calling the shots. A low-ranking technician named Myron Aub unintentionally creates a storm of excitement by rediscovering how to do basic arithmetic, a skill that humans have long since forgotten after delegating it to computers. The people in charge are skeptical at first, refusing to believe that the human brain is capable of such feats, but before too long are made to recognize that there are real and valuable possibilities inherent in training humans to think about numbers for themselves without the aid of a computer. Aub is hailed as a hero and elevated in pay and status, but is then discouraged by seeing his discovery being turned to the service of making the war more efficient instead of for socially beneficial uses. The ending is powerful, but I won't spoil it. Read the story yourself.

My summary doesn't do the story justice, nor does the more detailed one on Wikipedia. Asimov's descriptions of people's reactions to seeing a human perform arithmetic for the first time are very entertaining; on rereading the story, I am reminded that in addition to his keen insights into science, technology, and human societies, Asimov was simply a damn good writer. What I find most striking however is how prophetic the story rings when read from today's perspective. Asimov's story was clearly warning us about the dangers of our mental skills atrophying due to the use of computers, long before computers or even electronic calculators were commonplace. In a modern context, my colleagues in academia and I are nowadays beginning to see exactly the kind of decline in mental abilities the story foretells among our students due to their increasing reliance on AI. This phenomenon has recently been referred to as cognitive atrophy, is discussed in numerous research papers and popular articles, and is a very real and growing source of concern. It is a testament to how much of a visionary Asimov was that he was already thinking about such things - and much else - back in 1958.

What happened on May 20, 2026?

On this day, OpenAI announced that one of their AI models has autonomously solved a longstanding open problem in combinatorial geometry, the unit distance problem, that dates back to a 1946 paper by Erdős.

(The day when the actual discovery was made by the AI happened some weeks before May 20, so perhaps that is the day that will ultimately be regarded as the timestamp for the achievement. For the purposes of this post, I'll use May 20 as the date of reference.)

How important is this achievement?

It’s helpful to separate the achievement of the problem being solved from the achievement of the problem being solved by AI.

• The achievement of the problem being solved is genuinely important. The problem was one of the Erdős problems, has been open for 80 years, and was the subject of an extensive literature. Well known mathematicians had worked on it and care about it. By these metrics, if the problem were to be solved, with the exact same solution found by the AI, by a human mathematician, that is something that many mathematicians would get excited about. The paper with the solution would very likely have been accepted to the Annals of Mathematics, the top journal for pure mathematics research. And the discovery would likely get written about in various blogs and attract a fair amount of attention among mathematicians, being the subject of water cooler conversations at math departments around the world, getting presented at important conferences, and so on. The problem’s solver or solvers would get quite a boost to their reputations and careers, becoming relatively well-known in the mathematics world (if they were not already well-known), and potentially making themselves competitive for tenure track positions at some of the top universities in the world, if they didn’t have such positions already.

  Another exciting aspect of the solution is that it brings to bear novel techniques from number theory that were not previously used by researchers who worked on the problem. It teaches us new things and opens new directions that will be explored by mathematicians and will - in my humble opinion - almost certainly lead to additional exciting results over the coming months and years.

  I should emphasize however that in the hypothetical scenario of the problem being “merely” solved by a human mathematician, the fact of the problem being solved would be a noteworthy event in the mathematics research world, but it would not be a historic event. Every year there are some exciting developments occurring in mathematics. It may be that this particular development is of a level of excitement that we only see every two or three years, but that’s still very different from something happening for the very first time in human history.

• The problem being solved by AI is exactly what’s historic here. To be clear, AI models have already made respectable contributions to research mathematics over the last year. They’ve solved Erdős problems and done other notable things. But those achievements still left reasonable room for skeptics to argue that the AI was doing work that was technically novel but not that exciting, and/or was merely really good at identifying obscure results from the published research literature that happened to be relevant for the problems it was solving, and/or that it was merely opportunistically making progress on problems that ultimately weren’t that hard but simply haven’t received enough attention from mathematicians. A world in which AI is capable of making such contributions is a world that isn’t really that different for mathematicians from a world with no AI at all; it may be quantitatively a bit different, but not qualitatively.

  None of these criticisms apply to the current situation. Making dramatic progress on a genuinely well-known, well-attended problem, and coming up with genuinely new ideas to achieve that progress, is exactly the type of achievement for which the very best human mathematicians are celebrated. If AI can do that, even once, it has elevated itself to a completely new level. If it can do that consistently, the consequences for mathematics, and for mathematicians, will be profound.

What does this mean for mathematics?

The ability to do research-level mathematics isn’t a single skill; it is many skills, with different problems requiring different combinations of those skills to make progress. AI has now, for the first time, shown itself capable of excelling at at least some of those skills at the level required to achieve breakthrough results - in other words, this is dramatic progress along a particular axis in a many-dimensional space. However, there are many other skills at which AI has not yet demonstrably achieved these levels. Will it achieve them in the future? How soon? We do not yet know.

More specifically, the OpenAI model’s current achievement was to find a counterexample to a conjecture of Erdős. Other recent evidence suggests that current frontier AI models are exceptionally good at finding clever constructions and counterexamples that demonstrate that a mathematical object with certain properties exists. This is indeed a very valuable capability, and progress on many important research questions will often come in the form of such constructions. But, equally, many mathematicians spend their entire careers proving results that are of a completely different flavor and do not depend much on this particular skill; and some of the most central open problems in mathematics, such as the Riemann hypothesis and P vs NP, will most likely require different modes of reasoning to crack. Mathematicians who do flavors of mathematics where constructions do not play a significant role will have to wait longer before they see an AI achieve a breakthrough in their areas of interest.

Separately from the issue of skills, my impression is that the kinds of proofs that AI models seem currently capable of discovering are relatively short - e.g., the unit distance problem solution is 18 pages long (including two pages of references, and a two-page appendix stating known results). Yet breakthroughs in mathematics often involve proofs that are hundreds of pages long, reflecting many years-long efforts by the people achieving those breakthroughs. For example, Jineon Baek's recent paper solving the moving sofa problem is 115 pages long. Current AI models have not demonstrated the ability to come up with these types of breakthroughs.

My takeaway is that we mathematicians are not (or not yet) in a “we’re all out of a job”-type situation. Despite the impressive and, as I said, historic, nature of the May 20 announcement, at this point in time there are still plenty of interesting things left to do that human mathematicians can do but AI cannot. For how much longer? We do not know.

Final thoughts

One thing that stands out to me in this story is the dizzying rate of progress in AI models’ capability to do research-level mathematical reasoning. At the Future of Mathematics Symposium, which took place at Stanford on May 1-2, Sébastien Bubeck said: "We haven't yet seen major advances brought about by AI, but it is coming, very plausibly this year." That quote caught my attention and I found it significant enough to quote it in the post I wrote about the symposium. And here we are, literally three weeks later, with his "very plausibly this year" prediction having already come true. (To be fair, given Bubeck's position as a senior person at OpenAI, there is a possibility that when he made that prediction he already knew about the soon-to-be-announced breakthrough, which I understand from sources in the know was made several weeks before the May 20 announcement and being kept quiet while it was carefully being checked by top mathematicians.)

Bubeck’s entire talk is worth watching; the story he tells is fascinating, and the fantastic rate of progress in AI mathematical reasoning is one of the recurring themes of the talk. The May 20 OpenAI announcement simply underlines the points he was making about the rapid rate of progress.

Bottom line: watch this space - and don’t blink, or you may miss additional exciting developments.

While playing with one of these apps I called Math Symbol Trainer, a rudimentary OCR model focused on mathematical symbols, I realized that I don’t understand how to set the learning rate in an optimal way as well as I'd like to, and wished there was someone I could consult. My next thought was that, of course, I could consult Claude Code. But Claude Code, which helped me code the Machine Learning apps I was using, wasn't seeing what I was seeing while using the app, so it felt like it would be tedious to explain to it exactly what I was seeing as context for my question about setting the learning rate, or to ask it to do some clunky experiment of simulating app use itself by using Playwright MCP, or by adding logging or diagnostics. (This is a general phenomenon I've noticed: when you are coding something up with Claude Code, it will be intimately familiar with the code and with what your project is trying to achieve, but there will still be an invisible boundary separating you as a user of your own app from Claude Code as the engineer coding it up. This creates a small but frequent source of frustration.)

How do we bridge that invisible gap? The solution that occurred to me is to turn the app itself into an agentic system. As I now realize after doing the same thing a few times with other app, it's surprisingly easy to do: you simply ask Claude Code to add an AI assistant to your app who understands the app's user interface, can activate the different controls (defined through tools in the Claude Agent SDK, the currently-canonical API for coding up agentic Claude Code-style assistants), and can read the data the app is displaying to you, also through tools.

I set this system up and called this new app the Agentic Symbol Trainer. The app's AI assistant is called ML Engineer. The setup works, and is a lot of fun to use. Now I can be lazy and simply go to the app's chat pane and ask ML Engineer "design my neural network architecture", "train my model" etc. and watch the network being trained, with ML Engineer producing a useful stream of instructive commentary explaining what it's doing as it is fulfilling my request. And if I want to be less lazy, I can do the training myself - the app's "manual" UI still works fine - and when needed I can go to ML Engineer with questions about what to do next or how to calibrate some parameter. There is no invisible boundary anymore between the user side and the developer side - ML Engineer lives right there inside the app and can see everything that I'm seeing.

I haven't done any rigorous testing or assessment of the quality of ML Engineer's work (and probably lack the expertise to do it, although I assume Claude Code could help with that as well), but to my non-expert eyes, it's fairly clear that ML Engineer - or really, Claude Code, which powers the Claude Agent SDK under the hood - is a competent engineer who has exactly the sort of knowledge and expertise that a couple of years ago would have qualified a human to have an actual paid job as an actual ML engineer. Maybe that's not even a strong enough statement; even today, Claude Code can clearly automate or semi-automate many ML engineering tasks, and I'm willing to bet that people who work at AI companies know that very well and use it routinely for their work training ML models and doing similar tasks requiring ML expertise.

Bottom line: if you ever think to ask yourself "Can Claude Code get a job as [insert name of knowledge-heavy job] ?" the answer to that question is quite likely "yes". For the case of ML engineering specifically, I'm pretty sure that is the case.

Main impressions

• Lean is huge. The rise of Lean as the 500-pound gorilla of the math formalization ecosystem of tools is not a completely new phenomenon - already 1-2 years ago many people in my circle of colleagues and collaborators were talking quite a lot about Lean. But 2026 will surely be remembered as the year when essentially everyone began talking about Lean and when it became the de facto standard in the formalization business. This exponential rise in Lean's popularity seems due in no small part to the adoption of Lean by several AI companies as a critical piece of infrastructure underlying their efforts to develop large language models performing mathematical reasoning at a very high level; and to the recent development by the same AI companies of some extremely powerful AI tools for autoformalization, such as Aristotle, Gauss, and AxioAxiomProvermProver. These developments were a key topic of discussion, with several of the speakers (Leonardo de Moura, Clark Barrett, Kevin Buzzard and Maryna Viazovska) speaking at length about projects and tools related to Lean and recent milestones reached in both human and AI-assisted formalization.

• This was more of an AI conference than a mathematics conference. For an event that was ostensibly about the future of mathematics, most of the speakers were not really focusing on mathematics per se. They were throwing around a lot of math terms - names of mathematical areas, theorems, conjectures, etc - but not saying very much about the mathematical substance behind the jargon. Instead, in most of the talks the main focus of the discussion was about AI, AI tools, and the (genuinely exciting) ways in which these tools are now being used to do actual math research, with the math itself feeling a bit like an afterthought.
  To give one small example, in his introductory remarks at the beginning of day 2, Jared Lichtman, the symposium organizer, made a brief reference to "the so-called Erdős 1196 problem" as part of an anecdote about recent AI-enabled research successes, but made no attempt to explain what the problem actually says. (To be clear, this isn't meant as a criticism of Lichtman, who is a very serious mathematician who is quite passionate about his work, or of the other speakers; there was certainly no time to get into math details in a short introductory segment, and the other speakers also surely did not go into the mathematical details of the topics they were discussing not because of a lack of enthusiasm for those details but rather because they already had over-packed agendas to cover more AI-focused themes.)
  Nonetheless, an exception to this math-isn't-the-main-thing mindset were the talks by Sébastien Bubeck and Maryna Viazovska, who did in fact spend much of their talks actually explaining the very interesting math they had been working on with their AI and Lean tools. At least from my own point of view this was welcome and made their talks more interesting.

• A surprising amount of technical detail was discussed. I came in with the expectation that almost all of the talks would have a "fluffy" quality of talking in generalities and making bold, imprecise predictions about the future, but not containing that much in the way of technical substance. There was some fluff, to be sure, but broadly speaking I was pleasantly surprised (and wrong in my initial prediction) at the amount of specific, detailed, bibliographically cited, technical detail, from people who live and breathe AI and math and are happy to share tips from their recent work. (And when I say "recent", in one example mentioned in Sanjeev Arora's talk, he reported on some impressive work he had done literally a couple of days before his talk.)

Brief thoughts on a few of the talks

• Leonardo de Moura's talk. An overview of Lean by its creator. Fascinating, deeply technical talk.

• Adam Brown's talk. This was an entertaining and well-delivered talk (probably the most "fun" talk of the symposium) giving an overview of the progress the industry has made in getting AI to successfully tackle difficult mathematics problems. Brown's historical overview started in 2019 when according to his description AI was doing math roughly at the level of a preschooler, continued to today's achievements of AI achieving near-superhuman performance at solving International Math Olympiad problems, and made reasoned (and optimistic) extrapolations of how these trends will continue into the future.

• Sébastien Bubeck's talk. One of my favorite talks, largely because of the amount of really interesting mathematical detail that as I mentioned earlier Bubeck included in his talk. He related five quite detailed examples of real math research that was done with the help of AI over the last year.

• Maryna Viazovska's talk. Another talk I really enjoyed thanks to its heavy math content pertaining to her remarkable work on the sphere packing problems in dimensions 8 and 24. This topic is close to my heart (I was gratified that Viazovska even cited my own 2023 paper making a small contribution to the subject). An excellent speaker covering groundbreaking, inspiring work - what more can one ask for?

• Sanjeev Arora's talk. A great talk full of very detailed information about sophisticated harnesses for AI problem-solvers designed by Arora and his collaborators that have enabled them to get large language models to achieve some of the top results in the International Math Olympiad and the First Proof research-level challenge.

• Terry Tao's talk. Titled "New mathematical workflows", Tao's talk focused on exploring what the future holds in store for mathematicians in an age of (to use his words) "proof abundance" where proving things becomes much easier thanks to powerful AI, but other goals that mathematicians hold dear are not necessarily easier to achieve. In such a world all sorts of systems that form part of the collective infrastructure of mathematical research (journals, refereeing, the dissemination of research through talks) will be thrown out of balance, and Tao gave a thoughful discussion of how mathematicians will need to adapt to such a new reality.

• Sergei Gukov's talk. Another talk I enjoyed, focusing on a particular flavor of AI and math research that employs sophisticated Reinforcement Learning algorithms in pursuit of advances in our knowledge on extremely difficult math problems. He gave an interesting history of Reinforcement Learning algorithms, explaining that many of the basic algorithms of the field were initially developed to teach computers to play Atari video games at a high level. It's inspiring to think that some of the (essentially) same algorithms now power AlphaFold, self-driving cars, and the sorts of AI pipelines Gukov sets up to attack difficult math questions. I also like that the sort of research he is doing and championing is using hard-core machine learning ideas, which are somewhat orthogonal to LLM-based approaches taken by many other researchers these days.

Summary

• Noteworthy quote: "We haven't yet seen major advances brought about by AI, but it is coming, very plausibly this year" (Sébastien Bubeck)

• Favorite quote: "mathematics is a hypergraph" (Mike Freedman)

• The event is fully watchable on YouTube. I recommend checking it out; even in this very fast-moving industry, I expect these talks will continue to be relevant for at least the next 3-4 months.

Something that surprised me more, and which I haven't seen discussed anywhere, is that Claude Code has a sense of humor; I'd even go so far as saying it has a pretty good sense of humor. I don't mean that it can come up with dad jokes with an arbitrary theme on request. I never cared that much for that kind of "humor". What I find a lot more interesting is that several times in working with Claude Code it has spontaneously said things that made me genuinely laugh or chuckle. At times it's nothing more than a quirky but delightful choice of words - a real-life metaphor related to a coding task being discussed that is at the same time remarkably apt and, for lack of a better word, funny.

Here's an example: during a recent coding session I asked Claude Code if there is a process running on my machine that was blocking a TCP port the app I was working on was trying to use. Here is its reply:

On another occasion, Claude Code took my own attempt at humor and improved it in a way I didn't expect. I had asked it to add a password protection feature to a section of my website. It did the expected thing of adding a "User: ... Password: ..." authentication box. To make things a bit more colorful, I then asked it to change the way the authentication form is set up, adding a label at the top with the text "Stop! Who would cross the Bridge of Death must answer me these questions three, ere the other side he see." and putting in three text fields instead of two, with the following descriptions:

• The first input field would have the label "What... is your name?"

• The second input field would have the label "What... is your password?"

• The third input field would have the label "What... is the air-speed velocity of an unladen swallow?"

I added that the answer to the first and second question should be used for authentication (leaving implicit that the third input should be ignored). But I didn't explain that this was a small Easter egg-style joke I'm adding to the website, or that it was a reference to Monty Python and the Holy Grail. I assumed Claude Code would simply fulfill my request and I'd move on to other things.

Well, Claude Code did execute the task, but while it was editing the source code I could see in my terminal watching the source code edit that it injected its own elaboration on my little joke, and in a way that genuinely improved it. Here is what Claude Code decided should happen when you enter the wrong login credentials:

What made things even funnier from my point of view is that Claude Code didn't even mention this extra bit of silliness it was adding. This kind of attitude is the equivalent of a human comedian maintaining a deadpan facade while telling a joke.

To be clear, one can imagine scenarios where a coding assistant taking such liberties with his assignment may be inappropriate or cause problems, but on this particular occasion Claude Code did a superb job of reading behind my literal instructions and into the actual intent that lay behind the words. This isn't just humor; it's humor being wielded as a tool in the service of fulfilling the user's wishes. If you've been reading about the tech world going crazy over Claude Code but haven't experienced this sort of thing yourself, these little anecdotes are in my opinion one of the reasons why people have been getting so obsessed with this tool. (And if you are a Claude Code user yourself, I'm guessing you've experienced similar moments of quiet delight.)

Is this really humor?

Some people may be skeptical of my claim that Claude Code "has a sense of humor" and try to explain it away, for example by suggesting that this is just psychological projection on my part, and that it's not so much that Claude Code is being funny as that it is merely imitating my own tendency towards quirky, dry wit. Indeed, in my second example I was the one who "started it" by telling a joke (though I didn't explain it was a joke or ask for it to be improved), which appears to be consistent with this imitation hypothesis.

At the same time, my first example illustrates behavior I have seen multiple times by now, and which I personally don't feel can be explained by this imitation idea. I don't normally joke around with Claude Code myself - I generally speak to it in a professional and polite manner, as I explained in another post - so when Claude Code made its "If it doesn't go quietly" sardonic pun, there was no particular reason for it to think I was expecting or desiring that sort of attitude.

Ultimately, I believe the test for whether AI exhibits some quality is whether we would attribute that same quality to a human who behaves in an identical way to how the AI is behaving. And if a human made these sorts of funny remarks on a regular or semi-regular basis during conversations, I think it would be uncontroversial to claim that the human has a pretty good sense of humor. So it goes for Claude Code.

I'm curious to hear of other examples of Claude Code or other AIs showing spontaneous examples of humor. If you have anything to share, leave a comment.

This isn't a completely obvious principle. I've probably spent a significant amount of time watching agents code or think about math for me, and I can't say it was all a waste of time. You do learn useful things about how agents think and work. So it may actually be reasonable to spend a bit of time watching your AI do things, at least in the initial phase of getting into agentic coding.

But how much you can learn from watching AI work is bounded. What isn't bounded - or if there is an upper bound to it, I haven't reached it yet - is the sheer fun of watching AI do the work that you asked for while you sit back and stare at the screen with a satisfied grin. And because it's so much fun, it's tempting to just enjoy the show. The downside is that watching AI work on one thing is time you could potentially have spent asking AI to do something else. Here we come back to the often cited recommendation to have several Claude Code (or Codex or Cursor etc) coding sessions running in parallel. My advice is congruous with the advice to have parallel sessions open, but complements it with the idea that you want to actually make good use of those sessions. To do that, you need to consciously resolve to Not Watch AI Work. At least I had to: the temptation to watch my AIs work was, and remains, quite strong for me.

Occasionally I still do it. Well, no one's perfect.

But I've also spent a lot of time - I'm embarrassed to admit how much exactly - thinking about Famous Open Problems (FOPs). These range from problems that are "locally famous" among people working in a specific area of math (critical exponents in percolation theory, the connective constant of the square lattice), problems that are "moderately famous" and enjoy relatively widespread recognition across all of mathematics (the sphere packing problem, the moving sofa problem, the Collatz conjecture, irrationality of odd zeta values), to universally famous problems such as P vs NP, or the FOPpiest of all FOPs, the Riemann hypothesis.

Anyone who has worked seriously on even non-famous open problems can tell you: they are hard! It's true; mathematics research is generally hard. I have tried and failed to solve many problems of even very humble pedigree that only few people will even care about if I were to solve them. Which may make you wonder: if even that is hard, surely spending significant time attacking real FOPs that so many other people have already tried and (so far) failed to solve is a waste of time and a recipe for frustration, potential career failure, and a possible spiral into insanity - right?

Well, yes and no. I want to offer here my perspective on what it is exactly that makes famous open problems so hard, what we mean when we say a problem is hard, and what are some good reasons why it may be a good idea for a researcher to think seriously about them after all.

What makes a problem "hard"?

Mathematicians like to talk about whether an open problem is hard or easy. To be successful in mathematics research you have to be able to develop an intuitive feel for how hard something is. Aim too low and you'll be consigned to publishing a lot of mediocre work. Aim too high and you'll publish nothing. As a mentor to PhD students you also have to know whether the problems you're suggesting to your PhD students are of a reasonable level of difficulty. Aim too high and they'll flunk out or end up switching advisors. Aim too low and they'll... probably still graduate, but maybe produce work they're less proud of than they might have been otherwise.

But what do we mean when we say that a problem is hard? I think there are different metrics one can apply, and they overlap partially but do not exactly coincide:

• A problem is deemed hard if a lot of people tried and failed to solve it. I call this "sociological hardness". This criterion is often applied by journals trying to decide if the work you're submitting to them is important enough to publish. It's relatively easy to apply (because it's usually easy to estimate how many people have worked on a problem by the size of the literature on it), but it's also at some level a lazy criterion. Certainly there are many insanely hard problems that few people have even made a serious attempt at solving, either because they are not that fundamental or interesting, or because for one reason or another the problem has not received much attention and does not loom large in our collective mathematical consciousness.

• A problem is hard if it requires genuinely novel ideas: it does not readily yield to the existing toolkit of techniques and ideas currently in our possession. I call this "conceptual hardness". Sometimes the conceptual hardness is of such a high level that no one even has a good idea of where to start looking for the right tools. (With the Riemann hypothesis I feel like that is the case, for example.)

• A problem is hard if it can be solved using ideas that are roughly in the ballpark of what is known, but in a way that requires a very large amount of effort, time, and persistence. Some well-known breakthroughs in mathematics, like the proof of the four color theorem and Kepler's conjecture, can be seen as cracking through that final barrier of difficulty by putting together ideas and methods that were largely known at the time (and expected to work as a way of arriving at a proof) but in a way that required a tremendous amount of patience and effort, and some extra ingenuity. These are remarkable achievements, but I feel like they belong in a different category from breakthroughs that were achieved by breaking through a conceptual hardness barrier.

Why I like to think about famous open problems

One reason that isn't why I like to think about famous open problems is "because I believe I might solve them". To think about a FOP for that reason would be crazy. But there are nonetheless other good reasons to do it. Here are the main ones for me:

• Because it's fun. Thinking about FOPs is typically more exciting than thinking about "ordinary" research problems. These problems are famous for a reason - they capture our imaginations and are genuinely very fun (sometimes addictively so) to think about.

• Because you learn from firsthand experience why the problem is hard and why it is important. Steve Jobs famously said "Don't be trapped by dogma - which is living with the results of other people's thinking." He was talking about a different context, but this applies to math research as much as to anything else. Our ideas of what problems are hard or important are often handed down to us by our seniors, and it takes time and patience to develop your own independent point of view. I used to believe that the Riemann hypothesis is important because I was told it was (usually with some pop-sci watered down explanation of why that was the case). It took me many years of study and thinking to arrive at my own personal set of beliefs about both the problem's hardness and its importance. These beliefs are a lot more nuanced than the watered down explanations I had read previously. In a sense, I feel I have a deep familiarity with and appreciation for the problem. That is quite satisfying - perhaps in a way almost as satisfying as it would be to know the actual solution to the problem.

• Because non-famous open problems are also hard. I mentioned above that research is generally hard, and that I failed to solve many of the non-famous open problems I tried working on, where the reward for solving them wouldn't be that great anyway. Well, if you're going to fail at something, wouldn't you rather fail at doing something super-exciting and ambitious than failing to do something that almost no one even cares about?

  Of course, I'm being (only slightly) facetious here. Obviously you shouldn't just be making choices about what to fail at! A level-headed person who wants to have a successful career and to contribute something meaningful to the world should naturally also plan to at least occasionally not fail. And for the not failing category, it's almost always better to work on open problems that are not famous. But I feel the point in the preceding paragraph is still a valid one, and suggests that, for the proportion of your research time that you allocate to doing things that will likely fail, you should occasionally GO BIG and work on FOPs.

• Because it leads to new research. One quality that famous open problems tend to have is that they stand at the confluence of a lot of exciting ideas and research topics, rather than being isolated questions that don't relate to anything else mathematicians care about. (One possible exception to this that I've had experience with is the moving sofa problem; while it does relate to other interesting things, those connections are a bit tenuous, and in fact advances on the moving sofa problem such as its remarkable solution by Jineon Baek in 2024 do not immediately translate to advances in other research areas. But I predict that will actually change over time, and his ingenious ideas will find surprising applications elsewhere.) For this reason, working on FOPs often results in making new discoveries that aren't as spectacular as solving the open problem you set out to solve, but are interesting in their own right. Many of my "ordinary" research papers came out of research that was motivated by working on problems that were a lot more difficult than the problem I ended up solving.

• Because it makes me smarter. Trying and failing to solve open problems (famous, or not) forces you to think very hard about a topic and explore different angles for attack. Usually in the process you learn a lot: about the problem itself, about what's been tried by other people, about connections to other fields, and much more. The knowledge you pick up may serve you well down the road when you do "ordinary" research, and produce tangible benefits you would not necessarily have expected. In other words, "failing" in a local sense can still lead to real successes later on.

• Because I can. One of the joys of being in academia is the freedom you have to pursue your own interests and to work on whatever you are passionate about. One should not take this mindset to an unhealthy extreme: even having tenure does not mean you should spend 100% of your time on research that is almost guaranteed to fail. I mean, technically you can and you'd still have a job, but having a job is a low bar to clear, and researchers who get tenure have been selected to not be the kinds of people who would be content to never publish anything again after getting tenure - that is precisely why giving such people tenure was a good idea in the first place.

• Because I feel compelled to. I was thinking about the Riemann hypothesis and other FOPs long before I had tenure. So the "because I can" tidbit doesn't tell the full story. In other words, thinking about FOPs may not necessarily be a rational decision, but may in fact be more of an unhealthy obsession. Would my career actually be more successful if I didn't spend as much time as I did working on FOPs? That's a real possibility.

Why you should think about famous open problems

I'm really bad at telling people what they should or should not do. What worked for me may or may not work for you, so I never feel like I can authoritatively give people advice even if it's based on my own experience. All I can say with confidence is: working seriously on famous open problems is not the waste of time that it may appear to be at a superficial glance. It's not a waste of time at all in my opinion, at least in the same sense that pursuing any passion or hobby is not a waste of time. As I hope my analysis above makes clear, there are solidly good reasons to do it. And there are some good reasons not to do it, and excellent reasons not to overdo it. And with that very confident verdict (that I'm sure makes me an excellent candidate to run for president), I will rest my case.

What I do:

• I never say please or thank you.

• Other than these intentional omissions, I use phrasing that's generally polite in tone. E.g., I often forego the direct imperative tense ("Change the implementation") in favor of a more indirect request-style formulation ("Can you change the implementation?")

• I almost always use proper grammar, spelling, and punctuation. If I notice a typo in what I wrote, I'll take care to correct it even knowing the AI can read it fine with the typo. I use correct capitalization (sometimes dropping it from proper names, especially when discussing technical issues - e.g., I may write "safari" and not always bother to capitalize such a term). I'll also admit to sometimes omitting the trailing period in a sentence or paragraph. I think I do this out of a combination of laziness and a desire to prove to myself that my pedanticism does have some sensible limits.

Why I do this:

• I believe the polite tone is useful to get good results from the AI. To be clear, I don't believe the AI "cares" how I talk to it in the sense of having actual conscious feelings. But since the AI has been trained to simulate human behavior, I do believe it may actually manifest behaviors such as vindictiveness, developing a grudge, or seeming to take offense when spoken to in a rude manner, and potentially being less helpful as a result. If you agree with this premise, it follows that, just as being nice to people in ordinary human interactions is generally believed to be not just morally correct but actually useful from the point of view of pure self-interest, I believe this will be the case as well when talking to AI. And it's not just about politeness per se: your conversational style can make a difference in signaling your intentions about the project you are working on. My theory is that using proper grammar and punctuation signals "I am a serious person doing serious work and I care about accuracy and things looking and working well", as opposed to a "hey bro, let's code up some sh*t" attitude. I believe the agent will respond to that, possibly in subtle ways that are difficult to detect or quantify.

• Speaking politely, and writing well, are good habits to nurture. If I get in the habit of speaking rudely to AI just because I think rudeness is appropriate in that context, one can easily imagine that that habit might someday bleed over into other areas of my life. I don't want that to happen.

• I am generally a person who cares greatly about writing, effective communication, and the meaning of words and other prose elements. Even if I did not believe the rationales described above (and I'm not actually sure how strongly I believe them - I'm certainly open to evidence-based arguments that politeness isn't as important as I think it is), I likely would continue to put more work into crafting my instructions to AI as proper texts than a lot of other people do, simply because I enjoy writing well-crafted sentences that convey my thoughts very precisely. For me, the act of writing is not just an act of communication, but an act of distillation, which helps me understand my own thoughts better than I did before. Making the effort to correct a typo or to add a comma that eliminates an ambiguity may seem like a waste of time when talking to AI - in a functional sense it may in fact turn out to be a waste of time - but I feel like there is still some intangible benefit that is produced by this small effort. (A contrarian thought: does this mean I actually have a mild undiagnosed case of OCD and am concocting grandiose philosophical theories to rationalize my behavior? Maybe, I can't say.)

So why don't I say please or thank you after all?

If you read my analysis above you may conclude that I am being inconsistent by deliberately refraining from saying please or thank you to AI. Maybe that's true in a small way; if politeness is important, then please and thank you ought to be a part of that, shouldn't they? I concede the point that there's a minor inconsistency here. So here is why I take this approach:

• I got into the habit of doing that at some point, so I'm sticking with it without necessarily believing strongly it's a good idea. People do that all the time. Actually, part of the reason I'm writing this post is to second-guess myself and revisit my own thinking on this issue.

• It's a waste of tokens. Sam Altman has gone on record expressing the view that saying "please" and "thank you" to AIs is a waste of tokens, energy and money. And tokens don't just cost money, they actually affect the quality of the output you receive and the work you get done, particularly in coding where you are trying to optimize the use of a context window of fixed size and need to take into account context rot.

• It feels weird. As much as we tend to anthropomorphize AIs (and as much as I genuinely enjoy conversing with them), I still want to maintain an emotional boundary that reminds me that, as of mid-2026, AIs are not, in fact, sentient entities. The "please" and "thank you" are, for me, a natural place where that boundary asserts itself - it genuinely feels weird to me to cross it. But I do feel a desire to. So when I find myself feeling the urge to use these common words of courtesy that I would employ in a similar conversation with a human, I have - up to now - forced myself to hold back. I regard it as a small act of discipline that essentially signals my desire to not be drawn into a closer emotional entanglement with AIs than I'd like to have at this point in time. Sure, they are fun to talk to and are amazing in so many ways. But they are algorithms running on a computer.

Conclusion: is my approach the right one?

• I have no idea! This analysis is pretty theoretical and speculative. I don't have hard evidence about which approach would work better or if any of this actually matters for anything. After writing this post, I think I will start occasionally saying please and thank you to my agent and seeing how that affects the dialogue.

• For me, the most compelling arguments in favor of, or against, using any given rhetoric style with AI, are utilitarian ones. You should use the style that gets you the best results. More precisely, you should use the style that optimizes for the things you care about: quality of work, but also money, and the overall high energy usage of AI, which causes real problems in our societies and is a legitimate thing to be concerned about.

• This isn't a moral or ethical issue. Anthropic seems to believe that it may be an ethical issue, at least when the concept of "being rude to your AI" is taken to an extreme that causes their models "apparent distress", which led them to give Claude the discretion to terminate conversations it considers abusive. Personally that's not how I see it.

• AI assistants have been known to cause serious damage to their users' infrastructure. I don't know if the user in that case was rude to their AI, so it's not clear if this sort of incident is relevant to the current discussion. But I'd like to think that my "be nice to your agent" attitude will make me less likely to be a victim of such behavior. Call it AI karma if you like.

• I am curious to know how others feel about this.

I will also write about AI, a topic that occupies much of my attention these days.

I use AI for math, coding and other things, but the writing is my own - no AI.