Uri Bader, Professor at Maryland University and Weizmann Institute Math exists. We explore it.We do not invent math. We invent ways of understanding and communicating it. We choose which parts of it to study, and we develop an adequate language for that that suits us, humans. But we explore an objectively existing domain. Empirical fact: our abstract […]

[This is a guest post by Steven Strogatz. This blog post was initially written in a different file format and converted using AI. — T.] WIRED reported today that I began to cry while talking about AI and mathematics. But the article didn’t explain what moved me. The truth is, I’m not entirely sure myself. […]

[This is a guest post by Silvia De Toffoli and Eamon Duede. This blog post was initially written in a different file format and converted using AI. — T.] Silvia De Toffoli (University School for Advanced Studies IUSS Pavia) Eamon Duede (Princeton University and Purdue University) On September 8th, 2026, OpenAI announced that it had […]

Parallel to the other crowdsourced resource drive on this blog, I would like to collect books, writings and other resources that discuss or explain the purpose(s), value(s), and nature of mathematics. As seen in reactions to recent events, one can see many oversimplified or inaccurate perceptions of mathematics, such as The question of what mathematics […]

[This is a guest post by Claire Voisin. This blog post was initially written in a different file format and converted using AI. — T.] Hodge classes can be defined on any compact complex manifold . They are rational Betti cohomology classes on of even degree (eg combinations with -coefficients of classes of oriented codimension […]

AHM communications group This is a short announcement about the birth of the Association for Human Mathematics (AHM), an organization run collectively by mathematicians at all career stages. Its main goal is to protect mathematics and its practitioners from the jaws of AI companies, whose actions are increasingly destructive to our discipline, to the environment, and to […]

Manuel Rivera, Associate Professor at Purdue University I’ve been recently following some interesting debates about the future of mathematics in the age of AI. I think many discussions about how AI is “disrupting” or “changing” (pure) mathematics tend to conflate, or at least fail to distinguish between, three interlocked interpretations of mathematics: (1) Mathematics itself, […]

[This is a guest post by Burt Totaro. This blog post was initially written in a different file format and converted using AI. — T.] These are strange times for mathematicians. It now seems possible that AI companies will burn through vast resources in order to prove some new fact about the Hodge conjecture. I’d […]

I am proud to be among the list of 25 initial signatories — all Fields Medallists — to the declaration below, which grew out of discussions between ourselves over the last week. We have also posted our declaration on this web page, and (similarly to the Leiden declaration) invite further signatures. (It is unfortunate that […]

[This is a guest post by Andreas Thom. This blog post was initially written in a different file format and converted using AI. — T.] When I woke up on August 1st, 2026, I had received a few emails from colleagues asking for my opinion on a remarkable result that had circulated the previous day. […]

[This is a guest post by Lucas Fagan. This blog post was initially written in a different file format and converted using AI. — T.] I am excited to announce the Andrews–Curtis Conjecture Challenge, which opens today. This challenge is a collaboration between the SAIR Foundation and the Math-AI group at Caltech, organized by Sergei […]

Galyna V. Livshyts, associate professor at Georgia Tech When you realize that your opinion differs from seemingly everyone else’s, your first instinct is to keep quiet. But that is the wrong instinct. So I shall speak up. Imagine two high school kids taking a quiz that involves computing a derivative. They finish almost simultaneously. Suddenly, […]

Oren Hartstein, undergraduate at Columbia University and author of a blog LLMs have become shockingly good at math. They solve competition problems, assist with research, and are improving fast enough that some fear human mathematicians are becoming marginal. I want to make the opposite critique: LLMs aren’t nearly good enough at math, and the way […]

[This is a guest post by Anima Anandkumar. This blog post was initially written in a different file format and converted using AI. — T.] The last few days have been a flurry of activity. After months of sleepless nights and diving deep into different mathematical and computational tools to tackle the open problem of […]

Given current events, I think it is worthwhile to start collecting useful online resources with regards to AI and mathematics in general. I will start a list below, but ask contributors to contribute further links in the comments. Note that while I personally do not agree with 100% of the content of every single link […]

This is Jessica. If you’ve ever reviewed a heavily AI-generated paper, you know how unrewarding it can be. For example, last week I spent a morning reviewing a paper for a well-respected general science journal, which sounded interesting but was … Continue reading →

Robert Wegner, PostDoc at Karlsruhe Institute of Technology I have the impression that recent advances in the deployment of AI for the purpose of mathematical research have made many people uncomfortable. I want to present my personal view on these matters, which is more optimistic. Let me get the following assumption out of the way: […]

Mathathon Organizers: Alvan Arulandu, Sathvik Rodrothu, Caiman Moreno-Earle, Avni Garg, Brian Zhao. On September 4th, we announced Mathathon, an event allocating compute credits across 100 teams of mathematicians tasked with solving and presenting an open problem in 40 hours. As of September 10th, we have received over a thousand applications from undergraduates, Ph.D. students, postdocs, […]

A coalition of current and former Caltech mathematicians Note: The Mathathon organizers have published a response, available here. On October 30th, 2026, Caltech will begin hosting the Caltech Mathathon, a math hackathon in which participants are encouraged to solve open research problems by prompting LLMs. Two prominent AI companies, Anthropic and OpenAI, will supply participants […]

Orr Shalit, Professor at the Technion There is nothing new under the sun. Yes, we are living through a technological revolution of dramatic consequences. We who have lived through the birth of personal computers, the rise of the internet, the invention of smartphones, the disruptive eruption of social media, the promise of mass open online […]

Seewoo Lee, postdoc at EPFL. I want to share some thoughts on (auto)formalization based on personal experience, and propose some possible positive and negative usages of it. Disclaimer: I mostly focus on formalizations in Lean, due to my limited experience and knowledge. But most of the discussion can be applied to formalizations using other proof […]

There’s some exciting very recent work by Alpöge and Buckmaster, building upon prior work by Córdoba and Martínez-Zoroa, in the general topic around the infamous global regularity problem for the incompressible three-dimensional Navier-Stokes equations. It is now widely expected that it should be possible to construct smooth initial data and smooth forcing term that would […]

The θ(p꜀) = 0 conjecture is solved in all dimensions. In 2024 Gady Kozma and Shahaf Nitzan showed how to derive the dying percolation conjecture from a proposed conjecture about percolation on general graphs. (I briefly discussed it in this … Continue reading →

I woke up yesterday with the following thoughts, which are probably either obvious or dumb. A central thesis that many readers, including me, took from Douglas Hofstadter’s Gödel Escher Bach when young was that the secret of intelligence (and therefore, of AI) was going to have a lot to do with self-referentiality and “strange loops.” […]

Jeremy Horpedahl tells the story: Nicholas Polson has, by my count using his SSRN page, already written 258 working papers in 2026 alone. He’s already written (or at least published to SSRN), six papers today, August 26, 2026. OK, but … Continue reading →

Luke Ford writes: [Regarding] the replication crisis, researcher degrees of freedom, and the gap between what statistical methods claim to establish and what they can actually support . . . Looking at the current state of the social sciences, do … Continue reading →

Just send me your goddamn prompt.

Mathematical Discourse is a new online, peer-reviewed mathematics journal, publishing videos of mathematics research talks of the highest quality. Our inaugural scientific and editorial boards are listed at the end of this message. Our goal is to promote and nourish a culture which values communication as an essential part of the research process. Giving a […]

A frequent commenter pointed me to this paper, “Sport and longevity: an observational study of international athletes.” All I can say is . . . Wow! This paper is an absolute clinic in bad quantitative social science research. I’ll leave … Continue reading →

Ben Krause, Hamed Mousavi, Joni Teräiväinen, and I have just uploaded to the arXiv our paper Quantitative bounds for sets lacking polynomial progressions with shifted prime difference. The purpose of this paper is to obtain quantitative versions of this theorem of Wooley and Ziegler: Theorem 1 Let be polynomials of one variable with integer coefficients […]

For the sake of anyone who might read this blog post in the distant future (a month from now, say), let me mention that I am writing it a few days after OpenAI announced that it had solved ten major problems in mathematics and theoretical computer science, including the first construction of a non-sofic group, […]

Whatever you’ve been writing to me to ask if I’m aware of: yeah, I’m aware of it. In particular: Anyway, about the AI stuff. I don’t know whether this is literally our last year alive—I doubt it—but it’s pretty clearly the last year of math and theoretical computer science research in the style we’ve known […]

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 […]

The Ramanujan challenge Ido Kaminer shared with me the following information about The Ramanujan Challenge for AI, and I am happy to share it with the readers of this blog. The challenge page is at ramanujanmachine.com/ramanujan-challenge; Here is the The … Continue reading →

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 […]

Quadratic functions are the workhorse of the analysis of iterative algorithms such as gradient-based optimization. They lead, in discrete and continuous time, to closed-form dynamics that treat all eigensubspaces of the Hessian matrix independently. This leads to simple math for understanding convergence behaviors (maximal step-size, condition number, acceleration, scaling laws for gradient descen…

It’s been over three years since my last post on this blog and I have sometimes been asked, understandably, whether the project I announced in my previous post was actually happening. The answer is yes — the grant I received from the Astera Institute has funded several PhD students and a couple of postdocs, and […]

Among continuous optimization problems, convex problems (with convex objectives and convex constraints) define a class that can be solved efficiently with a variety of algorithms and with arbitrary precision. This is not true more generally when the convexity assumption is removed (see this post). This of course does not mean that (1) nobody should attempt...

I am very happy to say that I have recently received a generous grant from the Astera Institute to set up a small group to work on automatic theorem proving, in the first instance for about three years after which we will take stock and see whether it is worth continuing. This will enable me […]