— 1. The Hilbert transform — In this set of notes we begin the theory of singular integral operators – operators which are almost integral operators, except that their kernel just barely fails to be integrable near the diagonal . (This is in contrast to, say, fractional integral operators such as , whose kernels go […]

Ever since the dawn of the "LLMs do math" era, many have independently come up with what I'll call the "Convex Hull Hypothesis" (CHH). Roughly speaking, CHH states that every AI-generated proof, no matter how complicated it may seem at first glance, is a simple combination of known mathematical ideas.

Formal verification is having a moment.

This is something we strongly advocate (and provide as a service).

We've put a 6-part explainer series together explaining vericoding. Here's the first:

What is vericoding.

#vericoding #formalverification

“The popular youtuber 3blue1brown presented a beautiful IMO riddle which AI systems could not solve back in 2025 (link […]), but six talented problem-solvers got full marks on. I wanted to propose a challenge: let us attempt to formulate mathematical problems which clever human mathematicians (or perhaps even school kids) can solve, but AI systems stumble upon! […]”

Pavel Etingof, Slava Gerovitch, “Mathematics of the Future“ Timothy Gowers, “How worried should we be about extinction risk from AI?“ Elchanan Mossel, “LLMs, Reasoning and Plagiarism“. (See also this related petition.) nihilunbounded, “Who Watches the Watchmen“ Sung-Jin Oh, “What does it mean for a mathematician to understand? A case study in the age of AI“ […]

a second millennium prize has hit the stochastic parrot

Every maths-related "proof" released by an AI company should be treated as though posted to arXiv by someone unaffiliated with any math department anywhere and with no existing track record of math research. If the latter would be assumed to be faulty or a quack at worst, why are we assuming better from tech companies' text generating linear algebra systems #math #themathpocalypseisf…

It's not solving the Millennium Prize problem, which is to show that Yang-Mills theory without fermions makes sense and has a "mass gap", meaning that it predicts a glueball of some nonzero mass. But proving Yang-Mills theory makes sense is a big first step. And getting fermions into the theory is also good. Note how they're working with all the key bigshots in this particular su…

Whoa!!!! Showing that two quantum field theories in common use actually make rigorous sense: Yang-Mills theory, and QCD, which is Yang-Mills theory plus fermions (namely quarks). This will be HUGE if true. For many decades, mathematical physics has been stuck on this problem. (1/2)

An internal model at OpenAI has produced a claimed refutation of ETH! Russell Impagliazzo, Mohan Paturi, and I have been asked to help organize an open and independent examination of this claim. 1/4

In the midst of this turmoil, we attempt to make sense of it all by trying to salvage what is dearest to us. This post is about what I instinctively rushed to salvage, what appears to be dearest to me as a mathematician

My intention here is to contribute to the discussion by pointing at an aspect of mathematics which I believe is under threat and has been forgotten in the rubble of this ongoing earthquake: the mystical aspect of mathematics. I am aware that using the word “mystical” in the mathematics community comes with some risks; yet I will try to defend it, because I think that this is one of the crucial asp…

Even experts debate what a formal proof actually establishes.

If experts question what a proof guarantees, how can ordinary users judge a project's claims?

Verification should be independently accessible, not something users must simply trust.

Most discussions about LLMs in mathematics seem to focus on their (rapidly advancing) proof discovery abilities and the effects these abilities may have on professional mathematics. I write here about a different use of these technologies, to make the following point: we have a professional duty to use LLMs to proofread our work.

“In our September 29 recommendations, we outlined an approach to the release of AI-generated mathematical output by AI labs. Since then, we have been asked by people from multiple labs and by several mathematicians about a specific aspect of the recommendations — that the development of human understanding following a release must remain organic and community led — and what it says about the quest…

[This is a guest post by the Advisory Group on Mathematics and Artificial Intelligence, crossposted from their site. This blog post was initially written in a different file format and converted using AI. — T.] In our September 29 recommendations, we outlined an approach to the release of AI-generated mathematical output by AI labs. Since […]

“We mathematicians know that published papers and books often contain mistakes: sometimes small, like a missing hypothesis or a wrong constant, and sometimes more serious, like a theorem that is false as stated. We are kind of used to this. […]”

[This is a guest post by Abhinav Nellore, Johannes Schmitt, Soledad Villar, and Rachel Ward – T.] OpenAI’s math write-ups at https://github.com/openai/math can be difficult to understand and can have incomplete citations. https://github.com/mathcommunity/oai-math-revised/ seeks to be an inclusive effort to improve these write-ups, and the broader math community–students, postdocs, professors, and…

“The 100+ reactions post, and the ensuing discussion, made me think it would be helpful for the community to have a rough qualitative snapshot of not only how we’re feeling and what we’re thinking, but also how that depends on some material factors that intuitively seem relevant. […]”

We should preserve the human ability to understand and explore mathematics, not preserve unsolved problems merely to protect a familiar way of working. The goal should be to enlarge public mathematical knowledge and understanding, whether the discoveries are made by humans, AI, or a collaboration between them. I also discuss competition, credit for AI contributions, and why restrictions on AI in e…

“The arXiv is full of papers now that include a section, usually either after the acknowledgments or at the end of the paper, entitled ‘AI disclosure’. I strongly object to this phrase and I think it is a tremendous mistake to use it. […]”

My papers have certainly benefited from peer review. It can be a lovely form of anonymous collaboration. But it can also get out of hand to the point where the referee wants to be a coauthor

The IAS has opened its application page for its membership program for the upcoming year. This by itself would be unremarkable — the program does this every year — but there is an additional statement included in the announcement: We realize that applying to positions in the current environment is stressful. To help alleviate this, […]

“This is a response to a recent post on Proofs and Prompts titled ‘Who will become the new mathematicians of Math 2.0?’ written by an Anonymous PhD student (who I will refer to as ‘Anonymous’ below). I would urge particularly all tenured and tenure-track mathematicians to read it because there are many students that think the same way, and are planning to leave academia as a consequence. […]”

I completely disagree with the notion that AI companies have hijacked our mathematical culture and turned it into something new and unrecognizable. They have just taken our existing culture to its extreme.

“Mathematics is a creative and emotional endeavor. We are driven by challenge and curiosity. We express our ideas carefully to be precisely understood. Our highest moments can fill us with exhilaration and awe. But how we do mathematics has evolved over time. We have moved away from pictures and informal arguments; we now have axioms and proofs. […]”

[This is a guest post by Caroline Turnage-Butterbaugh. This blog post was initially written in a different file format and converted using AI. — T.] Evidence in favor of the Riemann Hypothesis comes in many flavors. OpenAI’s proof of the quasi-Riemann Hypothesis is definitely one such flavor; however, in this post I will focus my […]

Learn about the SAIR Foundation and its developing platform, its AI models, mathematical research tools, and how research teams can apply for access.

Learn more: www.openmathmodel.org

SAIR is also seeking collaborators to develop AI models tailored more specifically to mathematics and is working to make the platform available at no cost to researchers at academic institutions.

The platform offers tools to help researchers explore mathematical ideas, discuss and summarize complex arguments, and work on formalizing mathematics using Lean.

The American Mathematical Society (AMS) applauds efforts like SAIR’s to develop a platform that incorporates norms around research standards and broadens access for mathematicians to conduct AI-assisted research.

The Science and AI Research (SAIR) Foundation recently launched its Open Math OS, an emerging platform designed to expand access to open AI models for mathematical work.

Exploring New Possibilities in AI and Mathematics: SAIR Launches Open Math OS

“I am writing neither to defend a complete ban on AI nor to celebrate its unrestricted use, but because I no longer know what rules I am expected to follow, what work will count as my own, or how today's decisions will affect my career. […]”

Mathematicians are humans, and humans have emotions. This essay scratches the surface of an intricate issue behind a very simple question: in what ways and to what extent are mathematicians’ reactions to AI driven by emotion?

The post describes a workflow for using AI in theoretical research that I gradually developed while working on these projects. I am especially interested in whether experienced mathematicians use similar methods, even if they do not describe them as prompts or formal workflows.

@tristanbuckmaster I often agree with mathematicians on AI, and with a lot of your guest post on Tao's blog. But I traced the "distasteful" quote back through the timeline, and I'm not convinced OpenAI contradicted itself, read in its own terms. I'd be glad to hear if I've missed something. https://blog.kolen.dev/link/buckmaster-distasteful.html #math #AI

The AMS is committed to connecting our community and fostering thoughtful, informed conversations as the impact of AI on mathematics continues to unfold.

Have a resource to recommend? Email: ai@ams.org

Explore resources: www.ams.org/ai/ai

The web page brings together resources on AI-assisted research, proof verification, mathematics education, emerging tools, and expert perspectives with opportunities to explore new ideas, learn from others, and engage with the evolving role of AI in mathematics.

The American Mathematical Society (AMS) has launched AI and Mathematics: Selected Resources, a curated collection to help mathematicians explore the opportunities, questions, and evolving discussions surrounding AI and mathematical discovery.