— 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.

“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“ […]

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…

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. […]”

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 […]

“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.

[This is a guest post by Tristan Buckmaster. This blog post was initially written in a different file format and converted using AI. — T.] This blog post expands on a theme mentioned at the end of my recent talk at the Institute for Advanced Study on October 19. I would like to thank Matei […]

“We study mathematics for pretty much the same reasons that we study astronomy. The stars and planets are already out there, and, despite millennia of human observation, there is more we don’t know about them than we do. […]”

“Rapid advances in AI are transforming research, and for the growth of mathematical knowledge, this is thrilling. For our profession, however, this raises critical questions: Will there be a role for human mathematicians to play? How should mathematicians be trained and evaluated? […]”

What machines have: colossal knowledge and massive compute power. What humans have: intuition for beauty and elegance. The ultimate question is whether colossal knowledge and massive compute power wins over human intuition. If no, we only face some big changes. If yes, then human mathematics will degrade into exotic art.

In the spirit of my recent “Math 2.0” talk on new ways to use AI assistance to enhance the human understanding of mathematics, rather than substitute for it, I am using this post to propose an experimental challenge, inspired by the recent activity around excluding Landau–Siegel zeroes, as discussed in this previous post. There is […]

“I am mathematics PhD student. Up until recently, I fully intended to pursue a career in academic research. I deeply love mathematics, and I entered graduate school because of this. However I have made the decision to leave academia once I finish my doctorate and I am heartbroken; I would like to explain why. […]”

We present a theoretical framework to understand mathematics, and the disruptive effect of AI in our field. Treating mathematics as an economy, we borrow ideas from the theory of value to uncover the hidden dynamics of prestige, attribution, and the job market.

I don't know how it will all turn out, so I certainly can't promise you anything. But I do know that “understanding” is a human activity — something computers cannot do for us (even if they solve problems better than we do, it serves no purpose unless there are humans who actually understand).

For a given prime , the Legendre symbol is defined as follows: if is a quadratic residue modulo , then ; if is a quadratic non-residue modulo , then ; and if divides , then . Let be an integer that is not a perfect square. (One could require to be a fundamental discriminant if […]

[This is a guest post by Lukas Liehr, João Pedro Ramos and Mitchell Taylor – T.] We are launching the Bourgain Project, a community-driven initiative to revisit, expand upon, and explore the body of work of Jean Bourgain, one of the most prolific mathematicians of his time. The goal of this initiative is to develop […]

We provide a proposed set of guidelines for PhD students and their supervisors. Given that PhD students are the most affected, both professionally and psychologically, by the advent of AI in mathematics, such guidelines, even if they may soon be obsolete, could be useful to them, their advisors and graduate schools.

“This note is based on the assumption that the adoption of AI in mathematics is already inevitable. To paraphrase the principle of Chekhov’s gun, if we strive to resolve a conjecture while inhabiting a world that contains a button that, if pressed, will promptly resolve said conjecture, someone will indeed press the button. […]”

There are many ways in which AI is generating uncertainty in mathematics. We are not sure how it will impact our jobs, it is raising questions about how it might affect academic training, and it is not clear how it might change the job search process and how we evaluate research. This is by no means an exhaustive list. In this post, I will focus two another related questions, which are: “Is there…

“It has been two months since we started Proofs and Prompts. Once again thanks everyone for making it an active platform, we hope the conversations happening here will help build what mathematics will look like tomorrow. We take advantage of this new post to announce two new features that we would like to launch: […]”

[This is a guest post by Francesco Maggi. This blog post was initially written in a different file format and converted using AI. — T.] Large language models (LLMs) and the research systems built around them (and commonly known as “artificial intelligence”) have demonstrated remarkable abilities in generating mathematical proofs. While the quality and limits […]

“A small episode from this summer keeps coming back to me. At the National Academies meeting Organizing Mathematical Knowledge in the Age of AI and Formalization, David Spergel discussed Japan’s participation in the Department of Energy’s Genesis Mission. […]”

An open problem list is a bullseye painted on your field.

[This is a guest post by Shaoming Guo – T.] Several math colleagues and I are preparing to launch a new journal. As we considered which values we wanted the journal to uphold, we realized that there may be many other values worth exploring as a community, potentially through the creation of additional journals. We […]