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  I wrote recently about how the collection of good, fruitful open problems is now being mined in a non-renewable fashion, leading to the potential scenario of these problems becoming scarce. This may seem unintuitive at first, since the set of possible problems one could ask is infinite. Perhaps the following analogy can help: a country or region can suffer a critical shortage of drinking water while simultaneously being surrounded by a massive ocean. One can easily generate any number of open...

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  A group of 25 Fields Medalists, including myself, have made a joint declaration on Math and AI: https://mathandai.org/ . We welcome additional signatories (similar to the Leiden declaration), as can be seen on the page. See also this article in the Economist announcing the declaration: https://www.economist.com/science-and-technology/2026/09/11/top-mathematicians-are-outraged-by-openais-methods

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  In my recent ICM talk in https://teorth.github.io/tao-web/slides/age-of-ai-icm-2026.pdf, I highlighted the five stages of proof development and maturation: proof generation, proof verification, proof exposition, proof publication, and proof canonicalization. This developmental cycle is somewhat analogous to the developmental cycle of a living being, from an infant, to a child, to an adolescent, and finally to an adult; cf. Cedric Villani's popular book "Birth of a Theorem&q...

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  I decided to convert one of the points in my ICM 2026 slides https://teorth.github.io/tao-web/slides/age-of-ai-icm-2026.pdf into a meme format.

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  Announcing the Palomar registry of Lean formalized mathematics: https://palomar-registry.org/ . See also my blog announcement at https://terrytao.wordpress.com/2026/08/18/palomar-a-registry-of-lean-verified-mathematics/ and the Lean Zulip channel at https://leanprover.zulipchat.com/#narrow/channel/621638-Palomar.

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  I had recently made an analogy between the developmental cycle of a proof and the developmental cycle of a child, and noted the strong cultural preference for a "traditional" parenting model in which a single set of parents is responsible for all aspects of the childrearing process, from conception all the way to adulthood. In a similar vein, we have a traditional authorship model in which a single set of authors is responsible from a proof all the way from generation up to at...

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  As an experiment, I directed a coding agent to sort through recent #arXiv posts for similarity in topic or keywords to my own research papers, with additional weighting for papers authored by my former collaborators or mentees (such papers are marked with a star in the lists provided below). I also asked the agent to assess the degree of AI assistance in each paper using robot emojis, with one emoji denoting minor assistance, two denoting major assistance, and three denoting near-total automati...

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  #SAIR is launching a Lean Kernel challenge on September 15, with pre-registration now open at https://competition.sair.foundation/competitions/lean-kernel-challenge . The precise rules and format are still being finalized, but the competition is aimed at finding the fastest way to perform (verified) computations in Lean of benchmark mathematical challenges, such as computing digits of pi or computing a hash.

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  The advent of capable AI tools has highighted a variant of Simpson's paradox https://en.wikipedia.org/wiki/Simpson%27s_paradox : a technological advance can improve the quality and volume of each individual's output, and yet the average quality (signal-to-noise ratio) of the aggregate output can deteriorate as a result. I can illustrate this phenomenon with a toy numerical model (all numbers here are made up to simplify the exposition). Let's take the task of solving a...

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  The first phase of the #SAIR / #LFMDB inverse Galois challenge https://competition.sair.foundation/competitions/igp24/overview will close in a week. One notable milestone of the competition so far: the inverse Galois conjecture has now been verified for all 25,000 of the transitive groups acting on 24 elements, by explicitly matching each such group with a degree 24 polynomial with that group as its Galois group! Prior to the competition, only 79 of these groups had been so matched. In some w...

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  When multiple researchers accomplish the same result, who is credited with priority? When I was a graduate student, the primary yardstick for priority was the date of publication in a peer-reviewed journal. Every so often, this would lead to a sordid drama in which a referee of a submitted paper was accused of using the ideas in that paper to rush out a competing paper to a faster journal. Within a decade, though, the arXiv preprint server, with its trusted submitted timestamp, supplanted the...

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  The long-awaited "Leiden Declaration" on Artificial Intelligence and Mathematics is now live and seeking signatories: https://leidendeclaration.ai/ This declaration stemmed from a workshop in Leiden University last September on "Mechanization and Mathematical Research", where it became clear how important it was in the age of AI to make explicit the goals and values of the mathematical community. Many of these goals have long been left implicit, being dissem...

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  Alberto Alfarano, François Charton, Yongzheng Jia, Kristin Lauter, Cathy Li, Emily Wenger and I are launching a second challenge at SAIR, this time focused on seeing how efficiently neural networks can execute simple modular arithmetic operations : https://competition.sair.foundation/competitions/modular-arithmetic-challenge/overview . A more detailed blog post is at https://terrytao.wordpress.com/2026/06/08/modular-arithmetic-challenge/

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  I wrote a blog post on the proposed rule changes for the administration of federal grants: https://terrytao.wordpress.com/2026/06/09/on-the-proposed-rule-changes-to-the-administration-of-federal-grants/

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  The results of the First Proof "second batch" are now out: https://1stproof.org/assets/docs/report.pdf Ten research-level questions (many being part of forthcoming papers) were tested locally by the First Proof team against four AI harnesses, including one from my UCLA team, and then the submissions refereed by experts. In aggregate, 7 of the 10 problems were deemed to have at least one publication-level solution generated between the four harnesses. The UCLA harness had a...

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  John Jones, Jen Paulhus, David Roe, Andrew Sutherland, and I have launched the third SAIR challenge, this time aimed at attacking the notorious inverse Galois problem: https://competition.sair.foundation/competitions/igp24/discoveries https://terrytao.wordpress.com/2026/06/16/third-sair-competition-inverse-galois-challenge/ . We are focusing on the degree 24 case, which is almost the first unsolved case (excluding the notorious problem of whether the Mathieu group M_{23} is a Galois group). Th...

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  "I have only made this letter longer because I have not had the time to make it shorter." (Blaise Pascal, often misattributed to Mark Twain) I have previously written about the evolving impedance mismatch between proof generation, proof verification, and proof digestion in mathematics. This has led to the following unintuitive breakdown of monotonicity, already noticed by Pascal as far back as 1657: it is now easier to generate long correct proofs than it is to generate short...

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  A new AI benchmarking math challenge, inspired by FirstProof: the Ramanujan Challenge https://www.ramanujanmachine.com/ramanujan-challenge/ , which is a call for AI-generated proofs of 10 Ramanujan-type numerical identities whose proofs are known to the authors of the challenge, but are currently not public. Submissions are accepted until August 1, 2026.

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  I have uploaded my slides for the public #ICM lecture talk I just gave at https://teorth.github.io/tao-web/slides/age-of-ai-icm-2026.pdf . The recording will likely be forthcoming in a day or two. In the meantime, I have decided to use an AI to collate and summarize a hundred or more previous posts, interviews, or videos I gave on the topic of AI, and also to "interview" me about any topics not covered in that past material. I instructed it to be somewhat hard hitting with...

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  The Erdos problems repository at https://github.com/teorth/erdosproblems initially tracked such data as whether a given Erdos problem was considered "open" or "solved" (with some other technical variants such as "decidable" or "falsifiable" which I will ignore here). Later on, we also added additional subcategories such as "solved (Lean)" which indicated their formalization status. In response to the recent (and likely enduring) phenomenon of A...

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  Pre-announcement: the website for "Mathematical Discourse", a new peer-reviewed video journal that will referee and publish mathematical *videos* rather than papers, is now online: https://www.mathematicaldiscourse.org/ . I am serving as one of the editors. (A more detailed announcement will be forthcoming.)

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  To mark Hong Wang’s (very well deserved) honor as one of the four recipients of this year’s Fields Medal, I created (with AI assistance) a visualization applet for the Kakeya needle problem: https://teorth.github.io/tao-web/apps/kakeya.html

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  #IPAM seeks program proposals from the mathematical, statistical, and scientific communities for long programs, workshops, and summer schools. Most program proposals are reviewed at IPAM’s Science Advisory Board meeting, held in November each year. Programs are selected on the basis of their scientific impact and contribution to IPAM’s goals. IPAM is committed to supporting a community where people of all backgrounds and points of view can engage, learn, and thrive. If you would like to discu...

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  I created my homepage https://www.math.ucla.edu/~tao/ back in 1997 (web.archive.org's earliest version is at https://web.archive.org/web/19970521060812/http://www.math.ucla.edu/~tao/ ). Over time, I added to it substantially, for instance creating detailed subpages to hold 300+ of my papers, links to my past travel and teaching, my CV, and the like. But it has stayed the same "Web 1.0" architecture since 1997, updated by hand using vi in my departmental UNIX account. (The one co...

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  A few weeks ago, I recieved a detailed referee report on a lengthy paper my coauthors and I submitted, with many suggested corrections (both minor and major) spread out over many sections. We agreed to each claim some of the sections to revise based on the recommendations of the referee, and (because we were using Github as our version control system) were able to easily merge together all the revisions and have them done in a matter of days. Today, I received a response from the referee who w...

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  I have recently made the point that out of three major components of the mathematical problem solving process - proof generation, proof verification, and proof digestion - the first two are being automated far more successfully than the third, leading to a new experience of "proof indigestion" in which proofs are being generated and even verified without being digested. It has been proposed that the solution to this is simply to automate the third component as well - for insta...

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  Throughout my career, I have often made particular efforts to go through and digest recent solutions (or partial progress) in notable developments in my fields of research, partly for my own edification, and also to post on my blog or other social media as a service to these communities (and, in some cases, to highlight the work of early career researchers). This was a rational thing to do in the era of proof scarcity, in which such proofs were rare and valuable, compared to the time available...

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  My talk on "New Mathematical Workflows" at Stanford last week is now online: https://www.youtube.com/watch?v=Uc2zt198U_U One key recommendation in the talk is to now de-prioritize the historical emphasis on competing to be the first to provide a proof for a given unsolved mathematical problem. When we were in the proof scarcity era, the "local" goal of obtaining any proof at all for a problem was fairly well aligned with the more "global" g...

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  #IPAM (The Institute for Pure and Applied Mathematics) is soliciting applications from industry to sponsor projects in its Research in Industrial Projects for Students (RIPS) program, in which talented undergraduate students, with a postgraduate mentor, work on industry-sponsored projects over a 9-week program while at residence at IPAM. Details about the program can be found at https://www.ipam.ucla.edu/all-upcoming-programs/rips-sponsor-information/

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  I've been working (together with Javier Gomez-Serrano) with a group at Google Deepmind to explore potential mathematical applications of their tool "AlphaEvolve", a successor of their earlier tool "Funsearch" that was publicly announced today: https://deepmind.google/discover/blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/ . Very roughly speaking, this is a tool that can attempt to extremize functions F(x) with x rang...