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• Post #4480889

  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...

• Post #4409552

  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...

• Post #4296015

  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...

• Post #4296014

  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/

• Post #4296013

  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/

• Post #4296012

  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...

• Post #4296011

  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...

• Post #4296010

  "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...

• Post #4296009

  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.

• Post #4296008

  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...

• Post #4119488

  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...

• Post #4067785

  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.)

• Post #4038755

  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

• Post #3835415

  #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...

• Post #3707020

  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...

• Post #2581820

  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...

• Post #2581819

  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...

• Post #2581818

  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...

• Post #2581817

  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...

• Post #2558872

  #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/

• Post #2500974

  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...

• Post #2500627

  Earlier this year, I sat down for an extended interview with Lex Friedman on a variety of topics in mathematics, proof formalization, and AI. The edited (and nicely illustrated) form of the video, clocking in at over three hours, is now available: https://www.youtube.com/watch?v=HUkBz-cdB-k

• Post #2336583

  In the field of cybersecurity, a distinction is made between the "blue team" task of building a secure system, and the "red team" task of locating vulnerabilities in such systems. The blue team is more obviously necessary to create the desired product; but the red team is just as essential, given the damage that can result from deploying insecure systems. The nature of these teams mirror each other; mathematicians would call them "dual". T...

• Post #2153520

  I experimented with using a more powerful AI (ChatGPT5 Pro) to attempt an open problem https://mathoverflow.net/questions/425509/sphere-with-bounded-curvature which is not quite in my area of expertise (specifically, differential geometry): https://chatgpt.com/share/68e85cba-7228-800e-8804-a0f41aa64e14 . The outcome was surprisingly nuanced, and illustrates a point I made in an earlier post https://mathstodon.xyz/deck/@tao/114501120421010793 that one has to measure the usefulness of a tool at...

• Post #2142223

  #IPAM (the institute for pure and applied mathematics) is now fundraising to ensure continuity of operations during this suspension of NSF funding. Information can be found at https://www.ipam.ucla.edu/news/nsf-funding-to-ipam-suspended/ and donations can be made at https://giving.ucla.edu/Campaign/Donate.aspx?SiteNum=136&Fund=61766C

• Post #2142212

  Earlier this year, the second #AIMO (artificial intelligence mathematical olympiad) concluded, with the winning team solving 34/50 in the final set of math problems (that had been selected to be harder for AI than the first AIMO). The competition was restricted to open source models and run with a limited amoutn of compute. The AIMO has now conducted a retest of these problems both for the top two teams from that competition (NemoSkills and imagination research), as well as OpenAI's o3...

• Post #2066681

  The Communications of the Association for Computing Machinery (ACM) has compiled a list of articles from leaders in the profession on the importance of federal investment in basic research in the technology sector. https://www.acm.org/media-center/2025/december/cacm-federal-funding

• Post #2026629

  Over at the Erdos problem website, AI assistance is now becoming routine. Here is what happened recently regarding Erdos problem #367 https://www.erdosproblems.com/367 : 1. On Nov 20, Wouter van Doorn produced a (human-generated) disproof of the second part of this problem, contingent on a congruence identity that he thought was true, and was "sure someoneone here is able to verify... does indeed hold". 2. A few hours later, I posed this problem to Gemini Deepthink, which...

• Post #1762347

  A creative way by Michael Brenner to integrate LLMs into undergraduate mathematical education: ask the students to teach the #LLM to solve the homework and exam problems via prompt selection. https://www.youtube.com/watch?v=om7VpIK90vE I can also recommend looking up Brenner's work in creating the class "Science and Cooking" at Harvard, in which lectures on physics and mathematical modeling were alternated with lectures from renowned chefs to analyze physical problems...

• Post #1722096

  We are transitioning in mathematics from an era of proof scarcity to an era of proof abundance, but our mathematical infrastructure and culture has not yet adapted to this. As mentioned previously, there is now a strong (and growing) impedance mismatch between the three core components of mathematical problem solving: proof generation, proof verification, and proof digestion. An early sign of this transition could be seen in the chaotic response to the initial "First Proof" challenge...