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[This is a guest post by Sophie Huiberts , being an expanded version of a post on her own blog . This blog post was initially written in a different file format and converted using AI. — T.] When you prove a theorem, you are imparted a responsibility to the field of mathematics. At minimum you need to explain what you did, including which steps required new ideas and which steps are already well-known. More substanti
OpenAI’s massive new math drop: [This essay was written in extreme haste before a very long wifi-less flight; please forgive typos.] Part One: My take The real news here isn’t the result; it’s not what we were told. 1. AI once tried to be a science. Now we get stuff like the completely vague report from OpenAI below: “Same procedure”? “Using an unreleased model”? This would never pass peer review. We don’t know what
[This is a guest post by Ben Antieau , cross-posted from his blog . — T.] There was no personal problem, no world problem, whose eloquent solution did not exist— somewhere in some hexagon. —Jorge Luis Borges, “The Library of Babel” [1] New repository I am writing to announce Hexagon , a new repository for research works in the mathematical sciences and theoretical computer science. This has been a joint effort of man
[This is a guest post by Thomas Bloom , crossposted from the Erdős problems forum . This blog post was initially written in a different file format and converted using AI. — T.] AI is changing everything, for better and/or worse, and the rate of change is dizzying; in recent months this has been particularly evident in mathematics, where AI has gone from being essentially useless to helping solve some of the hardest
Terence Tao used a SAIR talk to criticize AI model companies for accelerating mathematics without knowing what comes next, saying, "We must slow down." He described a normal proof lifecycle—proposing a solution, mechanical verification, exposition, peer review, and textbook incorporation—and said AI is racing through generation and Lean formalization while the later stages stall, creating "proof indigestion": proofs
[This is a guest post by Diego Córdoba and Luis Martínez-Zoroa . This blog post was initially written in a different file format and converted using AI. — T.] Euler formulated the equations for an inviscid incompressible fluid more than 250 years ago. Published in his 1757 memoir , they belong to the earliest systems of partial differential equations in mathematical physics, following d’Alembert’s work on the wave eq
[This is a guest post by a coalition of Caltech mathematicians: the original organizers of Mathathon (Alvan Arulandu, Andrea Li, Avni Garg, Brian Zhao, Caiman Moreno-Earle, Sathvik Redrouthu), two coauthors of the Open Letter about the Mathathon (Dylan King, Tasmin Chu), and other members of the Caltech community (Mayla Ward, Merrick Hua, Robert Joseph George, Sergei Gukov, Shaowu Zhang, Tony Yue Yu, Vihaan Dheer). T
[This is a guest post by Tapio Schneider . This blog post was initially written in a different file format and converted using AI. — T.] [This will be cross-posted on the CliMA blog .] The apparent proof of finite-time blow-up of the forced Navier-Stokes equation, announced by OpenAI on September 8, has brought into focus a debate about the role of AI in mathematics. The proof was produced with a system of some 10,00
Last year I co-founded the Foundation for Science and AI Research (SAIR) with some private donors to create a non-profit organization that could support responsible uses of AI in mathematics and the other sciences, independent of the major AI companies. Initially, our resources were rather limited, and our main activities consisted primarily of podcasts , short events , and competitions . However, we have been slowly
On September 8, OpenAI said 10,000 of its agents, running for 88 hours, had produced a 166-page proof regarding the Navier-Stokes equations — which describe fluid movement and have remained an open question since 1934. It's one of the seven Millennium Prize Problems established by the Clay Mathematics Institute in 2000, each carrying a $1 million prize; only one has ever been awarded. OpenAI has said it won't be clai
A statement signed by 25 Fields medalists, posted by Terence Tao on September 11 under the title 《人工智能在数学中的严重错位》, argues that AI companies' focus on solving famous mathematical problems seriously diverges from the community's goals of conceptual understanding, method development, and knowledge transmission. Former OpenAI VP of post-training Liam Fedus responds that today's AI limits should not frame future possibilit
[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 produced an AI-generated solution to the Navier–Stokes existence and smoothness probl
Epoch AI says FrontierMath Tier 4 is now "saturated": by its accounting, when attempts across different models and different time periods are accumulated, every Tier 4 problem has been correctly solved by an AI at least once. OpenAI reports a score of 97.6% for GPT-6 Astra, which solved the one problem that had never been cracked before. When Tier 4 launched on July 11, 2025, the top score on the board was only about
Twenty-five Fields Medalists, with Terence Tao and Deng Yu among the first signers, issued an urgent public statement arguing that the way AI companies are pursuing mathematics is now in "severe misalignment" with the goals of the mathematical community. The statement explicitly does not call for stopping AI or deny its achievements. It raises three concerns: AI is solving problems faster than the community can diges
[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 Gukov, Terence Tao, and myself. The Andrews–Curtis conjecture is one of the most pro
[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 singularity in fluid dynamics, we were finally ready to send our work on Euler equations (without for
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 make these equations develop singularities
In recent months there has been a proliferation of AI-generated proofs of various old and new results, some of which have been formalized in the proof assistant language Lean. However, checking that a given Lean repository actually proves the claimed statement is somewhat non-trivial, especially for an audience which is not expert in the use of Lean: one has to first check that the claimed formal Lean statements have
DeepMind’s Pushmeet Kohli on AI’s Scientific Revolution Pushmeet Kohli leads AI for Science at DeepMind, where his team has created AlphaEvolve, an AI system that discovers entirely new algorithms and proves mathematical results that have eluded researchers for decades. From improving 50-year-old matrix multiplication algorithms to generating interpretable code for complex problems like data center scheduling, AlphaE
This post concerns the following conjecture of Sendov , as well as its strengthening by Phelps–Rodriguez : Conjecture 1 (Sendov’s conjecture) Let , and let be a degree polynomial with all zeroes in the unit disk. Then for every zero of , there exists a critical point of with . Conjecture 2 (Phelps–Rodriguez conjecture) Let , and let be a degree polynomial with all zeroes in the unit disk. Then for every zero of , the
Terence Tao discusses the Heil-Ramanathan-Topiwala (HRT) conjecture, which asserts that a nonzero function with reasonable decay cannot satisfy a finite linear relation among its time-frequency shifts. He notes that the conjecture holds when all shifts are frequency shifts or collinear. For the lattice case, commutativity allows a diagonalization that reduces the relation to a pointwise equation, using restrictions t
Tao, 51, an Australian and American mathematician recognized for his contributions to partial differential equations, combinatorics, harmonic analysis, and additive number theory, made the remarks in his lecture, "Mathematics in the Age of AI," delivered at the International Congress of Mathematicians (ICM) last weekend. His speech drew widespread attention from the global mathematics community. He began by reflectin
The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows. Conjecture 1 (Jacobian Conjecture) Let be a polynomial map in complex variables, whose Jacobian is a non-zero constant. Then is invertible (with polynomial inverse). The condition that the Jacobian is non-zero is equivalent to being locally invertible. (The implication of local invertibility from non-vanishing Jacobian
I believe that the creation of visualization apps to illustrate mathematical or scientific concepts is a particularly favorable use case for modern coding agents, as many of the downside risks attached to other LLM use cases are limited: Not mission-critical . As such apps are not authorative sources of truth and only used for secondary purposes, a small positive error rate in the output can be acceptable. Stand-alon
Terry Tao demonstrates how coding agents facilitate converting code-generated figures into interactive applets, using Figure 1 from his recent paper on the Gilbreath conjecture with Chase and Hunter. The figure plots exact and Monte Carlo simulated values of a sequence called the "Gilbreath expectation sequence," which arises from a Gilbreath array: an inverted pyramid with top entries as independent exponential rand
I am finding the newly revealed capability to code old applet ideas into reality to be very tempting to sink more time into, though I am certainly encountering the common “vibe coding” experience that the process can produce something that superficially resembles a finished product well before a satisfactory level of testing and review has been completed; indeed, it is the review process which is now the most time-co
I have been interested in machine-assisted ways to do and teach mathematics from as far back as 1999, when I started coding several applets in Java 1.0 , both for my complex analysis and linear algebra courses, as well as to visualize various mathematical objects I was interested in (such as honeycombs or Besicovitch sets). This was moderately successful; but the applets were time-consuming to program. Eventually, th
Terence Tao argues that generating and verifying proofs is still incomplete until results are digested and connected to the field.
In a Nature interview, Tao separates current research usefulness from hype and explains the growing role of formal verification and human judgment.
UCLA Computer Scientists, Mathematician Terence Tao Awarded Laude Moonshots Grant to Build AI System as Research Partner Project aims to unlock progress in physics, cryptography and other challenging fields The multidisciplinary UCLA team will lead the Accelerating the Queen of Sciences project to develop an AI system capable of thinking like a mathematician and accelerating scientific breakthroughs. A team of six UC
Tanya Klowden and Terence Tao examine how AI and formalization may change the purpose, practice and social role of mathematics.
"AI Has Overwhelmed Human Exam Questions," A Conversation Between Terence Tao and Mark Chen: Large Language Models Are No Longer "Math Underachievers" Recently, at a seminar hosted by the Institute for Pure and Applied Mathematics (IPAM), Fields Medalist Terence Tao and Mark Chen, the Chief Researcher at OpenAI, participated in a "fireside chat." The two discussed the leap - forward progress of AI in the past year an
Much of the excitement around the news has stemmed from the adjudicator of these AI-written proofs: Terence Tao, a UCLA professor who is widely considered to be the world’s greatest living mathematician, reported The Atlantic. When Tao was interviewed about what AI can offer mathematics, he was more tempered. The AI-generated Erdős solutions are impressive, he said, but not overwhelmingly so: The bots have functional
[This is a guest post by Boris Alexeev . Now over to Boris.] I’m here to tell you about various exciting developments centering on Erdős problems, especially involving the formalization of old and new mathematics using artificial intelligence. Background As is well known, Paul Erdős was a prolific mathematician of the 20th century who posed an extraordinary number of conjectures. Around May 2023 , Thomas Bloom set up
Google DeepMind’s AlphaEvolve AI is accelerating mathematical research at an unprecedented scale, according to a new paper published this week by collaborators including renowned mathematician Terence Tao. The research shows how the AI agent tackled 67 challenging problems, rediscovering top solutions and finding novel constructions for several long-standing challenges. The work demonstrates a powerful new method for