Terence Tao
Mathematician at UCLA, working mainly in harmonic analysis and partial differential equations. Writes What's new, a long-running blog on their research, open problems and expository notes.
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What they believe
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Most of the value that mathematics and mathematicians contribute to the world will not be replaced by AI, though it could be magnified by it.
Most of the value that flows into the world from mathematics and the work of mathematicians will not be replaced by AI, but could be magnified.
terrytao.wordpress.comSaid 26 Sept 2026This on korrents
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Mathematics is entering a period of turbulence that will stress-test its implicit values and practices rather than its foundations of truth.
I believe that we are now entering a era of comparable turbulence in mathematics. This time, though, what is being stress-tested is not our foundational framework for mathematical truth, but rather the largely implicit framework of mathematical values and practices: what we consider a contribution to be, what we reward, what we regard as understood, and who - or what - we regard as having done the work.
arxiv.orgSaid 17 Aug 2026This on korrents
The bold line is korrents’ wording; the quote is theirs. These 2: what they wrote before what they said aloud, one per source, newest first.
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From one piece A value-based approach to mathematics 3 beliefs · terrytao.wordpress.com
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korrents.com
The value of a body of knowledge comes from the process of engaging with it, not just from its end products.Their words
The value of math, like so many other forms of knowledge, also comes from the process, not just the product.
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korrents.com
Improper use of AI offloads a person's thinking and learning, which mortgages a field's future development.Their words
Improper usage of AI offloads thinking, understanding and learning, thus mortgaging future mathematical development.
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Their words
Most of the value that flows into the world from mathematics and the work of mathematicians will not be replaced by AI, but could be magnified.
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From one piece Mathematics in the age of AI 5 beliefs · arxiv.org
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But passing an automatic filter is not a substitute for community acceptance; I do not believe that human referees can be removed from the publication process.
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korrents.com
Excessively AI-polished proofs can remove natural friction and become easy to read but hard to learn from.Their words
An excessively AI-polished proof may sand away both the "artificial" friction (typos, awkward phrasing, disorganization) and the "natural" friction, leaving a text that is easy to read and hard to learn from.
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Their words
Consequently, excessive optimization for one or two goals may cause the many previously aligned goals of mathematics to diverge from one another;
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I argue that it will become necessary to make these unwritten goals of mathematics much more explicit; but once we have thoroughly examined and codified them, our community will emerge stronger and more resilient than before.
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Their words
I believe that we are now entering a era of comparable turbulence in mathematics. This time, though, what is being stress-tested is not our foundational framework for mathematical truth, but rather the largely implicit framework of mathematical values and practices: what we consider a contribution to be, what we reward, what we regard as understood, and who - or what - we regard as having done the work.
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From one piece From Solvers to Research: Large Language Model-Driven Formal Mathematics at the Research Frontier (with 18 co-authors) 4 beliefs · arxiv.org
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korrents.com
AI systems are now capable of meaningful contributions to formal mathematics, not only informal problem-solving.Their words
These developments inform our position that AI systems are now capable of meaningful contributions to formal mathematics, not merely informal problem-solving.
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Despite these strides, we argue that current AI4Math systems still largely operate as solvers, excelling at isolated, well-defined proof generation rather than as researchers capable of expanding the boundaries of mathematical knowledge.
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However, these LLM reasoners that generate informal reasoning in natural language are fundamentally limited by the lack of precise, machine-checkable semantics, making their outputs prone to hallucinations [Huang et al., 2025b] and precluding autonomous verification, a prerequisite for tackling open-ended mathematical research.
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We argue that the next leap in AI4Math systems requires a decisive shift from predefined problem-solvers to research agents that can address frontier mathematical challenges with rigorous formal mathematical reasoning.
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From one piece Terence Tao – How the world’s top mathematician uses AI 3 beliefs · youtube.com
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Their words
So, you you can't look at any given scientific achievement purely in isolation and give it an objective grade without being aware of the context both in the the past and the future. And so it it it may never be something that you can just reinforcement learn the same way that that you can for much sort of more localized problems.
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So, we're now in a situation where suddenly people can generate thousands of theories for a given scientific problem. And now we have to to verify them, evaluate them and this is something which we we have to to change our structures of science to actually sort this out.
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So, I think AI has basically driven the cost of idea generation down to almost zero in a very similar way to how the internet drove the cost of communication down to almost zero, which is an amazing thing, but it you know, it it it doesn't make it doesn't create abundance by itself.
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Beliefs
Korrents What they believe 54 beliefs — each backed by an exact quote.
Each is a — compiled by korrents.com, not by them: the one-line wordings are korrents', the quotes are theirs.
Recent
Most of the value that mathematics and mathematicians contribute to the world will not be replaced by AI, though it could be magnified by it.
Most of the value that flows into the world from mathematics and the work of mathematicians will not be replaced by AI, but could be magnified.
A value-based approach to mathematics Said 26 Sept 2026
Improper use of AI offloads a person's thinking and learning, which mortgages a field's future development.
Improper usage of AI offloads thinking, understanding and learning, thus mortgaging future mathematical development.
A value-based approach to mathematics Said 26 Sept 2026
The value of a body of knowledge comes from the process of engaging with it, not just from its end products.
The value of math, like so many other forms of knowledge, also comes from the process, not just the product.
A value-based approach to mathematics Said 26 Sept 2026
Show 51 more
Mathematics is entering a period of turbulence that will stress-test its implicit values and practices rather than its foundations of truth.
I believe that we are now entering a era of comparable turbulence in mathematics. This time, though, what is being stress-tested is not our foundational framework for mathematical truth, but rather the largely implicit framework of mathematical values and practices: what we consider a contribution to be, what we reward, what we regard as understood, and who - or what - we regard as having done the work.
Mathematics in the age of AI Said 17 Aug 2026
The unwritten goals of mathematics must be made explicit, and examining and codifying them will leave the community stronger and more resilient.
I argue that it will become necessary to make these unwritten goals of mathematics much more explicit; but once we have thoroughly examined and codified them, our community will emerge stronger and more resilient than before.
Mathematics in the age of AI Said 17 Aug 2026
Excessive optimization for one or two mathematical goals can cause previously aligned goals of mathematics to diverge.
Consequently, excessive optimization for one or two goals may cause the many previously aligned goals of mathematics to diverge from one another;
Mathematics in the age of AI Said 17 Aug 2026
Excessively AI-polished proofs can remove natural friction and become easy to read but hard to learn from.
An excessively AI-polished proof may sand away both the "artificial" friction (typos, awkward phrasing, disorganization) and the "natural" friction, leaving a text that is easy to read and hard to learn from.
Mathematics in the age of AI Said 17 Aug 2026
Automatic filters cannot replace community acceptance, and human referees cannot be removed from mathematical publication.
But passing an automatic filter is not a substitute for community acceptance; I do not believe that human referees can be removed from the publication process.
Mathematics in the age of AI Said 17 Aug 2026
The next advance in AI for mathematics requires shifting from predefined problem-solvers to research agents that address frontier challenges with formal reasoning.
We argue that the next leap in AI4Math systems requires a decisive shift from predefined problem-solvers to research agents that can address frontier mathematical challenges with rigorous formal mathematical reasoning.
From Solvers to Research: Large Language Model-Driven Formal Mathematics at the Research Frontier (with 18 co-authors) Said 8 Jul 2026
Natural-language LLM mathematical reasoners are fundamentally limited by lacking machine-checkable semantics, which blocks autonomous verification needed for open-ended research.
However, these LLM reasoners that generate informal reasoning in natural language are fundamentally limited by the lack of precise, machine-checkable semantics, making their outputs prone to hallucinations [Huang et al., 2025b] and precluding autonomous verification, a prerequisite for tackling open-ended mathematical research.
From Solvers to Research: Large Language Model-Driven Formal Mathematics at the Research Frontier (with 18 co-authors) Said 8 Jul 2026
Current AI4Math systems largely operate as solvers of isolated, well-defined proofs rather than as researchers that expand mathematical knowledge.
Despite these strides, we argue that current AI4Math systems still largely operate as solvers, excelling at isolated, well-defined proof generation rather than as researchers capable of expanding the boundaries of mathematical knowledge.
From Solvers to Research: Large Language Model-Driven Formal Mathematics at the Research Frontier (with 18 co-authors) Said 8 Jul 2026
AI systems are now capable of meaningful contributions to formal mathematics, not only informal problem-solving.
These developments inform our position that AI systems are now capable of meaningful contributions to formal mathematics, not merely informal problem-solving.
From Solvers to Research: Large Language Model-Driven Formal Mathematics at the Research Frontier (with 18 co-authors) Said 8 Jul 2026
AI has driven the cost of generating scientific ideas to almost zero, and just as with cheap communication before it, that by itself creates no abundance.
So, I think AI has basically driven the cost of idea generation down to almost zero in a very similar way to how the internet drove the cost of communication down to almost zero, which is an amazing thing, but it you know, it it it doesn't make it doesn't create abundance by itself.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
The structures of science now have to change, because the scarce thing is no longer proposing theories but verifying and evaluating the thousands anyone can now propose.
So, we're now in a situation where suddenly people can generate thousands of theories for a given scientific problem. And now we have to to verify them, evaluate them and this is something which we we have to to change our structures of science to actually sort this out.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
Whether a scientific idea was progress depends on a future and a culture that have not happened yet, so grading it may never be something you can reinforcement-learn.
So, you you can't look at any given scientific achievement purely in isolation and give it an objective grade without being aware of the context both in the the past and the future. And so it it it may never be something that you can just reinforcement learn the same way that that you can for much sort of more localized problems.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
We are living through a cognitive Copernican revolution: human intelligence is not the centre, and which tasks we count as requiring intelligence has to be re-evaluated.
you know, right now we're going through uh an in um an cognitive version of the Copernican revolution where we used to think that human intelligence is the center of the universe. And now we're actually seeing that there's there's very different types of intelligence um that that that are out there uh with very different strengths and weaknesses.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
The month in which AI one-shot solutions to open Erdős problems kept arriving has ended, and it was not for lack of people trying.
Um we are seeing a lot fewer sort of pure AI solutions now where um they are just one shots the problem. Um so so there was there was a month where that happened and and that has stopped.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
Almost every Erdős problem AI has solved was one with no literature behind it, which is the median level of what these tools can do.
Um So, like with the Erdős problems, you know, like almost all of the 50 problems that were solved by AIs were ones for which basically there was no literature.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
On any given problem an AI tool succeeds maybe one or two percent of the time; the headline results are what running it at scale and picking the winners looks like.
Uh but whenever we do a systematic study, um any given problem, an AI tool has a success rate of maybe 1 or 2%. Uh it's just that it's just that they can apply at scale and and you just pick the winners, it looks great.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
Today's AI tools either solve a problem or fail at it, and are really bad at partial progress or at identifying which intermediate step to attack first.
um you know, these tools they either succeed or they fail um and they they've been really bad at creating sort of partial progress or identifying intermediate um um stages that you should you should focus on first.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
What AI still cannot do in mathematics is build cumulatively on partial progress; it is trial, error and brute force rather than an idea improving through use.
There isn't this cumulative process which is, uh, sort of built up interactively. Um, it it it seems to be a lot more trial and error and just repetition, brute force, um, you know, which can see it scales and it can work amazingly well in in certain contexts, um, but yeah, this this idea this this sort of building up cumulatively from, um, from partial progress is kind of is what's still not quite there yet.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
A model that solves a hard problem has learned nothing from it: the next session has forgotten it, with no new skill to carry to related problems.
Yeah, you you you're on a new session it's forgotten what what it just did. Um, it it has you know, it has no new skills to to attach to to to build on on on related problems.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
AI excels at breadth and human experts at depth, which makes them complementary and makes today's depth-focused way of doing science the wrong shape.
Yeah, so they excel at breadth and humans excel at depth. Um like human experts at least. Yeah, so um I think they're very complementary. Um but our current uh way of doing math and science is focused on depth because that that's where the human uh expertise cuz humans can't do breadth.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
AI will revolutionise mathematics by giving it an experimental side: large-scale data on which methods work, rather than the careful solving of individual problems.
So, I think AI-type tools we really will will actually revolutionize the ex- the experimental side of math where where um you don't care so much about uh individual problems and and the process of solving them, but you you want to gather just large-scale data about about what things work, what things don't.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
On small tasks he can check himself, he catches about as many of the AI's errors as it catches of his.
So, they they they still make mistakes, but but um um I've I've tested these tools, you know, on on on on um like little tasks that I can do and and sometimes they pick up errors I make, sometimes I pick up errors that they make. It's it's about a tie right now.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
When no technique works and the AI starts suggesting things, chasing those suggestions down usually wastes more time than it saves.
Um and then they can kind of suggest random things and it it but it it it um often I find that trying to chase them down and make them work and find they don't work, it wastes more time than it saves.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
AI has made his papers richer and broader, but not deeper.
Uh yeah, so it's it's made papers sort of richer and broader, but not necessarily deeper.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
The hardest part of solving a mathematical problem is the part AI has not touched; he still does it with pen and paper.
Um so yeah, the the the um the core of what I do, like actually solving um the most difficult part of of a math problem, that hasn't changed too much. I still use pen and paper for that.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
The fear that a great theorem will arrive as an incomprehensible proof is misplaced, because once the proof exists as an artifact we can analyse it.
so so some people are concerned, you know, what if the Riemann hypothesis is proved with a completely incomprehensible proof? I I think once you have the artifact of a proof, we can do a lot of of of of of analysis on it.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
Some theorems may only ever be provable by unintelligible case-by-case brute force; the four colour theorem still has no conceptually elegant proof.
I mean, some problems have been basically solved by pure brute force. The four color theorem is is a famous example. Um, we have still not found a conceptually elegant proof of this theorem.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
Fully autonomous one-shot attempts are the wrong approach to the great unsolved problems; the mileage is in humans and tools working against each other.
I I I I do feel that, you know, fully autonomous one-shot approaches are not the right approach for these problems. I mean, I I think you you'll get a lot more mileage out of the interplay between between humans collaborating with these tools.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
If the Riemann hypothesis turned out to be false, prime-based cryptography would be abandoned very quickly, because one unknown pattern implies more.
Um and uh I think we would very rapidly abandon any cryptography based on the primes because if there was a one pattern that we didn't know about there's probably more.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
Modern society has become so good at optimising everything that it risks optimising away serendipity, which is a real cost and not a sentimental one.
Um And yeah, so I believe a lot in serendipity. Um And maybe there's a danger actually that you know, in the mo- modern society, it's not just AI, but we've become really good at optimizing everything.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
Within a decade AI will do the bulk of what mathematicians spend their time on, and we will discover that was never the important part of the job.
Um I think with within a decade, a lot of things that mathematicians currently do um we we spend a lot of the bulk of our time doing and a lot of stuff we put in our papers today can be done by AI. Um but we will find that that actually wasn't the most important part of what we do.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
Hybrid human-plus-AI teams will dominate mathematics for a lot longer, and a full replacement would need breakthroughs beyond anything we now have.
I mean, I I I I do believe that that hybrid um human plus AIs will will dominate mathematics for a lot longer. It it's It will depend It will require some additional breakthroughs uh beyond what we already have.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
Contributing to frontier mathematics no longer requires years of PhD training; with AI tools and Lean a high schooler can make a real contribution.
Um but now it's quite possible at the high school level or or whatever that that you could get involved in math project and actually make a real contribution because of all these AI tools and and and Lean and everything else.
Terence Tao – How the world’s top mathematician uses AI Said 20 Mar 2026
What sets mathematicians apart is caring whether something holds in 100 percent of cases, when 99.99 percent is good enough for everyone else.
If something holds 99.99% of the time, that's good enough for most things. But mathematicians are one of the few people who really care about whether really 100% of all situations are covered by it.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
He works as a fox, not a hedgehog: moving between fields, chasing analogies, and re-proving results he likes with the tools he favours.
I'm much more comfortable with the fox paradigm. Yeah. So yeah, I like looking for analogies, narratives. I spend a lot of time… If there's a result, I see it in one field, and I like the result, it's a cool result, but I don't like the proof, it uses types of mathematics that I'm not super familiar with, I often try to re-prove it myself using the tools that I favor.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
Physics will be unified again as it has been before; the obstacle is that relativity and quantum mechanics already explain almost everything we can measure.
But I have faith that we've been doing this for centuries and we've made progress before. There's no reason why we should stop.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
String theory, the leading candidate for decades, is slowly going out of fashion because it is not matching experiment.
Yeah, that was a leading candidate for many decades. I think it's slowly pulling out of fashion. It's not matching experiment.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
Formalising a proof in Lean currently takes about ten times the effort of writing it out: doable, but annoying.
So right now I estimate that the time and effort taken to formalize it, proof is about 10 times the amount taken to write it out. So it's doable, but it's annoying.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
When formalising a proof costs no more than writing it, mathematics will flip: papers written in Lean first, and journals refereeing only for significance because correctness is certified.
And that's a phase shift, because suddenly it makes sense when you write a paper to write it in Lean first, or through a conversation with AI, which is generally on the fly with you, and it becomes natural for journals to accept.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
Lean and tools like GitHub will let experimental mathematics scale far beyond what one mathematician's spaghetti code allows today.
I think the platform that Lean and other software tools, so GitHub and things like that will allow experimental mathematics to scale up to a much greater degree than we can do now.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
Solving one Olympiad problem with three days of Google's server time shows what is possible, but the approach does not scale.
Yeah, these are great work that shows what's possible. The approach doesn't scale currently. Three days of Google's server time can solve one high school math format there. This is not a scalable prospect, especially with the exponential increase as the complexity increases.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
AI could compete with human mathematicians once it acquires a mathematical sense of smell: knowing which way of splitting a problem makes it easier rather than harder.
Very rarely do you transform into a simpler problem. So if they can pick up a sense of smell, then they could maybe start competing with a human level of mathematicians.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
His prediction that research-level mathematics papers would be written in collaboration with AI by 2026 has already come true.
There are certainly math results which could only have been accomplished because there was a human authentication and an AI involved, but it's hard to disentangle credit. I mean, these tools, they do not replicate all the skills needed to do mathematics, but they can replicate some non-trivial percentage of them, 30, 40%, so they can fill in gaps.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
This decade, AI could propose a conjecture connecting two things nobody thought were related, with a real chance of it being correct and meaningful.
Yeah, this decade I can see it making a conjecture between two things that people would thought was unrelated.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
Current AI struggles even to rediscover old laws of physics from data, and when it does, contamination from training is the first suspect.
The dream is you just feed it all this data, and this is here is a new patent that we didn't see before, but it actually, even the current state of the art even struggles to discover old laws of physics from the data.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
The twin prime conjecture is certainly true, the random model gives overwhelming odds of it, and he just cannot prove it.
I can tell you with complete certainty the twin prime conjecture is true. The random model gives overwhelming odds it is true, I just can't prove it.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
In ten years there will be many results much closer to twin primes, perhaps not the whole thing; on the Riemann hypothesis he has no clue.
So I think in 10 years we will have many more much closer results, we may not have the whole thing. So twin primes is somewhat close. The Riemann hypothesis I have no clue.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
On P versus NP, the evidence leans towards no, and the problem is unusual in how many approaches have been proven not to work.
Certainly more on the no than on the yes. The funny thing about P equals NP is that we have also a lot more obstructions than we do for almost any other problem.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
The stable one-profession career is becoming a thing of the past; what will still be needed alongside AI is reasoning with abstractions and problem-solving when things go wrong.
It's becoming much more a thing of the past. So I think you just have to be adaptable and flexible. I think people will have to get skills that are transferable, like learning one specific programing language or one specific subject of mathematics or something. That itself is not a super transferable skill, but sort of knowing how to reason with abstract concepts or how to problem solve when things go wrong.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
Faced with a problem none of your techniques fit, try anything, the stupider the better, because the way it fails is the clue.
So the next step then is to try anything no matter how stupid and in fact almost the stupider, the better, which technically is almost guaranteed to fail, but the way it fails is going to be instructive.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
The mathematical community as a whole is a superintelligent entity that no single mathematician comes close to replicating.
so the mathematical community plural is incredibly super intelligent entity that no single human mathematician can come closer to replicating.
Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472 Said 14 Jun 2025
What is a korrent?
A korrent is a belief a person has stated in their own words: one sentence stating the claim, backed by a quote and a source, kept at korrents.com.
Under a name here, the quoted block is what they actually said. The korrent beneath it is the claim those words support, in korrents' wording — tap it to see the record, its source, and who else holds it.
Nobody here wrote their own korrents. They are compiled from public statements, and a person can change their mind, which is recorded too.
Where they publish
Site math.ucla.edu/~tao Their own faculty page: papers, preprints, books and teaching.
The academic front door they maintain themselves — papers and preprints, books, teaching, and links to the blog and the other accounts. Not a departmental page about them.
Link verified 1 Sept 2026.
Blog What's new Their long-running research blog. Has a feed.
Research announcements, open problems, expository notes and the occasional worked argument in the comments. Running since 2007 and still posting; the feed carries every post.
Recent
- SAIR Open Math OS released 16 Oct 2026 As announced a few weeks ago, the SAIR Open Math OS is now soft launched. It offers services and web interfaces similar to frontier AI models, though currently we are hosting existing open math models (DeepSeek V4.1 Fla…
- How to Formalize All of Mathematics 16 Oct 2026 Jeremy Avigad, Kevin Buzzard, Johan Commelin, Rémy Degenne, Kim Morrison, Oliver Nash and Adam Topaz have written a blog post over on Kevin Buzzard’s blog entitled “How to Formalize All of Mathematics”, regarding recent…
- Why are we even doing all this? 15 Oct 2026 [This is a guest post by Rikab Gambhir. This blog post was initially written in a different file format and converted using AI. — T.] First, they came for the artists, and I did not speak out, because I was not an artis…
Show 7 more
- Simons Foundation Launches $15M Effort To Help Mathematicians Navigate AI Era 15 Oct 2026 A new press release from the Simons Foundation, calling for workshops and other initiatives centered on mathematics in the era of AI.
- Call for stories: human–AI collaboration in recent mathematics 15 Oct 2026 [This is a guest post by Deanna Needell and Hayden Schaeffer – T.] [Edit: Oct 16, 2026 : we are in particular interested in ways mathematicians contributed to the results in the wave of recent AI generated preprints, e.…
- First Proof, Third Batch webinar 14 Oct 2026 Just a short note that First Proof is announcing the result of their “third batch” evaluation at a webinar at 3PM ET hosted by ICARM today. (Registration required.)
- 2026 AMS Committee on Education Workshop Report 13 Oct 2026 This is somewhat belated, but: on October 2, the American Mathematical Society released a report based on a recent workshop it organized on the topic of graduate education in the mathematical sciences. (If there have be…
- This is normal bad work 13 Oct 2026 [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…
- Additional recent pieces on math and AI 12 Oct 2026 I’m collecting here a list of some other recent blog posts or opinion pieces on math and AI beyond the ones posted here or at Proofs and Prompts. Readers are invited to contribute further such links in the comments. Kev…
- Illustration beyond Proof 11 Oct 2026 [This is a guest post by Kate Stange and Edmund Harriss. This blog post was initially written in a different file format and converted using AI. — T.] In January a group of mathematicians, historians, artists and educat…
Link verified 1 Sept 2026. Recent items update automatically from the channel.
Mastodon @tao@mathstodon.xyz Where the short mathematical notes and progress updates go.
Short notes on work in progress, formalisation projects and mathematics in the news. The profile's own linked fields are how the rest of these channels were confirmed.
Recent
- An update to the parody OpenAI announcement of their release of 500 film endings: 8 Oct 2026 x.com
- In traditional mathematics (or "Math 1.0"), a breakthrough proof of a long-standing open conjecture generates a large amount of subsequent activity and excitement in the field. The authors of the proof are invited to give many talks, and meet with other experts in the area. Workshops are set up to discuss the proof, as well as other recent developments. As a result, new connections and collaborations are formed, followup problems are shared, and other mathematicians (both junior and senior) are attracted to the field (including in some cases the authors themselves, if they initially came from… 6 Oct 2026
- I came across this parody of a rumored upcoming press release by OpenAI: "OpenAI Releases the Final Ten Minutes of 500 Previously Unreleased Films, Ushering in a New Era of Movie Watching" : 6 Oct 2026 x.com
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- [From Kaisa Matomaki] The university of Turku is advertising an analytic number theory postdoc which is open for applications until 20th of October. Note that the ideal starting day for the position is already January 2027, but there is a bit of flexibility. In any case there will be another position advertised later which will start in the autumn 2027. 26 Sept 2026 ats.talentadore.com
- Serre's 100th birthday conference starts tomorrow: https://serre100.sciencesconf.org/?lang=fr . I was on the organizing committee, but unfortunately am not able to participate in the event. 14 Sept 2026
- 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: 11 Sept 2026 economist.com
- "Toutes les grandes personnes ont d'abord été des enfants. (Mais peu d'entre elles s'en souviennent.)" [All grown-ups were once children... but only few of them remember it.] Antoine de Saint-Exupéry, "Le Petit Prince." An impromptu game of "who can name the largest number?" is an example I often point to as a way in which young children teach each other about non-trivial mathematical ideas and facts, such as infinity and the concept of proof by contradiction (embodied in this case by one child eventually realizing that they can defeat any other child's guess by adding "plus one" to that gues… 10 Sept 2026
- Javier Gomez-Serrano will give a special lecture at Harvard this Friday on recent developments in fluid mechanics: 9 Sept 2026 cmsa.fas.harvard.edu
- 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 problems in mathematics at will, such as working out the 10^10^10th digit of pi. But the vast majorit… 8 Sept 2026
- RE: https://mastodon.social/@tristanbuckmaster/117233413705701198 A remarkable achievement: Alpöge and Buckmaster have managed to push one of the major promising approaches towards constructing blowup solutions to fluid equations --- as developed by Cordoba and Martınez-Zoroa --- to establish finite time blowup for many key fluid equations, including 3D incompressible Euler, with a smooth forcing term. (Previous work by these authors on similar equations could obtain a smooth forcing term for the IPM equation, and a subcritical but not smooth regularity for other equations.) The arguments hav… 8 Sept 2026
- While working on the revision of the Equational Theories Project (ETP) report at https://arxiv.org/abs/2512.07087 I came across an observation that we quietly made at the back of that report (see Section 14.1) but has become more relevant today. The ETP was a successful project, running over several months, to resolve over 22 million implications in universal algebra (and formalize them in Lean) through a variety of methods, both human and automated, although in our case the automation was much more in the ``good old-fashioned'' form of utilizing automated theorem provers (ATP) and simple scr… 7 Sept 2026
- A new proposed competition for AI companies: rather than being the first to announce solutions to unsolved math problems, be the first to announce a new mathematical insight. 5 Sept 2026
- An elaboration of the previous post mentioning the bounded gaps between primes problem as an illustration of the opportunity cost of converting a fruitful problem such as this solely into a competitive benchmark. [Indeed, as I point out below, this already came close to happening back in 2013.] As with most other problems worth studying in pure mathematics, the particular bound one gets on these gaps is not of much intrinsic importance. Improving Zhang's bound of 70 million, to 246, 188, or 6 would not, in itself, have significant impact on any other mathematical problem, let alone any real-w… 5 Sept 2026
- In basic research, such as pure mathematics, one might naively expect that the natural question to ask with regards to a given problem X in a field is "What is the answer to X?". But in many cases the more valuable question is "What can be learned from studying X?" The answer to X itself can of course be one of the things learned in this process of study; but one can learn far more useful information besides, such as * What are the main difficulties to overcome to resolve X? * What new techniques can one discover in order to solve X? * Why are existing techniques insufficient to solve the pro… 3 Sept 2026
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Bluesky teorth.bsky.social The same short notes, on Bluesky.
A lower-volume mirror of the Mastodon account, carrying much the same short notes on work in progress.
Recent
- I came across this parody of a rumored upcoming press release by OpenAI: "OpenAI Releases the Final Ten Minutes of 500 Previously Unreleased Films, Ushering in a New Era of Movie Watching" : 6 Oct 2026 x.com
- A group of 25 Fields Medalists, including myself, have made a joint declaration on Math and AI: mathandai.org . We welcome additional signatories. See also this article in the Economist announcing the declaration: 11 Sept 2026 economist.com
- Announcing the Palomar registry of Lean formalized mathematics: palomar-registry.org . See also my blog announcement at terrytao.wordpress.com/2026/08/18/p... and the Lean Zulip channel at leanprover.zulipchat.com#narrow/chann.... 19 Aug 2026
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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: competition.sair.foundation/competitions... terrytao.wordpress.com/2026/06/16/t... . 16 Jun 2026
- Alberto Alfarano, François Charton, Yongzheng Jia, Kristin Lauter, Cathy Li, Emily Wenger, are launching a challenge at SAIR on how efficiently neural networks can perform modular arithmetic: 8 Jun 2026
- A #cosmicdistanceLadder post on the many ways we project our planet onto a plane. No projection can faithfully reproduce *all* the geometric features of a sphere, so each projection is a compromise; but some projections are still preferred for specific applications. 22 Feb 2026 instagram.com
- Tanya Klowden and I have uploaded a further sample chapter to our "cosmic distance ladder" book: terrytao.wordpress.com/books/climbi... . See also the blog post 22 Nov 2025 terrytao.wordpress.com
- A new paper with Bogdan Georgiev, Javier Gomez-Serrano, and Adam Zsolt Wagner: "Mathematical exploration and discovery at scale" arxiv.org/abs/2511.02864. Further discussion is at 6 Nov 2025 terrytao.wordpress.com
- I started a crowdsourced meta-project to list all the other crowdsourced mathematical research projects that are currently active and seeking participants: 23 Sept 2025 mathoverflow.net
- A first proof-of-concept outcome for the erdosproblems.com - OEIS linkage project: with a useful assist from AI, the OEIS was used to locate a proof in the literature for an Erdos problem that was previously marked as "open". 2 Sept 2025 mathstodon.xyz
- Thomas Bloom and I are launching a crowdsourced project to link up Thomas's erdosproblems.com site with the #OEIS, by systematically calculating the various integer sequences associated with the Erdos problems and crosschecking them against the OEIS database: 1 Sept 2025 terrytao.wordpress.com
- I wrote an op-ed on the world-class STEM research ecosystem in the United States, and how this ecosystem is now under attack on multiple fronts by the current administration: 18 Aug 2025 newsletter.ofthebrave.org
- #IPAM (the institute for pure and applied mathematics) is facing a critical shortfall for operating expenses due to an unexpected suspension of NSF funding www.ipam.ucla.edu/news/nsf-fun... . Donations for emergency continuity of operations funding can be made at 8 Aug 2025 giving.ucla.edu
- My thoughts on the crucial importance of methodology on self-reported AI performance on mathematics competitions, and my policy on commenting on such reports going forward: 19 Jul 2025 mathstodon.xyz
- The #SalemPrize for 2025 is now accepting nominations until September 15th. www.ias.edu/math/activit... (I am the chair of the Scientific Committee for the prize.) A bit more information in my blog post on this: 8 Jul 2025 terrytao.wordpress.com
- A new #CosmicDistanceLadder post to mark the summer solstice, on how astronomical measurements, from the time of Eratosthenes to the modern day, rely on the tireless (and often unsung) efforts of many careful and precise data collectors. 20 Jun 2025 instagram.com
- I have just launched a "Lean companion" to my textbook "Analysis I": github.com/teorth/estim... . This gives a Lean translation (or paraphrasing) of the various definitions, theorems, and exercises in the textbook into Lean. Further discussion at 31 May 2025 terrytao.wordpress.com
- A video on an actual formalization task from my Polynomial Freiman Ruzsa (PFR) project, which is sufficiently tricky (and sufficiently far from existing training data of current models) that one still has to largely formalize these proofs by "hand": 26 May 2025 youtube.com
- A new #CosmicDistanceLadder post on how the distance ladder can also be used to measure cosmic durations, as well as cosmic distances. 20 May 2025 instagram.com
- A third video in my occasional series on #Lean4 formalization workflows, this time focusing on how relying extensively on #GitHubCopilot fares against standard "epsilon delta" type problems in analysis. 17 May 2025 youtube.com
Link verified 1 Sept 2026. Recent items update automatically from the channel.
Papers ORCID Their ORCID record: peer-reviewed papers, newest first.
Academic papers, most recent first, restricted to open-access articles published in an identified journal. Fetched live from OpenAlex, keyed by this ORCID.
Recent
- New Nikodym set constructions over finite fields 25 Sept 2026 For any fixed dimension d ≥ 3 we construct a Nikodym set in ކ d q of cardinality q dd-2 log 2 + 1 + o(1) q d-1 log q in the limit q → ∞, when q is an odd prime power.This improves upon the naive random construction, w…
- Growth rates of sequences governed by the squarefree properties of their translates 10 Jul 2026 We answer several questions of Erdős regarding sequences of natural numbers A whose translates n+A intersect with the squarefree numbers in various specified ways. For instance, we show that if every translate only cont…
- On the number of exceptional intervals to the prime number theorem in short intervals 12 May 2026 For a fixed exponent $0 < θ\leq 1$, it is expected that we have the prime number theorem in short intervals $\sum_{x \leq n < x+x^θ} Λ(n) \sim x^θ$ as $x \to \infty$. From the recent zero density estimates of Guth and M…
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- New exponent pairs, zero density estimates, and zero additive energy estimates: A systematic approach 6 Nov 2025 We obtain several new bounds on exponents of interest in analytic number theory, including four new exponent pairs, new zero density estimates for the Riemann zeta-function, and new estimates for the additive energy of…
- Planar Point Sets with Forbidden 4-Point Patterns and Few Distinct Distances 28 Oct 2025
- Pointwise convergence of bilinear polynomial averages over the primes 1 Sept 2025 We show that on a $\sigma $ -finite measure-preserving system $X = (X,\nu , T)$ , the non-conventional ergodic averages $$ \begin{align*} \mathbb{E}_{n \in [N]} \Lambda(n) f(T^n x) g(T^{P(n)} x) \end{align*} $$ converge…
- On the distribution of eigenvalues of GUE and its minors at fixed index 25 Jul 2025 We obtain bounds on the distribution of normalized gaps of eigenvalues of [Formula: see text] GUE matrix in the bulk, that do not lose logarithmic factors of [Formula: see text] in the limit [Formula: see text]. As an a…
- Undecidability of translational monotilings 21 Jun 2025 In the 1960s, Berger famously showed that translational tilings of \mathbb{Z}^{2} with multiple tiles are algorithmically undecidable. Recently, Bhattacharya proved the decidability of translational monotilings (tilings…
- On several irrationality problems for Ahmes series 1 Apr 2025
- On a conjecture of Marton 1 Mar 2025 We prove a conjecture of K. Marton, widely known as the polynomial Freiman--Ruzsa conjecture, in characteristic 2. The argument extends to odd characteristic, with details to follow in a subsequent paper.
- Adjoint Brascamp–Lieb inequalities 14 Sept 2024 The Brascamp–Lieb inequalities are a generalization of the Hölder, Loomis–Whitney, Young, and Finner inequalities that have found many applications in harmonic analysis and elsewhere. In this paper, we introduce an “adj…
- Sumsets and entropy revisited 31 Jul 2024 The entropic doubling of a random variable taking values in an abelian group is a variant of the notion of the doubling constant of a finite subset of , but it enjoys somewhat better properties; for instance, it contrac…
- A counterexample to the periodic tiling conjecture 1 Jul 2024 The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ that tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently la…
- A lower bound on the mean value of the Erdős–Hooley Delta function 30 Jun 2024 We give an improved lower bound for the average of the Erdős–Hooley function , namely for all and any fixed , where is an exponent previously appearing in work of Green and the first two authors. This improves on a prev…
- Monotone Nondecreasing Sequences of the Euler Totient Function 23 May 2024 Let M(x) denote the largest cardinality of a subset of $$\{n \in \mathbb {N}: n \le x\}$$ { n ∈ N : n ≤ x } on which the Euler totient function $$\varphi (n)$$ φ ( n ) is nondecreasing. We show that $$M(x) = \left( 1+O\…
- The structure of arbitrary Conze–Lesigne systems 21 Feb 2024 Let Γ \Gamma be a countable abelian group. An (abstract) Γ \Gamma -system X \mathrm {X} - that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of Γ \Gamma - is said to be a…
- The convergence of an alternating series of Erdős, assuming the Hardy–Littlewood prime tuples conjecture 1 Feb 2024 It is an open question of Erdős as to whether the alternating series ∑ n = 1 ∞ ( − 1 ) n n p n \sum _{n=1}^\infty \frac {(-1)^n n}{p_n} is (conditionally) convergent, where p n p_n denotes the n t h n{\mathrm {th}} prim…
- 1. Does One Have to be a Genius to Do Maths? by Terence Tao 1 Jan 2024 and suggestions for graduate students, job seekers, early career academics of all types, and those who mentor them.Krystal Taylor and Ben Jaye serve as the editors of this section.Next month's theme will be Black Histor…
- Sums of GUE matrices and concentration of hives from correlation decay of eigengaps 28 Dec 2023
- Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions 12 Dec 2023 We establish quantitative bounds on the U^{k}[N] Gowers norms of the Möbius function \mu and the von Mangoldt function \Lambda for all k , with error terms of the shape O((\log\log N)^{-c}) . As a consequence, we obtain…
Link verified 9 Sept 2026. Recent items update automatically from the channel.
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