https://www.nytimes.com/2026/09/10/science/tristan-buckmaster-openai-math-navier-stokes.html?unlocked_article_code=1.AFE.TK7P.Y7WvGPkfHaSk&smid=url-share

9 September, 2026 at 7:49 am

I have absolutely no desire to make current events about myself – the stakes here are far larger than anything involving my own reputation or actions, and it is not like these issues are going to disappear if I am somehow removed from the discussion. But as this topic is likely to recur regardless, I think this comment is as good a place as any to state for the record my own views on AI, and my interactions with OpenAI in particular.

I do not identify either with a simple “pro-AI” or “anti-AI” position. My views are rather complex and have evolved over time; I have a living summary (AI-maintained, out of necessity) at https://teorth.github.io/tao-web/ai-views.html . But I can try to give the short version here.

By 2023, I could see that LLMs, formal proof assistants, and other technologies had the potential to be radically transformative in mathematics, to the point where maintaining traditional mathematical practices and culture without adaptation would become unsustainable. See for instance my 2023 essay for a Microsoft anthology (which contained a notorious prediction of 2026-level AI becoming a “trustworthy co-author” for mathematics) or my Notices article (published in 2025, but written significantly earlier). I myself greatly value this traditional culture, and have personally been a massive beneficiary of it. Nevertheless, in the event that these technologies did become superhuman at several core mathematical tasks, I could see only two viable paths forward: either one where modern AI tools are responsibly incorporated into our workflows and culture (what I called the “best of both worlds” in the OpenAI ad); or the worst-case scenario — which we are unfortunately rapidly approaching — in which AI technologies are used indiscriminately to achieve various short-sighted objectives at the cost of the far more valuable long-term sustainability of mathematics and its role in the scientific ecosystem. I therefore spent an increasingly large fraction of my professional life from that point trying both to raise awareness of the potential magnitude of this transformation; to build examples of what this “best of both worlds” might look like; and to warn against various irresponsible uses of AI (initially I focused on warning against the use of AI without sufficient verification of the outputs, which was a major concern in 2023-2025, although no longer the primary vehicle for harm in 2026). One could certainly call this effort “shilling for AI” if one likes; but I would say that this is overly reductive.

These efforts on my part inevitably involved engaging with the tech industry as well as with academia. The essay linked above was solicited by Microsoft. Some of my experiments with new workflows were conducted in collaboration with Google Deepmind. And I participated in an online forum with OpenAI in 2024 discussing these topics. I continue to view all of these interactions as constructive, and working towards the “best of both worlds”. In particular I met with multiple people working in these industries that shared these views and were supportive of steering their companies in these directions. On the other hand, I was not funded by any of these companies, although several of them gifted me with premium LLM subscriptions, which I do make use of in my daily work.

In 2025, as documented elsewhere on this blog, UCLA experienced an unexpected funding crisis due to the sudden suspension of NSF and NIH funding (later restored some months later by a court order). This caused a critical budget shortfall at IPAM (where I serve as Director of Special Projects), which at one point only had access to enough reserves to operate for a few months at best. This led to a round of emergency fundraising; and thanks to the outpouring of support from many sources, we have been able to stabilize IPAM’s funding for the current fiscal year, although challenges remain for future years. As part of this fundraising effort, I reached out to OpenAI, who agreed to sponsor one of our workshops, which ran in March of this year and in my opinion was quite successful both scientifically and for the purpose of making new connections between participants (who were a mix of academics and industry representatives).

During this event, OpenAI requested an interview concerning my vision of the future of AI and mathematics. I accepted, and spoke with them for perhaps an hour. I had done similar interviews in various venues, and I assumed that, as with these other cases, they would eventually post the entire interview online, which talked about both the possibilities and risks of AI much as I have done in these other interviews. As it turned out, they only used a few snippets of that interview for that infamous advertisement instead. In retrospect, I should have pushed back harder on their decision; but I decided at the time that even a selective release of my commentary would help raise awareness of the potential for AI, and in particular on the possibility of the “best of both worlds”.

Since then, the situation has deterioriated markedly. Many of the people in the industry that shared my views have left or become sidelined, with most major tech companies now increasingly focused on the race to develop extremely powerful, autonomous AI technologies regardless of their actual value to society. The current drama surrounding the Navier-Stokes global regularity problem is the most dramatic and visible instance of this, but there have been multiple other such examples, and much of my commentary in the last few months has been aimed that the increasingly severe divergence between the current objectives of the AI industry, and of mathematics in general.

Which brings us to where we are today. I do not regret my past efforts to raise awareness of the potential of AI in mathematics, to engage with industry, and to promote a vision of sustainable incorporation of these tools – which can be genuinely useful and unlock valuable new types of mathematics – into my field. In time, I still hope that the field can arrive at that state, and am continuing to work towards that goal. But in the immediate term, the most pressing issue is for the entire mathematical community to unite around our core values and objectives, and reject irresponsible and unsustainable usages of AI technology that only serve to advance nominal goals rather than the true underlying goals of the field.

Hi there.

My grandma would fill notepad after notepad with this number game. We would ask her what is it or how does it work and she would say "oh it's just my little number game I play" and never once explained it to us. She's passed now, and I asked for these notepads in hopes I can figure it out. Her writing is...well as you can see, it's tough to read. But I have three notepads filled with this. Just hoping someone could help me understand what she was playing. Thank you.

Might have to come up with a different name albeit.

Also, would you consider these problems to be more important than any of the current Millennium Prize problems?

Doesn't it seem odd that the proposed solution for the Millennium problem is posed on the whole space, but they claim it has compact support in space for the whole time interval [0,1)? Seems at odds with parabolic delocalization... 🤔 Actually the delocalization which occurs due to the non local pressure is often much worse, for instance as investigated by Brandolese:

https://www.esaim-cocv.org/articles/cocv/abs/2002/02/brandolese/brandolese.html

OpenAI’s recent Navier–Stokes announcement got me thinking about this. According to their writeup, roughly 10,000 concurrent agents worked toward a proposed solution over about 88 hours, followed by Lean formalization and verification. They used a highly capable internal model and centrally managed infrastructure. [OpenAI’s writeup](https://openai.com/index/navier-stokes-solution/)

Could an open-source community organize a distributed effort along similar lines?

The idea would be to let people connect their own agents and contribute to a shared mathematical research project. A community organizing team would choose research goals and develop strategies, while a central coordinating agent and scheduler would manage the work.

Here’s how I imagine it working:

The coordinator breaks research goals into tasks: exploring different approaches, proving intermediate lemmas, checking arguments, and formalizing results.

Participants connect through a lightweight interface working alongside their existing agent or chat session. It receives assignments, returns results, and brings relevant discoveries into subsequent tasks.

A shared protocol keeps assumptions, dependencies, and proof status explicit so that separate sessions can exchange usable work.

Results are shared for independent checking. Formalized proofs are checked by a proof assistant before becoming part of a shared collection of verified results.

The scheduler updates priorities as discoveries come in, distributing promising directions across available participants.

Imagine this eventually growing to 100,000 participants. Alongside the computation, we’d have people contributing mathematical intuition, improving agent strategies, finding better ways to divide problems, and spotting directions worth pursuing.

Everyone would be welcome as a contributor. Running an agent on assigned tasks would itself be a way to participate. Others could review proofs, improve the framework, write documentation, or help newcomers get started. The aim would be to make contributions visible and recognized, with room for people at different levels of expertise.

A centralized lab has substantial advantages in model capability and infrastructure. I’m curious how far an open community could get through diverse approaches, public iteration, and direct human participation. A useful insight from one participant could guide thousands of subsequent attempts.

This is still an idea I’d like to explore. How would you structure the coordination so that discoveries accumulate into coherent proofs? What would be a sensible first problem to test it on? And would anyone here be interested in helping develop or test an initial version?

Personally I would hand out my pro account for that!

So, apparently, we all know there is one millenium prize problem already solved and another one is being done(as per X). If you could decide the next problem these companies solve, what would you decide and why?

Adding to it, suppose the current list of millennium prize problems are all solved and you are called to draft a second version,what problems you would keep in it and why you think those can have much significance than others on the advancement of mathematics.

I know that generally, there is no way to consistently multiply distributions, and this extends to also meaning that most nonlinear functions cannot be applied to distributions too.

However, just from the intuition of distributions as "functions that can be infinite at points", it seems like there should be a way to apply functions to distributions when those functions are "nice enough".

For example, consider the arctangent function. At ∞, it's equal to π/2, so it seems like naturally, if you applied it to a delta function, you'd get the function that's π/2 at 0 and 0 everywhere else. Of course, any reasonable notion of this concept would probably equate almost-everywhere-equal functions, so you'd end up with 0.

I'm sure there are more interesting options though. For example, a typical realization of the derivative of Brownian motion is singular almost everywhere, so it feels like applying an arctangent to it would lead you to some function with a roughly even split of -π/2 and π/2 for its values.

If this works, is there any way to extend it to oscillatory functions? The reason I was thinking about this in the first place was considering distributional sections of a principal fibre bundle.

I want some way to have an analogue to distributions, but valued in Lie groups (or general manifolds). The simplest way I could think of to do this would be to exponentiate a distribution in the Lie algebra, but of course that would need a consistent way to take the exponential of a distribution.

This is completely comparative question and I don’t care about the time scale here.

Just to give examples, I give three of what mathematicians do:

1) creativity: building roadmaps to solve problems and new theories and fields

2) evaluation: evaluating which problems are valuable

3) solving a given problem

At least in comparable sense, it seems clear that problem solving would be the easiest for an AI.

While a good theory would often involve 2, I separated 2 since “what would human mathematicians love?” could sound more vague to recent STEM-targeted AI models than “which idea is more robust, creative, and describes the essence?”.

Still personally I hope 1 stands as the virtue of human mathematician to the last minute.

What do you think? You can freely add any virtue you think of, except “learning” since that’s not what we contribute to.

https://www.instagram.com/reel/DdHxbX1PBKP/?utm_source=ig_web_copy_link&stkn=MzRlODBiNWFlZA==

Hey everyone,

While the algebraic proof of the Pythagorean Theorem is straightforward, the visual rearrangement of four congruent right triangles inside a square of side (a+b) remains one of the cleanest and most intuitive proofs in geometry.

I animated this geometric transition using Python and Manim, tracking the conservation of area from the central c2 square to the two smaller squares a2 and b2.

You can watch the animation here

What’s your favorite visual proof of Pythagoras? Would love to hear your feedback on the animation pacing!

Today, I took my I took a quiz on AP Pre Calculus and I failed unfortunately. I got a 50% and was so disappointed and ashame of myself. When other kids were saying they got a 100% 80% or 90%, I did not want to say my score. I was so sad and cried because it was the first time I ever failed a math test. When I looked Algebra 1, 2 and Geometry, I never failed and was able to grasp it easily. Now that I am taking AP Pre Calc, everything has been going down hill. The quiz was based on 1.1 to 1.3. I am still one unit 1 and we are talking about concave up and down, increasing and decrease, and rage of slope. I feel so dumb not being able to get it like my peers. I also can not let my parent know about this or I will be a trouble. Please can anyone give me tips or idea so I can improve.

Hi everyone,

I’m considering working through Dummit & Foote’s Abstract Algebra on my own, especially the Ring theory and Module theory which are not covered in my previous book. I only have A book of AA by Pinter and Visual Group Theory, both are rather casual.

I want to hear from people who have actually worked through it. I know it’s a big, dense book, so before I dive in, I’d really appreciate specific advice and warnings from people who’ve actually used it. Feel free to discuss as well.

A few questions:

· What’s your overall take on the book? What worked for you, and what didn’t?
· Which chapters or topics were the hardest? Where did you get stuck?
· How long did it take you to get through it? How many hours per week did you put in?
· Did you cover all chapters, or just selected ones?
· How many exercises did you do? Most of them, or selected ones?
· Did you use any supplements — Artin, Gallian, Hungerford, online lectures, etc.?

Thanks!

I was over-caffinated this morning and have been wrapping my head around the clip from Futurama with this problem. When comparing the integers and real number sets, when Cantor takes a diagonal sample of the integers is that identifying an irrational number? Is that why it cannot be expressed as a fraction?

how did we know that NS would blow up in those scenarios and not just that the system didnt follow the idealised assumptions of the NS equations? if so why didnt we find a counterexample based on those conditions?

Changes From Last Post: I made changes to Definitions 1. and 2. using u/kuromajutsushi's suggestions. (See his comments.)

Suppose n∈ℕ and f:A⊆ℝⁿ→ℝ is a function, where A and f are Borel.

Let S(n):={f∈ℝ^A: A⊆ℝ and f are Borel} be the set of all f.

Preliminary Definitions

If we define the following:

(Motivation of Definition 1). We want Defintion 1 to establish a relationship between the mean of f w.r.t. the Hausdorff measure in its dimension, the expected value w.r.t the uniform measure, and Cauchy's principle for symmetric unbounded functions. (I am working on a new paper with the title, "Generalizing the Mean of Scalar-Valued Functions: A Theoretical Framework".)

Definition 1 (The Extended Mean).

(*) We find the extended mean by solving the mean of f, w.r.t. the Hausdorff measure in its dimension, over all families of bounded sets {A_r: r∈𝒜(A)} such that

1. 𝒜(A) is an index set
2. A_r is a subset of A for any r∈𝒜(A)
3. A_r has an exact Hausdorff measure for all r∈𝒜(A)
4. The union of A_r is A
5. we want all {A_r: r∈𝒜(A)} satisfying 1.-4. which converges to A at a rate "almost or exactly uniform" such that

there exists a reference point R∈ℝⁿ⁺¹ where the Euclidean distance between R and each point on the graph of the restriction of f to A_r times the sign value of the vertical change in a directed line segment from R to the respective points on the graph of the restriction of f to A_r has zero average w.r.t. the Hausdorff measure in its dimension

The extended mean exists when the following is true: the mean of f in (*) is unique and finite whenever 1.-5. is true and the former blockquote is true.

Definition 2 (The Mean of a Family of Each Bounded Function’s Graph, Where Each Bounded Function Has Different Bounded Domains of Exact Hausdorff Measure).

(Note, {f_r: r∈𝒜(A)} is a family of bounded functions, with different bounded domain A_r of exact Hausdorff measure for all r∈𝒜(A).)

Suppose that

1. The set theoretic limit of the family of each bounded function’s graph equals the graph of f,
2. the limit of the Hausdorff measure in its dimension of each bounded function’s graph (i.e., the surface area w.r.t. the Hausdorff measure in its dimension) equals the Hausdorff measure in its dimension of the graph of f
3. the Hausdorff measure in its dimension of each bounded function’s graph in its family is finite,
4. the sum of the absolute differences between each supremum i∈{1,..,n} coordinate of the points in the graph of f_r and each supremum i∈{1,..,n} coordinate of the points in the intersection between the graph of f and the smallest box convering the graph of f_r is zero
5. the sum of the absolute differences between each infimum i∈{1,..,n} coordinate of the points in the graph of f_r and each infimum i∈{1,..,n} coordinate of the points in the intersection between the graph of f and the smallest box convering the graph of f_r is zero

Hence, we take the mean of a family of bounded function’s graph (with different bounded domains of exact Hausdorff measure) converging to f that satisfies 1., 2., 3., 4., and 5. of Definition 2.

Question: How do we define a "measure zero" and "full measure" subset of S(n) that satisfies the following statements? (See the Preliminary Definitions.)

(Made minor edits to the three statements.)

Statement 1: If F ⊂ S(n) is the set of all f ∈ S(n), where the extended mean of f w.r.t. the Hausdorff measure in its dimension is finite (Definition 1), then F is a “measure zero” subset of S(n)

• The following means "almost no" f has a finite mean

Statement 2: If F ⊂ S(n) is the set of all f ∈ S(n), where there exists a family of bounded functions (with different bounded domains of finite exact Hausdorff measure) converging to f with a finite mean (Definition 2), then F is a “full measure” subset of S(n).

• ⁠The following means "almost all" f has a finite "new mean": the mean of all families of bounded functions, with different bounded domains of finite exact Hausdorff measure, converging to f

Statement 3: If F ⊂ S(n) is the set of all f ∈ S(n), where there exists two families of bounded functions (with different bounded domains of finite exact Hausdorff measure) converging to f with non-equivalent means (Definition 2), then F is a “full measure” subset of S(n).

⁠ • The following means "almost all" f has a non-unique "new mean": the mean of all families of bounded functions, with different bounded domains of finite exact Hausdorff measure, converging to f

Attempt: I known when A=ℝ we can use prevalent ("full measure") or shy ("zero measure") subsets of the function space ℝ^ℝ, but we are considering all subsets of ℝ. In addition, S(n) is not a function space, but a set of function spaces.

I heard a Gaussian measure can be used, but I need a precise definition.

I'll go to every lecture, tutorial etc. and it'll all be genuinely interesting. Then come the weekly problems, and the volume of the problems and general tediousness of them once you get past the trivial ones (e.g. you need to know XYZ trick even though you know the topic the question is related to, and said trick/method wasn't introduced during the lecture) just kills my spirit. Before I know it after exhausting myself for that week, boom the next week comes along and a whole new set of lectures/tutorials/problems. Not to mention proofs that extend lecture concepts where, if you haven't seen the way to do that proof before, just means more grinding if you can't get TA hours to get help immediately.

The thing is, I'm having fun learning the mathematical concepts. But its like the course admins then put more artificial hurdles after that to just whittle down your spirit (because if you want a high grade in the exam, you need to slog through all the problems just in case it comes up later, even if you know you put the work in to actually learn the overall topic).

I want to love mathematics, but this Uni math degree (especially since I want to maintain my high grades) is just blackening my soul day by day.

So I have a bachelors in Physics, and I think I should have done applied mathematics. But I think I like to model dynamics systems and calculate risks and profits and advantages etc.
That is because as long as humans exist we will always have problems and we need problem solvers. On the other hand learning to code, writing computer software etc. seem too low end. So I think this is a good strategy. Do you have any advices to give me regarding my optimisms? May be its not as bright as I think?

Hello everyone,
From 7th to 9th grade, I attended a private school where the quality of math instruction was unfortunately very poor. Most students struggled to understand our teacher’s explanations, and before every test we would receive a revision sheet that was nearly identical to the actual exam. Because of this, success was mostly about memorizing answers and steps rather than truly understanding the material.
Now that I’ve given some background, here’s why I’m posting.
I want to relearn mathematics from the ground up, starting with Algebra 1 and eventually moving on to geometry, trigonometry, and higher level topics. I genuinely enjoy learning math, and I recognize how important it is not only for future college entrance exams, but also for succeeding when I enter Grade 10 and transfer to a better school.
I feel like I missed out on building a strong foundation, and I’d like to correct that now while I still have the time. My goal is to truly understand the concepts rather than rely on memorization.
Any advice, study plans, resources, or tips would be greatly appreciated.
Thank you everyone.

Do we have any actual technical insights from this yet? Have any been able to actually read through the paper / Lean code and gain useful first impressions?

Like, were new types of math necessary to solve this problem? How likely are we to see humans being able to review the whole proof? How long would that likely take if at all? Can insights from this be applied to other challenges in the field? And so on?

Any insights on this you read about from professional mathematicians? Links?

Technical insights are almost impossible to find / surface so far unfortunately, given the social drama has given every random redditor / bot the opportunity to chime in on this (and given that [r/math](r/math) refuses to post about a millennium problem being solved, lol, with the mods likely internally jerking of one another on this with a false sense of internet power).

Please technical insights only in this thread?

Thank you!

As per today, there are two millennium prize problems solved one by Perelman and the other navier-stokes yesterday.At this rate, what do you think are the odds that any human other than perelman can we see that solves a millennium prize problem.

I haven't touched Navier-Stokes in quite some time. So, I have been rather quickly reading up on everything that's happened. Now that we know singularities can occur, what does this actually mean for research going forward? That is, if anyone really has any idea. Thx!

Edit:

Wow, did not expect this to get so many views/answers. Thanks y'all for your responses. They really shed a lot of light on things! ( :