Mark Sellke

New project: problemsilike.com, a website collecting open problems that I, personally, like, with comments on their context, difficulty, and interest.

Can AI do Theory? Some of my friends are hosting a workshop on this topic at #STOC2026 in Salt Lake City Speakers include Scott Aaronson, @CarinaLHong, @MarkSellke, David Woodruff, @SebastienBubeck, & Prabhakar Raghavan (@WittedNote) Call for posters ddl: 5/29 Check it out!

When GPT-5.5 misses on a Frontier Math question

GPT-5.5 leaps over GPT-5.4 and becomes #1 on MathArena! - Massive jump on BrokenArXiv (+34%) - Solid performance increase on all other benchmarks - Only slightly more expensive: twice the cost per token, but much lower output token count.

Some dismiss Erdős problems as trivialities - this couldn't be further from the truth! While many are amusing novelties, some of them are the most central problems in number theory and combinatorics. A blog post with, in my view, the 10 most important: erdosproblems.com/forum/thread/b…

The world of mathematics is rapidly changing. But more importantly look at this TikZ figure 😍

We’ve just released another paper solving five further Erdős problems with an internal model at OpenAI: arxiv.org/abs/2604.06609. Several of the proofs were especially enjoyable to digest while writing the paper. My personal favorite was the solution to Erdős Problem 1091. The question asks: if a graph G has chromatic number 4, while every small subgraph has chromatic number at most 3, must it contain an odd cycle with many diagonals? The internal model gives a very enlightening counterexample to this conjecture, and the proof was a pleasure to understand. For those so inclined, a really fun exercise is to try to reconstruct the proof from Figure 5 of the paper, which was of course produced by Codex.

In a beautiful recent paper, Vishesh Jain and Clayton Mizgerd used GPT-5.4 Pro to prove a striking result in the theory of Markov chains: arxiv.org/pdf/2604.03937 They study the adjacent transposition Markov chain on the symmetric group. A conjecture of Fill, recently settled by Greaves and Zhu, determined which parameters of this chain maximize the spectral gap, a natural quantity controlling how fast the chain mixes. Jain and Mizgerd go further and characterize exactly when this extremal spectral gap is achieved, answering another question of Fill. As they explain in the paper, once the first part was in place, GPT-5.4 Pro was able to one-shot generate the second part of the main result. From talking with the authors, my understanding is that this would likely have taken substantial effort without GPT. Furthermore even given the first part, several ingredients, such as the piecewise eigenvector construction in Proposition 6.6, were new to them. Just another example of how AI is already changing the everyday practice of research mathematics.

We are excited to share a new paper solving three further problems due to Erdős; in each case the solution was found by an internal model at OpenAI. Each proof is short and elegant, and the paper is available here: arxiv.org/pdf/2603.29961

I’ve just submitted a paper that resolves a long-standing open problem in spatial statistics. But what excites me even more than the result is the method: the estimation approach seems to be genuinely new. What’s striking is how this came together. Started as a weekend project, several of the key breakthroughs emerged from sustained interaction with GPT-5.4 Pro. It didn't one-shot the solution, but it was a true collaborator in the loop, helping refine ideas and push through technical barriers. I haven't tried other models yet, and I would love to know what they'd do on this problem. Even more surreal: the entire paper came together in under a month! Usually, this would have taken a year. It’s not about Erdos or IMO problems anymore. I have long believed this to be the case. Folks interested in AI + math collab, please reach out. More to come. arxiv.org/abs/2603.23959 @kevinweil @SebastienBubeck

My good friend Christian Coester solved a 50 years old open problem in self-organizing lists (was it really open? did people even care? yes and yes), and, you guessed it, ChatGPT-Pro made the key step in the proof*! (*a Christian Coester on the other side of the screen was needed to see through the full proof!) The problem is extremely simple: you are organizing (online) a list of items, and when item in position i is requested you pay i. Requests arrive i.i.d., and so the best thing to do is to organize the list in order of decreasing probability of appearance of items. But now what happens if the probabilities are not known, and you want a memoryless algorithm (note that with memory one could just estimate frequencies and approximate the optimal ordering)? 40 years ago it was shown that the most stupid memoryless rule of moving the requested item to the front achieves a competitive ratio of pi/2 (and this is tight). But empirically there seems to be a better idea, which is to be a bit less aggressive and simply move up the requested item by one position (the "Transposition Rule"), see Codex's experiment below. In the very first paper on this topic 50 years ago, Rivest conjectured that indeed this should be a very good rule. What did Christian prove? He showed that Transposition Rule's average cost is at the most the optimal fixed list plus ONE. Just a regret of one. Beautiful. In fact the proof shows something more delicate, and this is where ChatGPT comes in: Denote p_i for the probability of the i^th item (in the optimal ordering). Now under the stationary distribution of the Transposition Rule, denote s_i for the "excess cost" that item i incurs from wrong placement, namely: s_i = sum_{j < i} (p_j - p_i)*Prob(i appears before j) The regret of Transposition Rule is exactly the sum of those slack terms s_i. Now here is the magic: ChatGPT suggested that s_i might be smaller than p_i. If true that concludes the proof, since then we have sum s_i <= 1. Note that it is quite magical, I don't see a reason a priori why s_i would be smaller than p_i ... ChatGPT proposed a an argument to prove this for small n, and Christian found an actual proof for all n.

Some thoughts on AI and mathematics, inspired by "First Proof."

Our recent preprint on gluon amplitudes has sparked a lot of discussion, so I want to share the backstory — including how AI helped crack a problem that had stumped us for a year. I'll also be giving a public lecture at Harvard this week. Details at the end.

1/ Technical thread on #1stProof Problem 6: finding “spectrally light” vertex subsets in a graph, and how its solution fits into the landscape of spectral sparsification + restricted invertibility. Original thread: x.com/yangpliu/statu…

My thoughts on #1stProof Problem 6 (closely related to areas I've worked in): OpenAI’s solution is essentially correct, and the difficulty feels consistent with AI capabilities over the past several months. More detail in the thread.

Solution attempts from our model: cdn.openai.com/pdf/a430f16e-0…

Very excited about the "First Proof" challenge. I believe novel frontier research is perhaps the most important way to evaluate capabilities of the next generation of AI models. We have run our internal model with limited human supervision on the ten proposed problems. The problems require expertise in their respective domains and are not easy to verify; based on feedback from experts, we believe at least six solutions (2, 4, 5, 6, 9, 10) have a high chance of being correct, and some further ones look promising. We will only publish the solution attempts after midnight (PT), per the authors' guidance - the sha256 hash of the PDF is d74f090af16fc8a19debf4c1fec11c0975be7d612bd5ae43c24ca939cd272b1a . This was a side-sprint executed in a week mostly by querying one of the models we're currently training; as such, the methodology we employed leaves a lot to be desired. We didn't provide proof ideas or mathematical suggestions to the model during this evaluation; for some solutions, we asked the model to expand upon some proofs, per expert feedback. We also manually facilitated a back-and-forth between this model and ChatGPT for verification, formatting and style. For some problems, we present the best of a few attempts according to human judgement. We are looking forward to more controlled evaluations in the next round! 1stproof.org #1stProof

GPT 5.2 derived a new result in theoretical physics. For decades it's been assumed that certain gluon amplitudes ("single minus") were zero, and that the maximally helicity violating amplitudes had two gluons of one helicity and n-2 of the other. It turns out that isn't necessarily true! Andy Strominger realized this a year ago, and with Alfredo Guevara, David Skinner, and @ALupsasca they had shown this up to n=6 by hand. The expressions were getting incredibly complicated though, and Alex invited them to OpenAI to see what we could do together. In short order, GPT-5.2 Pro suggested a beautiful and general formula for arbitrary n—but couldn't prove it. An internal scaffolded model, thinking continuously for over 12 hours, proved it. This is exciting because when complicated calculations reduce to something simple, it implies there is yet-to-be-understood physics waiting to be discovered. There are multiple follow-ons to this paper, and we hope other physicists (maybe with AI!) explore the implications as well. One particular fun thing for me was getting to work with Andy Strominger, someone I looked up to throughout my time as a physics student in undergrad and grad school. To hear him talk about how he's been accelerated by AI was incredibly motivating. AI 🤝 Physics. Here's to bringing the science of the future into the present!

@weijie444 Congrats Weijie!

It is my tremendous honor and privilege to receive the 2026 COPSS Presidents' Award. Statistics is powerful and will only grow more vital in the AI age. Grateful to my mentors, collaborators, colleagues, and students who made this journey possible.

I've recently gone on leave from Columbia to join OpenAI, working on OpenAI for Science. Over the past few months, AI-including GPT 5.2-has become an increasingly important part of my workflow as a mathematician. I'm excited to contribute to efforts to accelerate progress in mathematics and scientific research.