Seminar on OpenAI's 722 Manuscripts

I will discuss OpenAI's proposed solution of the unramified Langlands conjecture for function fields (problem #014 on the list). Recently, the geometric Langlands conjecture in characteristic 0 has been resolved, leading to substantial progress towards the geometric Langlands conjecture in characteristic p (aka the unramified Langlands conjecture for function fields). In this talk, I will explain the statement of the Langlands conjecture for function fields and its status until recently. Moreover, I will give an overview of the methods used in OpenAI's proof and give appropriate credit to the mathematicians whose work they build on.

The nearby Lagrangian conjecture asks whether every closed exact Lagrangian in the cotangent bundle of a closed manifold is Hamiltonian isotopic to the zero section. Although relatively few results on this conjecture are known, some progress has been made over the years. In files recently released by OpenAI, a counterexample to the conjecture is proposed using the machinery of tube-type generating functions and the K-theory of spectra (with Waldhausen’s stable tube space providing a model), based on a sequence of earlier works. In this talk, I will survey the history and previous discussions on this topic and discuss how the proposed counterexample works. Then I will try to explain why this is simply the start of the entire field.

The Kakeya Conjectures concern overlapping properties of direction-separated tubes in $\mathbb{R}^n$. They are elementary to state, but difficult to prove. In 2025, Wang-Zahl proved the Kakeya Set Conjecture for $n=3$. On October 6, OpenAI claimed progress on Kakeya Conjectures in dimensions 3 and 4. Then on October 26, OpenAI claimed to have resolved these conjectures in all dimensions by proving Sogge's Local Smoothing Conjecture for free waves. I will introduce the Kakeya Conjectures and various human attempts, showing its many faces and why some approaches appeared difficult. I will also discuss how the Kakeya Conjectures connect to harmonic analysis and which directions mathematicians hoped a complete solution would unlock. Then I will share some personal first impressions on OpenAI's proposed proof for the Local Smoothing Conjecture on October 26.

Let X be a (nice) topological space, and suppose that we wish to know if X is homotopy equivalent to a simplicial complex of dimension $\leq n$. An obvious necessary condition is that the cohomology of X (with arbitrary coefficients) must be concentrated in degrees $\leq n$. In the 1960's, Wall showed that this condition is also sufficient when $n \geq 3$, and asked if the same result holds in low dimensions. Last month, OpenAI gave a counterexample in the case $n=2$. In this talk, I'll review some of the mathematics connected to Wall's question and explain a simplified version of the counterexample (following even more recent work of Bartholdi-Bridson-Mikhailov-Nicholson).