Abstract: Spin glasses are disordered magnetic systems that display remarkably complex collective behavior. Understanding these systems has challenged both physicists and mathematicians for decades. In the late 1970s, Giorgio Parisi proposed a revolutionary solution to the Sherrington–Kirkpatrick model. At the heart of his theory is the Parisi measure, an object that reveals the hidden organization of disorder. In this talk, I will survey what is now rigorously known about the Parisi measure and highlight the ​major open problems in the field.

Abstract: The “Convexity Conjecture” by Talagrand asks (very roughly) whether one can “create convexity” in constant steps regardless of the dimension of the ambient space. Talagrand also suggested a discrete version of the Convexity Conjecture and called it “my lifetime favorite problem,” offering $1,000 prize for its solution. We introduce a reformulation of the discrete Convexity Conjecture using the new notion of “k-thresholds,” which is an extension of the traditional notion of thresholds, introduced by Talagrand. Some ongoing work on understanding k-thresholds, along with a (vague) connection between the Kahn-Kalai Conjecture and the discrete Convexity Conjecture, will also be discussed. Joint work with Ascoli, He, and Talagrand.

Machine learning is transforming mathematical discovery, enabling advances on longstanding open problems. In this talk, I will discuss AlphaEvolve, a general-purpose evolutionary coding agent that uses large language models to autonomously discover old and new mathematical constructions and potentially go beyond them. AlphaEvolve tackles a wide variety of problems across analysis, geometry, combinatorics, and number theory. In some instances, AlphaEvolve is also able to generalize results for a finite number of input values into a formula valid for all input values. Furthermore, we are able to combine this methodology with Deep Think and AlphaProof in a broader framework where the additional proof-assistants and reasoning systems provide automated proof generation and further mathematical insights. This illustrates how general-purpose AI systems can systematically successfully explore broad mathematical landscapes at an unprecedented speed, leading us to do mathematics at scale.