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.

The”Schur index” is typically defined as a protected operator count in 4d N=2 superconformal field theories. It turns out in fact that one can define it for a generic 4d N=2 theory, conformal or not, by using the holomorphic-topological twist. Its categorification, namely the space of holomorphic-topological local operators, is expected to be a Poisson vertex algebra. However, for a general non-conformal theory, not much is known about the shape of this PVA. For 4d N=2 gauge theories with matter, I will formulate this PVA as a (relative) Lie algebra cohomology problem and then for the case of pure SU(2) Seiberg-Witten theory propose an explicit answer for the cohomology.

I will explain some results on the localization and deformation theory of vertex algebras, algebraic objects encoding a class of topological associative algebras generalizing the enveloping algebra of an affine Kac-Moody Lie algebra. I will also explain how these results can be used to give geometric constructions of free field realizations, embeddings of these algebras into infinite dimensional Weyl algebras, motivated by the physics of 4d N=2 superconformal field theories. All new results that will be presented are in joint work with Sujay Nair.

We discuss the strategies and results on the determination of index and nullity of minimal surfaces. In particular, we prove that for any large enough $m$, the genus $\gamma=m+1$ equator-poles minimal surface doubling of the equatorial two-sphere in the round three-sphere, which has two catenoidal bridges at the poles and  $m$ bridges equidistributed along the equatorial circle and was discovered in earlier work of Kapouleas, has index $2g+5=2m+7$ and nullity $6$.  We also discuss the progress in the study of indices of minimal surfaces from similar constructions.

https://yale.zoom.us/j/92334178441?pwd=FmjkR0LeihzxyR7AaNzsvTIPa11tmK.1 [13]

A Tate-Shafarevich twist of a (proper) fibration modifies it by a 1-cocycle of automorphisms given by flows of (holomorphic) vector fields relative to the base, locally in the analytic topology. In general, the total space of a twist does not even have to be homeomorphic to that of the original fibration. Nevertheless, it was conjectured by Saccà that if one started with a Lagrangian fibration of an irreducible hyper-Kähler variety, then the total space of the resulting twist should always be deformation-equivalent to that of the original fibration, provided that it is also algebraic. I will introduce evidence towards this conjecture, including coincidences of certain cohomological invariants, as well as a proof under further topological constraints.

We prove that in any dimension n there exists an origin-symmetric ellipsoid of volume c n^2 that contains no points of Z^n other than the origin. Here c > 0 is a universal constant. Equivalently, there exists a lattice sphere packing in R^n whose density is at least c n^2 / 2^n. Previously known constructions of sphere packings in R^n had densities of the order of magnitude of at most n log n / 2^n. Our proof utilizes a stochastically evolving ellipsoid that accumulates at least  c n^2 lattice points on its boundary, while containing no lattice points in its interior except for the origin.

https://yale.zoom.us/j/99948057179 [14]

We will have our first introductory and organizational meeting  to study the recent advances where matroids are used to show the failure of the integral Hodge conjecture.