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: Waves are ubiquitous in our daily life. Two best-known linear models are the free wave and free Schrödinger equations, whose simplest forms are very amenable to Fourier analysis. Still, a basic question—how large can a solution be, and where can it be large?—is surprisingly subtle and only partly understood, especially in higher dimensions. Over decades, it transpired that in order to answer this fundamental question, one often needs to understand whether and how much the solution can concentrate on important subsets of $\mathbb{R}^n$. I will discuss three kinds of such subsets (convex sets, semialgebraic sets and lattices) and their importance based on sample problems. Some of them have nice connections to nearby areas such as number theory, geometry and combinatorics.

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.