We show log-supermodularity of densities on a Euclidean space is preserved under convolution with log-concave product densities, confirming a generalized form of a conjecture of Zartash and Robeva. Additionally, this stability yields a "conditional entropy power inequality" for log-supermodular random vectors. The proof relies on a known continuous form of the Ahlswede-Daykin four-function theorem, for which we give a new proof. The talk is based on joint work with James Melbourne and Cyril Roberto.

The Gaussian width of a set is a fundamental quantity in high-dimensional phenomena. I will present some new bounds for the width as well as for the quermassintegrals of a convex body. This is joint work with Reese Pathak.

I will discuss recent results at the intersection of free probability and geometry of polynomials. Concretely, we establish finite free analogues of the Stam and entropy power inequalities for real-rooted polynomials, which, in the large-degree limit, provide fundamentally new proofs of their known free probability counterparts. Equivalently, our results may be interpreted as potential-theoretic inequalities for the zeros of real-rooted polynomials under differential operators that preserve real-rootedness. This is joint work with Nikhil Srivastava and Zachary Stier.

We discuss recent developments in probabilistically checkable proofs (PCPs) that use high-dimensional expansion. We will focus on the construction of low-soundness size efficient PCPs, and their relation to fault-tolerant distributed protocols.