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Randomness extractors are beautiful objects at the heart of pseudorandomness. Over the past 40 years, their influence within (and "without") pseudorandomness has only grown stronger, as they have proven to be strikingly powerful tools for the construction of pseudorandom generators, lower bounds, and beyond. Today, the field remains as vibrant as ever, and many open problems remain. In this talk, I’ll tell you about my top 3.

Linear maps are among the most fundamental objects in mathematics. Yet surprisingly little is known about the extent to which random linear maps behave pseudorandomly, giving rise to intriguing questions in pseudorandomness and coding theory. In this talk, I will discuss some recent work on random linear maps and highlight several interesting directions for future research.

In this talk we will give a brief informal introduction to discrete periodic operators. Much of their spectral theory can be translated into the study of characteristic polynomials of matrices with Laurent polynomial entries. The zero sets of these polynomials encode the operator's spectrum. We will also see a few examples and mention some recent results on flat bands, reducibility, and isospectrality.

Coboundary expansion generalizes the isoperimetric inequalities of graphs to higher-dimensional complexes. Recent work by Chapman–Lubotzky, Kaufman–Oppenheim–Weinberger, Oppenheim–Valentiner-Branth, and others has produced algebraic constructions of bounded-degree coboundary expanders.

The Riemannian analogue of coboundary expansion is the uniform waist inequality. Recent work by Bader–Sauer and Fraczyk–Lowe gives families of Riemannian manifolds satisfying this inequality.

Are these discrete and continuous constructions manifestations of the same underlying phenomenon? I will explain why an answer in either direction would be interesting.

I'll be discussing some work on efficient tensor decomposition and a new thing called functional stochastic localization -- these are unrelated to each other. I'm most interested in the foundational frameworks these ideas may produce, and somewhat in specific applications (e.g., sampling for the latter).

In this talk, I will briefly introduce my work on optimal list decoding and list recovery of algebraic codes, their connections to expanders and pseudorandomness, and some future directions I am interested in exploring.

Abstract not available

Random matrix theory (RMT) is the area of probability theory concerned with spectral behavior of large matrices with random entries. Historically, RMT originates from statistics through the work of Wishart in 1920s and, independently, from mathematical physics thanks to Wigner in 1950s. Wigner first had the brilliant idea to model complicated operators arising in the study of many-body quantum systems as random matrices. It has since been rigorously established that spectra of certain classical random matrix ensembles do, in fact, exhibit the behavior that is expected from deterministic quantum chaotic operators (the latter is mostly out of reach of formal mathematical treatment at this time). I will briefly mention some classical results in RMT, in particular as they relate to quantum chaos. I will also discuss an alternative, yet related, approach to modeling quantum chaos using graph theory and share related open problems.