Georgia Institute of Technology College of Sciences

Consider a sequence of independent and identically distributed SL(2, R) matrices. There are several classical results by Le Page, Tutubalin, Benoist, Quint, and others that establish various forms of the central limit theorem for the products of such matrices. I will talk about a recent joint work with Anton Gorodetski and Victor Kleptsyn, where we generalize these results to the non-stationary case. Specifically, we prove that the properly shifted and normalized logarithm of the norm of a product of independent (but not necessarily identically distributed) SL(2, R) matrices converges to the standard normal distribution under natural assumptions. A key component of our proof is the regularity of the distribution of the unstable vector associated with these products.

Volume polynomials constitute a distinguished class of log-concave polynomials with remarkable analytic and combinatorial properties arising from convex bodies and projective varieties. I will introduce new entropy inequalities satisfied by volume polynomials, discuss applications to the combinatorics of algebraic matroids, introduce the new class of analytic matroids, and pose several open questions (based on joint with Lukas Grund, Mateusz Michalek, Henrik Süss, and Botong Wang).

The Fox trapezoidal conjecture is a longstanding open problem about the coefficients of the Alexander polynomial of alternating links. In this talk, we will discuss recent work which settled this conjecture for “special alternating links”. The first tool is a graph theoretic model of the Alexander polynomial of an alternating link discovered by Crowell in 1959. The second is the theory of Lorentzian polynomials, developed by Brändén and Huh in 2019 and a key part of Huh’s Fields medal work. We will show how a version of Crowell’s model produces a refinement of the Alexander polynomial of special alternating links that is Lorentzian, from which the result follows quickly.

The brain performs efficient, adaptable, and robust computations of noisy sensory information in changing environments. While artificial neural networks have achieved remarkable successes in recent years, the brain's computational capacity is yet to be matched. To understand mechanisms underlying the exquisite computational efficiency and flexibility of the brain, complex architecture and dynamics of the biological neural networks should be studied. In this talk, I will give a broad overview of recent research projects from my group, that investigate links between neural coding and network structures using data-driven modeling.