A set of matrices can be defined as uniformly hyperbolic if products of the matrices have a norm that grows exponentially. A paper written by Avila, Bocci, and Yoccoz in 2008 has expanded on this concept and posed a variety of questions on this subject. In this talk we will go over some of the concepts covered in this paper, a few additional tools developed to help study this subject, and ways the tools are being used to address the questions posed.

Between 2013 and 2015, Marcus, Spielman and Srivastava wrote a sequence of papers where they famously solved the Kadison-Singer problem and proved the existence of Ramanujan graphs of all sizes. For the latter, they used a convolution of polynomials introduced by Walsh, which they showed to have surprising connections to free probability theory. This discovery gave rise to finite free probability, which studies polynomial convolutions from a free probability perspective. With the aim of unifying the results of Marcus, Spielman and Srivastava, and developing general machinery for deducing root bounds, we extend the framework of finite free probability to multivariate polynomials. We show that this extended framework has interesting parallels with the theory of freeness with amalgamation, and can potentially be used to obtain important results in diverse areas, ranging from algebraic combinatorics to operator theory. This is work in progress with Nikhil Srivastava.

Let F be a finite field with q elements. A (projective) algebraic set over F is the set of common zeros in the projective m-space over F of a bunch of homogeneous polynomials in m+1 variables with coefficients in F. Fix positive integers r, m and d with d < q.  We consider the following question:

What is the maximum number of points in an algebraic set in the projective m-s[space over F given by the vanishing of r linearly independent homogeneous polynomials of degree d with coefficients in F?

The case of a single homogeneous polynomial (or in geometric terms, a projective hypersurface) corresponds to a classical inequality proved by Serre in 1989. For the general case, an elaborate conjecture was made by Tsfasman and Boguslavsky, which was open for almost two decades. Recently significant progress in this direction has been made, and it is shown that while the Tsfasman-Boguslavsky Conjecture is true in certain cases, it can be false in general. Some new conjectures have also been proposed. We will give a motivated outline of these developments. If there is time and interest, we will also explain the close connections of these questions to the problem of counting points of sections of Veronese varieties by linear subvarieties of a fixed dimension, and also to coding theory.

This talk is mainly based on joint works with Mrinmoy Datta and with Peter Beelen and Mrinmoy Datta.

Abstract:   Motived by the notion of the algebraic hyperbolicity introduced by X. Chen, we introduce the notion of Nevanlinna hyperbolicity for a pair of (X, D), where X is a projective variety and D is an effective Catier divisor on X. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard type extension theorem (more generally the Borel hyperbolicity). It also implies the algebraic hyperbolicity. The key is to use the Nevanlinna theory on parabolic Riemann surfaces recently developed by Paun and Sibony. This is a joint work with Yan He.