University of Oxford

The joint spectral radius of a finite family S of matrices measures the rate of exponential growth of the maximal norm of an element from the product set S^n as n grows. This notion was introduced by Rota and Strang in the 60s. It arises naturally in a number of contexts in pure and applied mathematics. I will discuss its basic properties and focus on a formula of Berger and Wang and results of J. Bochi that extend to several matrices the classical for formula of Gelfand that relates the growth rate of the powers of a single matrix to its spectral radius. I give new proofs and derive explicit estimates with polynomial dependence on the dimension, refining these results. If time permits I will also discuss connections with the Tits alternative, the notion of joint spectrum, and a geometric version of these results regarding groups acting on non-positively curved spaces.

In his pioneering and celebrated 1968 paper on the elementary theory of finite fields Ax asked if the theory of the class of all the finite rings $\mathbb{Z}/m\mathbb{Z}$, for all $m>1$, is decidable. In that paper, Ax proved that the existential theory of this class is decidable via his result that the theory of the class of all the rings $\mathbb{Z}/p^n\mathbb{Z}$ (with $p$ and $n$ varying) is decidable. This used Chebotarev’s Density Theorem and model theory of pseudo-finite fields.

I will talk about a recent solution jointly with Angus Macintyre of Ax’s Problem using model theory of the ring of adeles of the rational numbers.

I'll briefly explain quantum groups and $R$-matrices and why they're the same thing. Then we'll see how to construct various $R$-matrices from Nakajima quiver varieties and some possible applications.

Wreath-like product groups were introduced recently and used to construct the first positive examples of rigidity conjectures of Connes and Jones. In this talk, I will review those examples, as well as discuss some ideas to construct examples with other rigidity phenomena by modifying the wreath-like product construction.

This is a continuation of https://www.maths.ox.ac.uk/node/61240

The Dedekind zeta function generalises the Riemann zeta
function to other number fields than the rationals. The analytic class number
formula says that the leading term of the Dedekind zeta function is a
product of invariants of the number field. I will say some things
about the class number formula, about L-functions, and about Stark's
conjecture which generalises the class number formula.

Let $p \ge 3$ be a prime integer. The density of a non-empty solution set of a system of affine equations $L_i(x) = b_i$, $i=1,\dots,k$ on a vector space over the field $\mathbb{F}_p$ is determined by the dimension of the linear subspace $\langle L_1,\dots,L_k \rangle$, and tends to $0$ with the dimension of that subspace. In particular, if the solution set is dense, then the system of equations contains at most boundedly many pairwise distinct linear forms. In the more general setting of systems of affine conditions $L_i(x) \in E_i$ for some strict subsets $E_i$ of $\mathbb{F}_p$ and where the solution set and its density are taken inside $S^n$ for some non-empty subset $S$ of $\mathbb{F}_p$ (such as $\{0,1\}$), we can however usually find subsets of $S^n$ with density at least $1/2$ but such that the linear subspace $\langle L_1,\dots,L_k \rangle$ has arbitrarily high dimension. We shall nonetheless establish an approximate boundedness result: if the solution set of a system of affine conditions is dense, then it contains the solution set of a system of boundedly many affine conditions and which has approximately the same density as the original solution set. Using a recent generalisation by Gowers and the speaker of a result of Green and Tao on the equidistribution of high-rank polynomials on finite prime fields we shall furthermore prove a weaker analogous result for polynomials of small degree.

Based on joint work with Timothy Gowers (College de France and University of Cambridge).

Perverse sheaves are an indispensable tool in representation theory. Their stalks often encode important representation theoretic information such as composition multiplicities or canonical bases. For the nilpotent cone, there is an algorithm that computes these stalks, known as the Lusztig-Shoji algorithm. In this talk, we discuss how this algorithm can be modified to compute stalks of perverse sheaves on more general varieties. As an application, we obtain a new algorithm for computing canonical bases in certain quantum groups as well as composition multiplicities for standard modules of the affine Hecke algebra of $\mathrm{GL}_n$.

This course will serve as an introduction to the theory of gradient flows with an emphasis on the recent advances in metric spaces. More precisely, we will start with an overview of gradient flows from the Euclidean theory to its generalisation to metric spaces, in particular Wasserstein spaces. This also includes a short introduction to the Optimal Transport theory, with a focus on specific concepts and tools useful subsequently. We will then analyse the time-discretisation scheme à la Jordan--Kinderlehrer-Otto (JKO), also known as minimising movement, and discuss the role of convexity in proving stability, uniqueness, and long-time behaviour for the PDE under study. Finally, we will comment on recent advances, e.g., in the study of PDEs on graphs and/or particle approximation of diffusion equations.

PhD_course_Esposito_1.pdf

This course will serve as an introduction to the theory of gradient flows with an emphasis on the recent advances in metric spaces. More precisely, we will start with an overview of gradient flows from the Euclidean theory to its generalisation to metric spaces, in particular Wasserstein spaces. This also includes a short introduction to the Optimal Transport theory, with a focus on specific concepts and tools useful subsequently. We will then analyse the time-discretisation scheme à la Jordan--Kinderlehrer-Otto (JKO), also known as minimising movement, and discuss the role of convexity in proving stability, uniqueness, and long-time behaviour for the PDE under study. Finally, we will comment on recent advances, e.g., in the study of PDEs on graphs and/or particle approximation of diffusion equations.

PhD_course_Esposito_0.pdf