Georgia Institute of Technology College of Sciences

Many numerical methods have been developed in the past years for computing weak solutions (with shock waves) to nonlinear hyperbolic conservation laws. My research, specifically, concerns the design of well-balanced numerical algorithms that preserve certain key structure of these equations in various applications, including for problems involving moving phase boundaries and other scale-dependent interfaces. In particular, in this lecture, I will focus on the evolution of a compressible fluid in spherical symmetry on a Schwarzschild curved background, for which I have designed a class of well-balanced numerical algorithms up to third-order of accuracy. Both the relativistic Burgers-Schwarzschild model and the relativistic Euler-Schwarzschild model were considered, and the proposed numerical algorithm took advantage of the explicit or implicit forms available for the stationary solutions of these models. The schemes follow the finite volume methodology and preserve the stationary solutions and, most importantly, allow us to investigate the global asymptotic behavior of such flows and determine the asymptotic behavior of the mass density and velocity field of the fluid. Blog: philippelefloch.org

The Johnson filtration is a filtration of the mapping class group induced by the action of the mapping class group on the lower central series of the fundamental group of a surface.  A theorem of Johnson tells us that the first term of this filtration, called the Torelli group, is finitely generated for surfaces of genus at least 3.  We will explain work of Ershov-He and Church-Ershov-Putman, which uses Johnson's result to show that the kth term of the Johnson filtration is finitely generated for surfaces of genus g at least 2k - 1.  Time permitting, we will also discuss some extensions of these ideas.  In particular, we will explain how to show that the terms of the Johnson filtration are finitely presented assuming the Torelli group is finitely presented.

In recent years, algebraization theorems arising from model theory, in particular o-minimality, have been a crucial ingredient in several breakthroughs in arithmetic geometry and Hodge theory. In this talk, I'll present some of my recent work on p-adic versions of these model theoretic algebraization criteria, with a focus on two different applications of this circle of ideas. The first being an algebraization theorem in the context of Shimura varieties, which are vaguely speaking moduli spaces of Hodge structures. The second being in the context of non-abelian Hodge theory, in the setting of moduli spaces of flat connections and local systems.

Zoom: https://gatech.zoom.us/j/95425627723

 The classical theorems of Perron and Frobenius, which explore spectral properties of nonnegative matrices, have been extensively examined and generalized from various perspectives, including a cone-theoretic (geometric) viewpoint. Concurrently, in the past decade, there has been a notable effort to fuse the techniques of algebraic geometry and combinatorics in an exploration of Lorentzian polynomials by Brändén and Huh, also known as completely log-concave polynomials (CLC) by Anari et.al. or strongly log-concave polynomials by Gurvits.

In this talk, I will discuss my ongoing joint work with Greg Blekherman regarding the class of polynomials with Lorentzian signature (PLS) defined over closed convex cones. This class encompasses various special polynomials, including Lorentzian polynomials over the nonnegative orthant and hyperbolic polynomials over hyperbolicity cones. We establish a compelling connection between PLS over a self-dual cone K and the generalized Perron Frobenius theorem over K. This connection enables us to provide an alternative necessary and sufficient condition to characterize the Lorentzian polynomials.