Texas Analysis and Mathematical Physics Symposium 2019/20 / Jan 31-Feb 2
Paul Bourgade (Courant Institute, New York University)
Branching processes in random matrix theory and analytic number theory
Fyodorov, Hiary and Keating have conjectured that the maximum of the characteristic polynomial of random matrices behaves like extremes of log-correlated Gaussian fields. This allowed them to predict the size of local maxima of L-function along the critical axis. I will explain the origins of this conjecture and some rigorous understanding, for unitary random matrices and the Riemann zeta function, relying on branching structures.
Dmitry Dolgopyat (University of Maryland)
Randomness of toral translations
We will describe recent results about limit theorems for toral translations and present some open questions.
Konstantin Khanin (University of Toronto)
On stationary solutions to the stochastic heat equation
Bryna Kra (Northwestern University)
Symmetries of shift systems
Symbolic dynamics is a tool for studying general dynamical systems, by coding iterates of points with some alphabet. The symmetries of a symbolic, or shift, system form a countable group, and can range from being quite complicated to very simple, reflecting the underlying dynamics. I will give an overview of what is known about these groups, including the definition of a new version of a symmetry, and highlight numerous open problems in the area.
Jared Speck (Vanderbilt University)
A New Formulation of Compressible Euler Flow: Miraculous Geo-Analytic Structures and Applications
I will discuss my works (some joint) on the compressible Euler equations with non-trivial vorticity and entropy. We derived a new formulation of the equations exhibiting miraculous geo-analytic structures, including I) A sharp decomposition of the flow into geometric "wave parts'' and "transport-div-curl parts;'' II) Null form source terms; and III) Structures that allow one to propagate one additional degree of differentiability (compared to standard estimates) for the entropy and vorticity. We were inspired to search for such a formulation by Christodoulou's groundbreaking 2007 monograph on shock formation for relativistic Euler solutions in irrotational and isentropic regions. I will then describe how the new formulation can be used to derive sharp results about the dynamics, including results on stable shock formation and the existence of low-regularity solutions. I will emphasize the role that nonlinear geometric optics plays in the framework and highlight how the new formulation allows for its implementation. Finally, I will connect the new formulation to the broader goal of obtaining a rigorous mathematical theory that models the long-time behavior of solutions that can develop shocks. Various aspects of this program are joint with J. Luk, M. Disconzi, C. Luo, G. Mazzone, and L. Abbrescia.
Balint Virag (University of Toronto)
The random planar geometry of the directed landscape
Perturbations of Euclidean geometry in the plane are conjectured to have a universal scaling limit. The resulting random geometry is given by a random directed metric on the plane. I will explain the settings in which this limiting behavior has been proven, and show some properties of the limiting geometry, and how it is connected to models in the KPZ universality class. Joint work with Duncan Dauvergne, Mihai Nica and Janosch Ortmann.