CBMS Conference: Dyson-Schwinger Equations, Topological Expansions, and Random Matrices

Columbia University, New York NY

Investigators

Abstract

This award provides support for the CBMS conference "Dyson-Schwinger Equations, Topological Expansions, and Random Matrices," which will run from August 28 to September 1, 2017 at Columbia University in New York City. The conference features a series of ten lectures by mathematician Alice Guionnet. Probability, as a field, seeks to understand how large, complex random systems manifest deterministic behaviors, and universal fluctuations around them. Matrices are fundamental objects in mathematics and all of science -- they encode transformations of space, descriptions of atoms in physics, chains of chemical reactions and probabilities in chemistry, and series of observations in experimental sciences, and they are vital in signal processing. In many applications, there is noise present in the system under study, and the associated matrices end up having structured but random entries. The fundamental challenge becomes to understand the effect of noise on these structures. How does it change the key properties of these objects, and how can statistical methods be developed to separate the signal from the noise? The principal lecture series will report on some of the most exciting new advances in tackling these types of questions. The meeting also will include presentations by other experts in the subject area. There will also be daily tutorials and problem sessions, with lecture notes provided in advance to help students prepare adequately. Understanding the large-dimension asymptotics of random matrices or related models such as random tilings has been of great interest for the last twenty years within probability, mathematical physics, and statistical mechanics. Because such models are highly correlated, classical methods based on independent variables fail. Special cases have been studied in detail thanks to specific forms of the laws, such as determinantal laws. These lectures will discuss a general class of models using the so-called Dyson-Schwinger equations and generalizations. The ten lectures cover the following subjects: law of large numbers in random matrices and concentration of measure; central limit theorem in the one-cut case; generalization to the discrete setting of tiling models; topological expansions for large-N asymptotics of partition functions; and generalization to several-cut models. Additional talks will be presented by Charles Bordenave (University of Toulouse), Gaetan Borot (Max Planck Institute for Mathematics, Bonn), Paul Bourgade (New York University), Vadim Gorin (Massachusetts Institute of Technology), and Antti Knowles (ETH Zurich). The website for this conference is: http://www.math.columbia.edu/department/probability/seminar/Guionnet.html