MIT Probability Seminar

Building 2, 147

Speaker: Pierre Patie (Cornell University)Title: A spectral and algebraic algorithm: the centralizer and the fixed points, scaling and universality classesAbstract:Over the last few decades, the exploration of scaling limits and universality classes has unveiled a spectrum of intriguing results, alongside complex and fascinating challenges. In this talk, we present a comprehensive framework designed to address these challenges in a constructive and solvable manner. It is based on an appropriate combination of group representation theory, group actions, spectral theory and operator algebras. Relying on the Stone-von Neumann Theorem, we identify a canonical setting for this framework, the so-called canonical G-module, and design a constructive algorithm. This formalism not only highlights the fundamental role played by the choice of the representation of mathematical objects but also offers constructive perspectives and connections into classical mathematical topics such as the spectral theory of self-adjoint operators, Lie point symmetry, von Neumann algebras, and the fundamental Stone-von Neumann theorem. We will illustrate this framework by describing the universality classes of the LUE ensemble, emphasizing the canonical role of its limit to the Bessel ensemble in the comprehensive framework. Finally, we shall discuss the role played by the Fourier transform, the Laplacian, and Brownian motion in this formalism.