Course Abstract

Course: SDEs and the Propagation of Randomness: An Introduction

Abstract: This summer school offers an elementary introduction to stochastic differential equations (SDEs) . We will begin with the foundations of discrete time Markov processes and tools to study their long-time behavior. We will then explore the concept of a stochastic differential equation (SDE) and their connection to partial differential equations (PDEs). Key topics will include Brownian motion, Itô calculus, stationary measures, recurrence and transience, and the impact of degenerate noise. The course will feature several practical examples such as stochastic gradient descent and Ornstein-Uhlenbeck processes.

Course: Python and Dynamical Systems

Abstract: This summer school course offers an elementary introduction to the Hamiltonian formalism, symplectic integrator and Python programming. We will begin with a simple introduction to Hamiltonian dynamics to set a common ground for the next two classes. In the second class, we will introduce the basics of python (variables, loop, dictionaries and math libraries).
In the last class, we will see how to use Python to analyze some data, built a neural network and program a symplectic integrator for Hamiltonian systems.

Course: Random Matrix Theory

Abstract:  We will provide an introduction to random matrices through the lens of “universality”, a concept originating with the notion that sums of random variables tend to behave like Normal (Gaussian) random variables as the size of the sum grows to infinity.  From this first example of universality (the central limit theorem), we will introduce the fundamental examples of random matrices and the central objects of study – their eigenvalues.  To see universality in action, we will study the limiting behavior of the distribution of eigenvalues as the matrix size grows to infinity in the simplest example, and then explain how the same behavior emerges for many different examples of random matrices. 

Random matrices are an important instance of integrability.  First, the ability to compute very detailed eigenvalue statistics is due to the discovery of a connection to Riemann-Hilbert problems, which will let us explore the limits to the universality of the eigenvalue density, as well as study the behavior of the largest eigenvalue.  Second, random matrices are  directly connected to 2-dimensional models of quantum gravity through the combinatorics of graphs embedded in Riemann surfaces.  These ideas will be explored in the lectures on Thursday and Friday, and will be adjusted based on the interests of the participants.