Lecture 4: Dynamical systems in high dimension: methods and applications

This series of lectures aims at reviewing the research activity on the study of dynamical systems in high dimension. This subject appears in several contexts: in physics, understanding the many body dynamics of complex systems is essential to characterize their equilibration properties and/ or lack thereof. Beyond physics, understanding the dynamics of optimization algorithms is essential when the corresponding optimization problems are high-dimensional and non-convex. For example this is the typical case of the training dynamics of articial neural networks. The purpose of this course is to review what we know about dynamical systems in high dimension in several contexts and to discuss dynamical mean eld theory which is the main toolbox to study these problems.