The Geometry of Nonlinear Function Spaces

Abstract: Topologists have long been interested in the topology of function spaces -- indeed homotopy is essentially an understanding of the connected components of function spaces. In these lectures, I'd like to ask questions about the geometry of these function spaces -- a topic that grows out of classical approximation theory, and has interesting applications to variational problems and quantitative geometry -- and (I hope) describe some of the beautiful discoveries of the past several years.

14:00 on Tuesday January 2, 2024, Room 614 Science and Education Building

Lecture 1. Entropy

The first geometric invariant we will consider is the entropy of function spaces. We will give applications to topology and also to the existence of closed geodesics on certain manifolds.

12:00 on Wednesday January 3, 2024, Room 614 Science and Education Building

Lecture 2. Persistent homology

Persistent homology was a tool introduced into computational topology to study proteins and arctic snow fields. It is a useful tool for measuring some non-topologically invariant properties of functionals and can be viewed as a distillation of Morse theory. I will explain this formalism and apply it to some interesting functionals.

12:00 on Thursday January 4, 2024, Room 614 Science and Education Building

Lecture 3. Diameter

Algebraic topology tells us (when it succeeds) whether two functions can be deformed into one another, i.e. can be connected by a path in a function space. We will see that many function spaces are "small worlds", i.e. any two points in the same components have only a few "degrees of separation". We will see that this has signficant geometric implications.

See this link for the history of this lecture series.