Aidan Backus will give a talk at 04/09/2024.

A map between metric spaces is said to be Lipschitz if it only stretches line segments by a bounded factor. In these lectures I will discuss some answers to the following question: “Given a domain $U$ and a Lipschitz map $f: \partial U \to \mathbb R^D$, what is the best way to extend $f$ to a Lipschitz map $u: U \to \mathbb R^D?$” Aside from its intrinsic interest, this question arises naturally if one wishes to study stochastic games, deformations of Riemann surfaces, or the $p$-Laplacian.

When $D = 1$, the optimal extension is a viscosity solution of the infinity-Laplacian, so we will have the opportunity to also talk about the method of vanishing viscosity in some generality. When $D \geq 2$, the situation becomes much murkier and I will discuss the pros and cons of the approaches of Sheffield and Smart ‘10 and Daskalopoulos and Uhlenbeck ‘22. I will also discuss some possible future research directions, and some open problems waiting to be snapped up by an eager graduate student.