Topics in Geometry: Lipschitz Extension, Reverse Isoperimetry and Rounding

Suppose that we are given a Lipschitz function f from a subset S of a metric space X to a metric space Y. Can we extend f to a Y-valued Lipschitz function that is defined on all of X? This depends on geometric properties of X,Y,S,f, and it is typically impossible, but for over a century a range of creative methods were devised to prove such extension theorems in many settings. The course describes results on extending Lipschitz functions, starting from the classical and arriving to current research. This includes links to questions on how one can reverse the isoperimetric inequality and ways to round continuous space to a discrete subset.