Spaces: from Analysis to Geometry and Back

There are many problems in analysis which involve constructing a function with desirable properties or understanding the properties of a function without completely precise information about its structure that cannot be easily tackled using direct “hands on” methods. A fruitful strategy for dealing with such problems is to recast it as a problem concerning the geometry of a well-chosen space of functions, thereby making available the many techniques of geometry. For example, one can construct solutions for a large class of ordinary differential equations by applying the “contraction mapping principle” from the theory of metric spaces to an appropriate space of continuous functions. The application of geometric techniques to spaces of functions proved so successful that it led to the birth of an independent area of mathematical research known as functional analysis. This subject has taken on a life of its own, but the deep interplay between geometry and analysis is still very relevant. The goal of this course is to investigate some of the basic ideas and techniques which drive this interplay.