Concentration and group representations in Banach spaces

Abstract

The theory of Banach spaces and group representations are both well-known areas of mathematics, but the phenomenon of concentration of measure, despite (or maybe because of!) its variety of settings and applications, is less well-known. Throughout the introduction we will see many different examples of the concentration phenomenon manifesting itself, and look at some of the motivations behind the phenomenon. What is concentration of measure? The phenomenon of concentration of measure has only been discovered relatively recently. A common setting for concentration is in a set equipped with both a metric and a measure (an mm-space), thought of as being 'high-dimensional', about which our intuition seems to be often misleading. Often, concentration is linked to Ramsey-type results, and could informally be thought of as being the continuous version of Ramsey theory. To what extent this is true has not been explored in any real depth.