Ball will study geometric properties of metric spaces, in particular the Lp-spaces, which describe the rate at which random walks in these spaces can wander. He has shown that these properties are intimately connected with questions about factoring and extending Lipschitz maps. The aim is to develop a non-linear theory which parallels the powerful theory of type and cotype in linear spaces. Such a theory can be expected to have numerous applications to the analysis of the various types of metric spaces which arise in all areas of mathematics. Banach space theory is that part of mathematics that attempts to generalize to infinitely many dimensions the structure of 3-dimensional Euclidean (i.e.ordinary) space. The axioms for the distance function in a Banach space are more relaxed than those for Euclidean distance (For example, the "parallelogram law" is not required to hold.), and as a result, the "geometry" of a Banach space can be quite exotic. Much of the research in this area concerns studying the structure theory of Banach spaces.
