Abstract. Metric spaces and nonexpansive maps are ubiquitous in mathematics (geometry, topology, several complex variables, operator theory, group theory, ergodic theory, ...) In spite of this, one seldom encounters studies with this general perspective in mind. (This is in contrast to the situation for the basic theory of linear operators and vector spaces.) There are however nontrivial general results and open problems here.
In this talk I will focus on metric methods in ergodic theory. More precisely, I will present a conjectural metric ergodic theorem, a partial result which simultaneously extends the ergodic theorems of von Neumann, Birkhoff, Beck-Schwartz, Oseledec, and with certain consequences for Brownian motions on manifolds (first proved by Ballmann and Ledrappier using the difficult so-called rank rigidity theorem).
This is in part based on joint work with Margulis.