Spaces with metric and measure: on their geometry and dynamics
by
Vladimir Pestov
Victoria University of Wellington

Spaces with metric and measure (mm-spaces) form the subject of a new and interesting theory that has grew out of, and is extensively overlapping and interacting with, such subjects as probability theory, asymptotic geometric functional analysis, combinatorics, statistical physics, information theory, and a number of other disciplines. The emergence and development of the theory is linked to the names of Paul L'evy, Vitali Milman, Michel Talagrand, Mikhail Gromov, and others. mm-Spaces in particular provide a natural setting for studying the phenomenon of concentration of measure on high-dimensional structures. In this talk, we will outline the basics of the theory, discuss the dynamical aspects of the situation where an mm-space is acted upon by a group of transformations, and then proceed to some recent applications, especially dwelling on Bekka's concept of an amenable representation.

Date received: April 24, 2001