Hillel Furstenberg
May 14 (colloquium), May 15 & May 16, 2003
Math Sciences 6627: 1:00 pm

Title:
Non-conventional Ergodic Theorems, Nilpotent Groups, and the Long-Term Memory of Dynamical Systems (I and II)

Abstract
The "conventional" ergodic theorems deal with long-term time averages of dynamic parameters. Recently there has been interest in more complex expressions involving the present state, a far-off future state, as well as still more remote future states of a system. Such expressions play a role in combinatorics - specifically, in Ramsey theory, but are of independent interest in ergodic theory. Quite remarkably, it has been found that the "long-term" constraints on very general dynamical systems are of an algebraic character. More specifically nilpotent Lie groups and their homogeneous spaces have been shown to play a key role in identifying these constraints. With this information it has been possible to establish a variety of non-conventional (mean) ergodic theorems. In these lectures we shall try to clarify this interplay of dynamical theory, group theory and combinatorics, giving, at the same time, an introduction to the elements of ergodic theory.

Background
Hillel Furstenberg holds the Maurice and Clara Weil Chair of Mathematics at the Hebrew University of Jerusalem. Professor Furstenberg has research interests across a wide range of areas: combinatorics, number theory, probability theory, ergodic theory, and group theory. In his work, he has discovered entirely novel interrelations between many of these areas, leading to simplified proofs of old results and to new results not obtainable otherwise.

Furstenberg's work has been recognized by the award of the Harvey Prize (1993), the Israel Prize (1993) and the Rothschild Prize. He has been invited to give talks at the International Congresses of Mathematicians in 1970 and 1990. In 1989, Furstenberg was elected to the US National Academy of Sciences.