SCHOOL OF MATHEMATICS

Ordway Lectures

February 24-26, 2004

Hillel Furstenberg

Hebrew University, Jerusalem, Israel
Non-Conventional Ergodic Averages, Nilpotent Groups, and the Long-Term Memory of Dynamical Systems

Abstracts of Lectures

Lecture 1: Patterns in the Stars, Recurrence in Dynamical Systems, and the Combinatorial Background for Non-Conventional Ergodic Theorems

Ramsey Theory - a branch of combinatorics - treats the phenomenon that rich structures often must contain certain patterns.  This can be regarded as a form of "recurrence", and often can be tied to the phenomenon of recurrence in dynamical systems. We shall give examples of this and show how the investigation of recurrence phenomena has motivated the the study of non-conventional ergodic averages.

Lecture 2: Ergodicity, Mixing and the Long-Term Memory of Dynamical Systems

We discuss briefly the historical background for Ergodic Theory and the notion of ergodic systems which form the building blocks for this theory.  Ergodic systems can be classified by the degree of forgetfulness - or randomness - which they display. We'll discuss notions of mixing and higher order mixing which will culminate in Bergelson's "polynomial ergodic theorem" for weakly mixing systems. We will begin our analysis of non-weakly-mixing systems.

Lecture 3: Ergodic Geometry and the Role of Nilpotent Groups and Nilmanifolds

We will describe the basic ideas underlying the recent work of B. Host and B. Kra in which they were able to establish a very general non-conventional ergodic theorem.  "Ergodic Geometry" enables one to isolate "geometrically meaningful" objects in the context of ergodic (rather than transitive) group actions,and the rudimentary study of these leads to a "completion" of the acting group.  The fact that this completion is nilpotent for certain cases will play a crucial role.

For more information: see the web page of the School of Mathematics, University of Minnesota, at http://www.math.umn.edu
10/29/2003