Paul Turán Memorial Lectures 2006

This year's Paul Turán Memorial Lectures will be delivered by

Hillel Fürstenberg

of the Einstein Institute of Mathematics , The Hebrew University of Jerusalem

From Arithmetic Progressions to Nilmanifolds: A Chapter in Ergodic Theory

The lectures are organized by the János Bolyai Mathematical Society and will take place in the Main Lecture Hall of the Rényi Institute.

Program of the lectures:

Abstract: 1. We present the historical background that has led to the interplay of ergodic theory with Ramsey theory. On the combinatorial side the "Erdos-Turán" conjecture was crucial, and by way of a certain "correspondence principle" the notion of "recurrence", familiar in dynamical phenomena, can be brought to play.

2. The ergodic theoretic approach to Ramsey-theoretic questions has the advantage that it allows for very general structure theorems available for measure preserving dynamical systems to enter the picture. The fact that both systems exhibiting a great deal of randomness as well as systems exhibiting some generalized periodicity display recurrence phenomena enables one to deduce these in great generality.

3. The ergodic theoretic analysis shows that quite surprisingly, the obstruction to randomness that is relevant to our goal always has an algebraic source. Another way of putting this is that any correlation between the past, the present and long range future is due to underlying "group-related" behavior. The groups that enter here turn out to be nilpotent. We will try to explain this and give the application to Ramsey theory. In all of this the work of Bergelson, Kra, and Host plays a crucial role.