##
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

titled
##
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:**

**Lecture 1 (Tuesday, November 21st, 4pm):**
Number Theory, Combinatorics and Recurrence in Dynamical Systems;
the Correspondence Principle
**Lecture 2 (Wednesday, November 22nd, 4pm):**
Ergodicity, Mixing, Conventional and non-Conventional Ergodic Theorems
**Lecture 3 (Thursday, November 23rd, 4pm):**
The Long Term Memory of Dynamical Systems and the Strange Role of
Nilpotent Groups and Nilflows

**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.

ALL ARE WELCOME TO ATTEND!