*The 2001 Bowen Lectures will be delivered by Yakov G. Sinai on October 2, 4, and 5.*

Department of Mathematics, University of California, Berkeley, presents

The 2001 Bowen Lectures

Yakov G. Sinai

Professor of Mathematics

Princeton University

and

Member of the Landau Institute of

Theoretical Physics, Moscow**"Statistical (3x+1)-Problem"Tuesday, October 2**

10 Evans Hall - 5:10pm-6:00pm

Let x be an odd number. Then 3x+1 is even and there is a unique integer k > 0 such that (3x+1)/2^k is again odd. This defines a transformation t where t(x) = (3x+1)2^k. The (3x+1)-problem asks whether it is true that for each odd integer x there is a positive integer n(x) such that the n(x) iteration of t applied to x equals to one. The problem was introduced more than 70 years ago and is very popular in combinatorics. Despite the efforts of many authors no real progress has been achieved. I shall explain a new approach to this problem and outline some results obtained using this approach.

**Reception in 1015 Evans at 6:15pm after Tuesday's lecture.**

**"Adiabatic Piston as a Dynamical System"**

Thursday, October 4

Thursday, October 4

Sibley Auditorium, 4:10pm-5:00pm

Adiabatic Piston is a macroscopic body that moves under the action of collision with gas particles. The problem of adiabatic piston appeared in statistical mechanics almost a hundred years ago and is connected with foundations of statistical mechanics. Finite-dimensional approximations of this system lead to interesting examples of dynamical systems. In this lecture we shall discuss properties of this system in the thermodynamical limit.

**"On Some Problems in Mathematical Hydrodynamics"**

Friday, October 5

Friday, October 5

Sibley Auditorium - 4:10pm-5:00pm

It is believed that for reasonable initial conditions the 3D-Navier-Stokes system has a nice strong solution. However, the corresponding problem seems to be very hard. I shall discuss new approximations of the 3D-Navier-Stokes system which lead to new systems of quasilinear equations with non-local coefficients. Finite-dimensional versions of this system have remarkable first integrals but it is an open question whether they are integrable.