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
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
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.