Fri Apr 22, 2011
60 Evans Hall, 4–6 PM
Matthew D. Foreman (Professor of Mathematics, University of California, Irvine )
Classifying Measure Preserving Diffeomorphisms of the Torus
The statistical behavior of deterministic dynamical systems can be completely random. The ergodic theorem tells us that the statistical behavior of a system carrying an invariant ergodic measure is captured by the isomorphism class of that measure. Motivated by this type of phenomenon, in 1932 von Neumann proposed the program of classifying the ergodic measure preserving systems.
In this talk we will describe applications of the tools of descriptive set theory to show that this program is impossible. In earlier joint work with Weiss, it was shown that the isomorphism relation is “turbulent” and hence no generic class can be reduced to an S-infinity action. With Rudolph and Weiss it was shown that the isomorphism relation is a complete analytic set.
But what about concrete systems: C-infinity measure preserving transformations of the torus? Recent work with Weiss extends these results to such diffeomorphisms. Finally a very recent result shows that the “graph-isomorphism problem” can be reduced to isomorphism of measure preserving diffeomorphisms. As a corollary the isomorphism relation is complete for S-infinity actions.
The biweekly LOGIC TEA will be held in the Alfred Tarski Room (727 Evans Hall) immediately following the Colloquium (with support from the Graduate Assembly).