Series: Penn State Logic Seminar Date: Tuesday, April 15, 2003 Time: 10:10 - 11:00 PM Place: 116 McAllister Building Speaker: John D. Clemens, California Institute of Technology, Mathematics Title: Classifying Borel Automorphisms up to Conjugacy Abstract: Given a standard Borel space, we may consider the group of all Borel automorphisms of the space together with the equivalence relation of conjugacy. We investigate the complexity of this conjugacy relation in two respects. First, we can ask what sort of complete invariants are necessary to classify automorphisms up to conjugacy. The theory of definable equivalence relations can be used to formulate this precisely. We use techniques from ergodic theory to show that conjugacy is very complicated in this sense; it is not possible to effectively compute "simple" complete invariants such as reals (in particular, this relation is not "smooth"). Second, we may ask how complicated the conjugacy relation is in a descriptive set-theoretic sense. Here we use a result about uniformizations of closed sets to show that the conjugacy relation is Sigma^1_2-complete.