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.