Talk Title:

Group actions and the descriptive classification of equivalence relations

Abstract:

In developing the theory of definable equivalence relations, we look at the possible sets of complete invariants for a variety of classification problems from all areas of mathematics, and by applying the techniques of descriptive set theory to these invariants we get a reasonable framework for comparing the complexity of these classifications. One particularly interesting case is when the equivalence classes are realizable as the orbits of a reasonably definable action of a separable completely metrizable (i.e., Polish) group. For example, isomorphism of finitely-generated groups, isomorphism of countable groups, isomorphism or isometry of separable Banach spaces, conformal equivalence of Riemann surfaces, and even the isomorphism and isometry relations between the separable completely metrizable spaces themselves are all of this type. We will give a brief introduction to the basic ideas and techniques of the research program as well as a couple of interesting examples. If time permits, we will discuss how (under the notion of complexity we are most interested in) the orbit equivalence of an action of any compactly-generated locally compact abelian Polish group is comparable to the isomorphism of rank 1 torsion-free abelian groups which is realizable as an action of the integers.