Research Proposal: Borel Equivalence Relations

The study of definable equivalence relations on complete separable metric spaces (i.e., Polish spaces) has emerged as a new direction of research in descriptive set theory over the past twenty years. Quotients of such equivalence relations include the orbit space of an ergodic group action, the set of Turing degrees of subsets of natural numbers, and the space of isomorphism classes of countable models of some Lω1, ω-sentence, among many others. For all but the simplest equivalence relations E on a Polish space X, the collection X/E of E-classes cannot be viewed in any reasonable way as a definable set inside a Polish space, and so the standard techniques of descriptive set theory do not apply in the usual way to their study. The field of Borel equivalence relations seeks to understand the structure of these “singular” spaces X/E. Its methods, as they have been developed over the past decade, draw upon diverse areas of mathematics such as model theory, topology, ergodic theory and orbit equivalence theory, as well as the theory of various classes of countable groups and their definable actions, including especially the free groups, amenable groups, Kazhdan groups, Polish groups, and Lie groups together with their lattice subgroups. One reason for studying the moduli spaces X/E is that they appear as sets of invariants for classification problems arising throughout mathematics. Indeed, a wide range of naturally occurring classes of mathematical objects may be given the structure of a standard Borel space — ie, a Polish space equipped only with its σ-algebra of Borel sets — and in many cases the corresponding notion of classification turn out to be a definable equivalence relation on that space. Consider, for instance, the problem of classifying countable graphs up to isomorphism. Letting C be the set of graphs of the form Γ = 〈N, E〉 and identifying each graph Γ ∈ C with its edge relation E ∈ 2N2 , one may check that C is a Borel subset of the Polish space 2N2 , and hence is itself a standard Borel space. Furthermore, the isomorphism relation on C is simply the orbit equivalence relation arising from the natural action of the infinite symmetric group Sym(N) on C. More generally, if σ is any Lω1, ω-sentence of a countable language L, then