Descriptive set theory equivalence relations, and classification problems in analysis

We apply descriptive set-theoretic techniques to analyze the complexity of various sets and relations arising in areas of analysis. In particular, we use the theory of Borel reducibility among definable equivalence relations to quantify the difficulty of classifying various mathematical objects up to some notion of equivalence.; We first analyze the group of Borel automorphisms of a Polish space, and show that its isomorphism relation is quite complicated: It is a $S12$-complete relation, and reduces the relation of equality on Borel sets.; Next we consider the notion of a weakly wandering sequence for a transformation and show that the set of sequences which are weakly wandering for some transformation is a $S11$-complete set, as are several related sets. We apply our techniques to produce specific sequences of interest, for example, a sequence which is exhaustive weakly wandering for some transformation but which is not weakly wandering for any ergodic transformation.; We then briefly consider equivalence relations which reduce all Borel equivalence relations, and ask whether there can be any minimal such relations. We show that the equality of Borel sets is not minimal, nor is it a universal $P11$ equivalence relation.; Next we analyze Polish metric spaces. We first show that a set of non-negative reals is the set of distances of some Polish metric space if and only if it is either countable or it is analytic and has 0 as a limit point. We also characterize the distance sets for certain classes of metric spaces.; We then consider the equivalence relation of isometry of Polish metric spaces and present a technique for reducing the orbit equivalence relation of a Polish group action to this isometry relation. We also give lower bounds for the complexity of isometry restricted to certain classes of spaces.; Finally, we consider the isometry of spaces with large isometry groups. We prove several results about the isomorphism relation on various classes of countable structures which are of independent interest, in particular that the classification of countable vertex-transitive graphs up to isomorphism is Borel-complete.