Analytic Equivalence Relations and Ulm-Type Classifications

Our main goal in this paper is to establish a Glimm-Effros type dichotomy for arbitrary analytic equivalence relations

Incomparable, non isomorphic and minimal Banach spaces

A Banach space contains either a minimal subspace or a continuum of incomparable subspaces. General structure results for analytic equivalence relations are applied in the context of Banach spaces to show that if $E_0$ does not reduce to isomorphism of the subspaces of a space, in particular, if the subspaces of the space admit a classification up to isomorphism by real numbers, then any subspace with an unconditional basis is isomorphic to its square and hyperplanes and has an isomorphically homogeneous subsequence

Descriptive Dynamics

The purpose of the following informal lectures is to give a brief introduction to descriptive dynamics, which I understand here to be the descriptive theory of Polish group actions. I will concentrate on the foundations, and hopefully at a level accessible to anyone with a basic knowledge of descriptive set theory. I will illustrate some of the main methods used in this area, including Baire category arguments and various implementations of the "changing the topology" technique. A general reference for the results discussed in this paper is Becker-Kechris [1996]

Actions of Polish Groups and Classification Problems

We will discuss in this paper some aspects of a general program whose goal is the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related study of definable equivalence relations. This work is motivated by basic foundational questions, like understanding the nature of complete classification of mathematical objects up to some notion of equivalence by invariants, and creating a mathematical framework for measuring the complexity of such classification problems. This theory, which has been growing rapidly over the last few years, is developed within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of the broad scope of this theory, there are natural interactions of it with other areas of mathematics, such as the theory of topological groups, topological dynamics, ergodic theory and its relationships with the theory of operator algebras, model theory, and recursion theory

New Directions in Descriptive Set Theory

I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical examples are R^n, C^n, (separable) Hilbert space and more generally all separable Banach spaces, the Cantor space 2^N, the Baire space N^N, the infinite symmetric group S_∞, the unitary group (of the Hilbert space), the group of measure preserving transformations of the unit interval, etc

The complexity of the classification of Riemann surfaces and complex manifolds

In answer to a question by Becker, Rubel, and Henson, we show that countable subsets of ℂ can be used as complete invariants for Riemann surfaces considered up to conformal equivalence, and that this equivalence relation is itself Borel in a natural Borel structure on the space of all such surfaces. We further proceed to precisely calculate the classification difficulty of this equivalence relation in terms of the modern theory of Borel equivalence relations. On the other hand we show that the analog of Becker, Rubel, and Henson's question has a negative solution in (complex) dimension n ≥ 2

Nonstandard models and analytic equivalence relations

Abstract. We improve a result of Hjorth[93] concerning the nature of thin analytic equivalence relations. The key lemma uses a weakly compact cardinal to construct certain nonstandard models, which Hjorth obtained using #'s. The purpose of this note is to improve the following result of Hjorth [93]. Theorem. (Hjorth) Suppose that for every real x, x # exists. Let E be an analytic equivalence relation, Σ 1 1 in parameter x 0 . Then either there exists a perfect set of pairwise Einequivalent reals or every E-equivalence class has a representative in a set-generic extension of Hjorth's proof makes use of his analysis of nonstandard Ehrenfeucht-Mostowski models built from #'s. Instead, we construct the necessary nonstandard models using infinitary model theory, assuming only the existence of weak compacts. Theorem 1. Suppose that for every real x there is a countable ordinal which is weakly compact in L [x]. Then the conclusion of the Theorem still holds. The main lemma is the following. It is not known if the conclusion of Lemma 2 holds in ZF C alone, for arbitrary x (with ZF replaced by an arbitrary finite subtheory). Proof of Theorem 1 from Lemma 2. Suppose that E is an analytic equivalence relation, Σ 1 1 in the parameter x 0 and choose an x 0 -recursive tree T on ω×ω×ω ω that xEy ←→ T (x, y) has a branch. For each countable ordinal α we define xE α y ←→ rank(T (x, y)) is at least α; then E α is Borel in (x 0 , c) where c is any real coding α and E is the intersection of the E α 's. We may assume that each E α is an equivalence relation. A theorem of HarringtonSilver says that a Π 1 1 -equivalence relation has a perfect set of pairwise inequivalent reals or each equivalence class has a representative constructible from the parameter defining the relation. As E α is Borel in (x 0 , c) where c is a real coding α and as we may assume that E and hence each E α has no perfect set of pairwise inequivalent reals, we know that each E α -equivalence class has a representative in L[x 0 , c] where c is any real coding α. Now let x be arbitrary and by Lemma 2 choose a countable nonstandard ω-model M x of ZF containing (x 0 , x) such that L(M x ) has an isomorphic copy in a set-generic extension N of L[x 0 ]. Let a ∈ ORD(M x ) be nonstandard and let c be a code for a, generic over M x ; then by applying Harrington-Silver in M x [c] we conclude that there is y in L(M x )[x 0 , c] such that yE a x. By choosing c to be generic over N as well we get that y belongs to a set-generic extension of L[x 0 ]. Finally, yEx since if not, yE α x would fail for some α admissible in (y, x) and hence for some (standard) α < a