Abstracts

Applications of descriptive set theory to classical dynamical systems

In 1932 von Neumann proposed the project of classifying smooth measure preserving transformations. As part of the project he raised the question of whether every ergodic measure preserving transformation of the unit interval is isomorphic to a diffeomorphism of a manifold.

Despite deep progress on both questions, they remained open until recently. The lecture presents joint work with B. Weiss that shows that the classification problem is impossible to solve--because the associated equivalence relation is not Borel (and moreover is strictly more complicated than any $S^\infty$-action). Along the way the authors made progress on the second problem, by showing that a quasi-generic class of transformations can be realized as diffeomorphisms of the 2-torus. This class is the source of the complexity of the classification problem.

Compactness for chromatic numbers and other cardinal sins

A compactness principle is a statement of the form: If every small substructure of a given structure has a certian property, then the whole structure has this property. In this tutorial we shall deal with the property "The graph G has chromatic number <= \kappa". We shall connect this property with other set theoretical principles, like reflection of stationary sets, give some consistency results using large cardinals and list some interesting open problems.

Iterated forcing and the Continuum Hypothesis

One of the great successes in set theory in the 1970s and 80s has been the isolation of an optimal hypothesis for iterating forcings while preserving uncountablity. It turns out that while there is a well developed theory of iterating forcings which do not introduce new reals, this theory is necessarily more ad hoc in nature. This tutorial will discuss Shelah's preservation theorems for not adding reals as well as recently discovered examples which illustrate that these results are, in some sense, sharp.

The distance between HOD and V

The pursuit of better understanding the universe of set theory V motivated an extensive study of definable inner models M whose goal is to serve as good approximations to V. A common property of these inner models is that they are contained in HOD, the universe of hereditarily ordinal definable sets. Motivated by the question of how ``close" HOD is to V, we consider various related forcing methods and survey known and new results. This is a joint work with Spencer Unger.

Forcing analytic determinacy

The earliest-known tight connection between determinacy and large cardinals is the theorem of Martin and Harrington that $\Sigma^1_1$ determinacy is equivalent to the existence of $0^{\#}$. All known proofs of the forward implication go through Jensen's Covering Lemma; Harrington asked whether the theorem can be proved just in second-order arithmetic. We discuss progress on Harrington's question, building in particular on work of Cheng and Schindler showing that the standard proofs of Harrington's theorem cannot be carried out in any system substantially weaker than fourth-order arithmetic. We also describe a connection with the proper class games recently described by Gitman and Hamkins.

Weak Squares and Very Good Scales

The combinatorial properties of large cardinals tend to clash with those satisfied by G\"odel's constructible universe, especially the square property (denoted $\square_\kappa$) isolated by Jensen in the seventies. Strong cardinal axioms refute the existence of square, but it is possible with some fine-tuning to produce models that exhibit some large cardinal properties together with weakenings of square. In this talk we will exhibit some results along these lines and will outline the techniques used to produce them.

Space decomposition techniques in Borel dynamics

In recent years a substantial progress has been achieved in the field of Borel dynamics. A part of this progress is due to the development of space decomposition methods. The goal of the talk is to make an overview of the old and new results that have been proved along this path. In particular, we will discuss in various degrees of details the following: Dougherty-Jackson-Kechris classification of hyperfinite Borel equivalence relations, Multi-Tower Rokhlin Lemma for Borel automorphisms and regular cross sections of Borel flows, Lebesgue orbit equivalence of multidimensional flows, and Hochman's proof of existence of finite generators for compressible automorphisms.

Compactness of $\omega_1$

We investigate various aspects of compactness of $\omega_1$ under ZF + DC. We say that $\omega_1$ is X-supercompact if there is a normal, fine, countably complete nonprincipal measure on $\powerset_{\omega_1}(X)$ (in the sense of Solovay). We say $\omega_1$ is X-strongly compact if there is a fine, countably complete nonprincipal measure on $\powerset_{\omega_1}(X)$. We discuss various results in constructing and analyzing canonical models of $AD^+$ + $\omega_1$ is (X)-supercompact. We also discuss whether the theories "$\omega_1$ is X-supercompact" and "$\omega_1$ is X-strongly compact" can be equiconsistent for various X.

Integer cost and ergodic actions

A countable Borel equivalence relation $E$ on a probability space can always be generated in two ways: as the orbit equivalence relation of a Borel action of a countable group and as the connectedness relation of a locally countable Borel graph, called a \emph{graphing} of $E$. Assuming that $E$ is measure-preserving, graphings provide a numerical invariant called \emph{cost}, whose theory has been largely developed and used by Gaboriau and others in establishing rigidity results. A well-known theorem of Hjorth states that when $E$ is ergodic, treeable (admits an acyclic graphing), and has integer or infinite cost $n \le \infty$, then it is generated by an a.e. free measure-preserving action of the free group $\mathbf{F}_n$ on $n$ generators. We give a simpler proof of this theorem and the technique of our proof, combined with two other new tools, yields a strengthening of Hjorth's theorem: the action of $\mathbf{F}_n$ can be arranged so that each of the $n$ generators acts ergodically. This is joint work with Benjamin Miller.

The poor man's tree property

Motivated by producing a model where no regular cardinal greater than $\aleph_1$ carries a special Aronszajn tree, we prove that from large cardinals it is consistent that $\aleph_{\omega^2}$ is strong limit and there are no special Aronszajn trees on any regular cardinal in the interval $[\aleph_2,\aleph_{\omega^2+3}]$.