The last two decades have seen the emergence of a theory of set theoretic complexity of classification problems in mathematics. In this workshop, I will present recent developments concerning the application of this theory to classification problems in ergodic theory.

The first lecture will be devoted to a general introduction to this area. The next two lectures will give the basics of Hjorth's theory of turbulence, a mixture of topological dynamics and descriptive set theory, which is a basic tool for proving strong non-classification theorems in various areas of mathematics.

In the last three lectures, I will show how these ideas can be applied in proving a strong non-classification
theorem for orbit equivalence. Given a countable group Γ, two free, measure preserving, ergodic actions
of Γ on standard probability spaces are called *orbit equivalent* if, roughly speaking,
they have the same orbit spaces. More precisely this means that there is an isomorphism of the underlying measure
spaces that takes the orbits of one action to the orbits of the other. A remarkable result of Dye and Ornstein-Weiss
asserts that any two actions of an amenable group are orbit equivalent. My goal will be to outline a proof of a dichotomy theorem which states that for any
non-amenable group, we have the opposite situation: The structure of its actions up to orbit equivalence is so complex that it is impossible, in a vey strong
sense, to classify them (Epstein-Ioana-Kechris-Tsankov). Beyond the methods of turbulence, an interesting aspect of this proof is the use of many diverse
tools from ergodic theory. These include: unitary representations and their associated Gaussian actions; rigidity properties of the action of SL_{2}(Z) on
the torus and separability arguments (Popa, Ioana), Epstein's co-inducing construction for generating actions of a group from actions of another, quantitative
aspects of inclusions of equivalence relations (Ioana-Kechris-Tsankov) and the use of percolation on Cayley graphs of groups and the theory of costs in proving
a measure theoretic analog of the von Neumann Conjecture, concerning the "inclusion" of free groups in non-amenable ones (Gaboriau-Lyons). Most of these tools
will be introduced as needed along the way and no prior knowledge of them is required.

For participants who wish to do some background reading before the workshop, here are some suggestions:

- For the first three lectures:
- H. Becker and A. S. Kechris, The descriptive set theory of Polish group actions, Cambridge Univ. Press, 1996
- G. Hjorth, Classification and orbit equivalence relations, Amer. Math. Soc., 2000
- C. W. Henson, J. Iovino, A. S. Kechris and E. Odell, Analysis and Logic, Cambridge Univ. Press, 2002

- For the last three lectures:
- A. S. Kechris, Global aspects of ergodic group actions, Amer. Math. Soc., 2010
- A. Ioana, Orbit inequivalent actions for groups containing a copy of F
_{2}[link] - I. Epstein, Orbit inequivalent actions of non-amenable groups [link]
- D. Gaboriau and R. Lyons, A measurable-group-theoretic solution to von Neumann's problem. Invent. Math. 177 (2009), no. 3, 533-540
- A. Ioana, A. S. Kechris and T. Tsankov, Subequivalence relations and positive-definite functions. Groups Geom. Dyn. 3 (2009), no. 4, 579-625

Another conference webpage [link] has travel information.

The Vanderbilt Mathematics Department also has a webpage for visitors [link].

- Your name, university affiliation, mailing address, phone number and email address
- Your professional status and
- undergraduate students: please describe your background in set theory
- graduate students: please tell us your year and the name of your thesis advisor if you have one
- faculty: please tell us whether you hold a federal research grant

- A brief statement about your interest in the workshop
- An itemized estimate of your expected transportation expenses