Alekos Kechris : "The complexity of classification problems in ergodic theory"

List of participants in this workshop

Lecture notes by Alekos Kechris and Robin Tucker-Drob (PDF; Revised 8/11)

Workshop description

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 SL₂(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 F₂ [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

Pre-registration

This is optional but if you are planning to attend, it would be helpful to hear from you! [email]

Lodging

We have set aside blocks of rooms at Extended Stay America Vanderbilt (615-383-7490) and Holiday Inn Vanderbilt (615-327-4707). The rooms will be held until October 8, 2010. Mention VU Math Dept/Appalachian Set Theory when making your reservation. These and other hotels within walking distance of the mathematics department are listed at [link] but the rates given there are different.

Travel information

See [link] for a campus map.

Another conference webpage [link] has travel information.

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

Participant travel support

Funds provided by the National Science Foundation will be used to reimburse some participant transportation and lodging expenses. Priority will be given to students and faculty who do not hold federal research grants. Please request such funds as far in advance of the meeting as possible by sending the following information to James Cummings and Ernest Schimmerling by email.

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

Acknowledgement

Financial support for this workshop has been provided by the NSF and Vanderbilt University's Shanks endowment.