From eschimme@andrew.cmu.edu Fri Jul 2 20:57:00 2010 Date: Fri, 2 Jul 2010 20:57:40 -0400 (EDT) From: Ernest Schimmerling To: eschimme@math.cmu.edu Subject: workshop (fwd) -- Ernest Schimmerling Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 www.math.cmu.edu/~eschimme ---------- Forwarded message ---------- Date: Fri, 2 Jul 2010 17:00:09 -0700 From: Alexander S Kechris To: Ernest Schimmerling Cc: robin tucker-drob Subject: workshop Dear Ernest, Here is the information about the October Workshop: ------------------------------------------------------------------------------------------ Title: The Complexity of classification problems in ergodic theory 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 $\Gamma$, two free, measure preserving, ergodic actions of $\Gamma$ on standard probability spaces are called \emph{orbit equivalentl} 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 of tools from ergodic theory. These include: unitary representations and their associated Gaussian actions; rigidity properties of the action of $SL_2(\bbf{Z}$ 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 N eumann 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, math/0701027 Orbit inequivalent actions for groups containing a copy of $\Bbb F_2$. I. Epstein, arXiv:0707.4215 Orbit inequivalent actions of non-amenable groups. 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. ---------------------------------------------------------------------------------------- With best wishes, Alekos