Fall 2010

- Date:
**9/3/10** **Thomas Sinclair, Vanderbilt University**- Title:
**Strong solidity of factors from lattices in SO(n,1) and SU(n,1)** - Abstract: Generalizing techniques found in Ozawa and Popa, ``On a class of II_1 factors with at most one Cartan subalgebra, II'' (Amer. J. Math., 2010), we show that the group factors of ICC lattices in SO(n,1) and SU(n,1), n \geq 2, are strongly solid.
- Date:
**9/10/10****Michael Brandenbursky, Vanderbilt University**- Title:
**Introduction to Khovanov homology** - Abstract: We will discuss a definition and some properties of the unnormalized Jones polynomial and Khovanov homology.

- Date:
**9/17/10****Michael Brandenbursky, Vanderbilt University**- Title:
**Introduction to Khovanov homology, II** - Abstract: We will discuss a definition and some properties of the unnormalized Jones polynomial and Khovanov homology.

- Date:
**9/24/10****Ionut Chifan, Vanderbilt University**- Title:
**Von Neumann algebras with unique group measure space Cartan subalgebras.** - Abstract:
In this talk I will introduce a class of groups $\mathcal {CR}$ satisfying the following property:

If $\Gamma \in \mathcal {CR}$ then any free, ergodic, measure preserving action of $\Gamma$ on a probability space gives rise to a von Neumann algebra with unique group measure space Cartan subalgebra.

I will also discuss some applications of this result to W*-superrigidity. This is joint work with Jesse Peterson.

- Date:
**10/8/10****No Meeting, Special Sessions on Rigidity in von Neumann Algebras and Ergodic Theory, and Free Probability and Subfactors**, AMS Sectional Meeting, at UCLA

- Date:
**10/15/10****No Meeting, Fall Break.**

- Date:
**Monday 10/18/10, 4:10 - 5:30pm****Adrian Ioana, Clay Research Fellow**- Title:
**A class of superrigid group von Neumann algebras.** - Abstract: I will present a recent result joint with Sorin Popa and Stefaan Vaes showing that any group G in a fairly large class of generalized wreath product groups is von Neumann superrigid. This means that if the group von Neumann algebra LG of G is isomorphic to the von Neumann algebra LH of an arbitrary countable group H, then G and H must be isomorphic.

- Date:
**10/22/10****No Meeting, East Coast Operator Algebras Symposium**, at Dartmouth College

- Date:
**Monday 10/25/10, 4:10 - 5:30pm, Joint seminar with the Physics Department****Roman Buniy, Arizona State University**- Title:
**An algebraic classification of entangled states** - Abstract: We propose a classification of entangled states that is based on the analysis of algebraic properties of certain linear maps associated with the states. An iterative procedure that uses the kernels of the maps defines new discrete measures of entanglement, which lead to a new method of entanglement classification. We proved a theorem on a correspondence between new algebraic invariants and sets of equivalent classes of entangled states. The new method works for an arbitrary finite number of state spaces of finite dimensions. As an application of the method, we considered a large selection of cases of three spaces of various dimensions. We also obtained the complete entanglement classification for the case of four qubits.

- Date:
**10/29/10****J. Owen Sizemore, UCLA**- Title:
**W* and OE Rigidity Results for Action of Wreath Product Groups** - Abstract: To a measure preserving group action one can associate 3 structures: the action, the equivalence relation, and the von Neumann algebra. This leads to three notions of equivalence for group actions: conjugacy, orbit equivalence (OE), and W* equivalence (W*E). It is easy to see that conjugacy implies OE implies W*E. Rigidity results are when the implications can be reversed. We will explain recent rigidity results for actions of wreath product groups. This is joint work with I. Chifan and S. Popa.

- Date:
**Saturday 10/30/10** **Appalachian Set Theory Workshop**, at Vanderbilt University- Date:
**11/26/10****No Meeting, Thanksgiving Break.**

- Date:
**12/3/10****John Williams, Indiana University**- Title:
**Decomposition and Tightness in Free Probability** - Abstract: In this talk I will discuss recent results proving the existence of a decomposition of a random variable into an infinite sum of freely independent, "prime" random variables. This will lead naturally into a discussion of tightness theorems in free probability and, more generally, qualitative observations regarding the free convolution operation. That is, given two measures, we will try to summarize what is known about their free convolution, short of actual calculation (as this may be quite difficult in practice). An example of such in observation is that, based on work by Bercovici and Voiculescu, one may state that the free convolution operation "destroys" atoms, in a sense that will be made precise. We will discuss other general principles that may be gleaned from these tightness results as well as previous work.

- End of Fall Semester.