Fall 2009

- Date:
**8/28/09****Richard Burstein, Vanderbilt University**- Title:
**Automorphisms of the bipartite graph planar algebra**

- Date:
**9/4/09****Richard Burstein, Vanderbilt University**- Title:
**Automorphisms of bipartite graph planar algebras, II** - Abstract: In 2000, Jones described a diagrammatic calculus, or planar algebra, acting on the closed loops of a locally finite bipartite graph. Continuing from last week, I will compute the automorphism group of this planar algebra, and describe the subalgebras obtained as fixed points under groups of automorphisms. Using results of Jones and Popa, I will then give applications to the construction of subfactors.

- Date:
**9/11/09****Richard Burstein, Vanderbilt University**- Title:
**Constructing low-index subfactors with multicolored jellyfish** - Abstract: Bigelow's jellyfish procedure has revolutionized the construction of singly-generated planar algebras. Subfactors with certain principal graphs exist if and only if the planar algebra associated to the graph contains a jellyfish which can rise to the surface. I will provide some simple examples of this subfactor construction, including a multicolored generalization of the jellyfish procedure.

- Date:
**9/18/09****Mikhail V. Ershov, University of Virginia**- Title:
**Kazhdan quotients of Golod-Shafarevich groups** - Abstract: Informally speaking, a finitely generated group G is said to be Golod-Shafarevich (with respect to a prime p) if it has a presentation with a "small" set of relators, where relators are counted with different weights depending on how deep they lie in the Zassenhaus p-filtration. Golod-Shafarevich groups are known to behave like (non-abelian) free groups in many ways: for instance, every Golod-Shafarevich group G has an infinite torsion quotient, and the pro-p completion of G contains a non-abelian free pro-p group. In this talk I will extend the list of known "largeness" properties of Golod-Shafarevich groups by showing that they always have an infinite quotient with Kazhdan's property (T). An important consequence of this result is a positive answer to a well-known question on non-amenability of Golod-Shafarevich groups.

- Date:
**9/25/09****No Meeting.**

- Date:
**10/2/09****Mrinal Raghupathi, Vanderbilt University**- Title:
**Representations of logmodular algebras** - Abstract: In this talk I will provide some background on the problem of when a contractive representation of a nonselfadjoint operator algebra has a dilation. I will then discuss the class of logmodular algebras, which were originally studied by Hoffman, and describe their basic structure. Finally, I will try to sketch a proof of the fact that a 2-contractive representation of a logmodular algebra does dilate. The talk will assume only a basic knowledge of operator theory. It will also serve as a tiny introduction to operator spaces and nonselfadjoint operator algebras.

- Date:
**10/9/09****Mrinal Raghupathi, Vanderbilt University**- Title:
**Representations of Logmodular Algebras II** - Abstract: In this talk I will give a proof of the fact that a two-contractive representation of a logmodular algebra has a positive extension to the C-star envelope. The proof is based on ideas of Foias-Suciu and some basic techniques from operator space theory.

- Date:
**10/16/09****Thomas Sinclair, Vanderbilt University**- Title:
**Cocycle superrigidity for Gaussian actions** - Abstract: This talk will cover joint work with Jesse Peterson. In this talk I will discuss cocycle superrigidity within the context of Gaussian actions of countable, discrete groups. In particular, I will demonstrate that Bernoulli actions of L^2-rigid groups are U_fin cocycle superrigid. The class of L^2-rigid groups contains both groups with Kazhdan's property (T) and direct products of infinite groups with non-amenable groups, recovering Popa's cocycle superrigidity theorem for Bernoulli actions. Moreover, I will show that certain generalized wreath products of groups are L^2-rigid, giving new examples of cocycle superrigid groups. I will also establish that groups with non-zero first L^2-Betti number are not U_fin cocycle superrigid.

- Date:
**10/23/09****No Meeting, Fall Break.**

- Date:
**10/30/09****Remus Nicoara, University of Tennessee, Knoxville**- Title:
**A finiteness result for commuting squares with large second relative commutant** - Abstract: We prove that there exist only finitely many commuting squares of finite dimensional *-algebras of fixed dimension, satisfying a "large second relative commutant" condition. When applied to lattices arising from subfactors satisfying a certain extremality-like condition, our result yields Ocneanu's finiteness theorem for the standard invariants of such finite depth subfactors.

- Date:
**11/6/09****No Meeting, Special Session on Operator Algebras**, AMS Sectional Meeting, at UC Riverside

- Date:
**11/27/09****No Meeting, Thanksgiving Break.**

- Date:
**12/4/09****Hanfeng Li, SUNY at Buffalo**- Title:
**Entropy and Fuglede-Kadison determinant.** - Abstract: Given a countable amenable group G and an element f in the integral group ring ZG, one may consider the shift action of G on the Pontryagin dual of ZG/ZGf. I will discuss the relation of the entropy of this action and the Fuglede-Kadison determinant of f.

- End of Fall Semester.