Center for Noncommutative Geometry and Operator Algebras

Subfactor Seminar – Spring 2014

Subfactor Seminar
Spring 2014

Organizers: Dietmar Bisch, Vaughan Jones, and Jesse Peterson

Fridays, 4:10-5:30pm in SC 1432

  • Date: 1/13/14 – 1/17/14

  • Date: 1/15/14 – 1/16/14

  • Date: Thursday, 1/30/14, SC 1312, 4:10-5:30
    • Mark Sapir, Vanderbilt University
    • Title: The Tarski numbers of groups
    • Abstract:The Tarski number of a non-amenable group is the minimal number of pieces in a paradoxical decomposition of the group. It is known that a group has Tarski number 4 if and only if it contains a free non-cyclic subgroup, and the Tarski numbers of torsion groups are at least 6. It was not known whether the set of Tarski numbers is infinite and whether any particular number >4 is the Tarski number of a group. We prove that the set of possible Tarski numbers is infinite even for 2-generated groups with property (T), show that 6 is the Tarski number of a group (in fact of any group with large enough first L_2-Betti number), and prove several results showing how the Tarski number behaves under extensions of groups. This is a joint work with Mikhail Ershov and Gili Golan.

  • Date: Wednesday, 2/26/14, SC 1310, 4:10-5:00, joint seminar with Topology & Group Theory.
    • Alex Furman, University of Illinois at Chicago
    • Title: Classifying lattice envelopes of some countable groups.
    • Abstract: In a joint work with Uri Bader and Roman Sauer we study the following question: given a countable group L describe all locally compact groups G which contain a copy of L as a lattice (uniform or non-uniform). I will discuss the solution of this problem for a large class of countable groups L.

      The proof involves a somewhat unexpected mix of tools, including: Breuillard-Gelander’s topological Tits alternative, Margulis’ commensurator superrigidity, arithmeticity, and normal subgroup theorems, quasi-isometric rigidity results of Kleiner-Leeb, and Mosher-Sageev-Whyte.

  • Date: Thursday, 2/27/14, Departmental Colloquium, 4:10-5:00.

  • Date: 2/28/14
    • Zhengwei Liu, Vanderbilt University
    • Title: Noncommutative uncertainty principle
    • Abstract: We will talk about the uncertainty principle of subfactor planar algebras. The noncommutative generalization of two classical uncertainty principle of finite abelian groups are stated and proved. Furthermore we will describe the element which achieves the minimal value of the uncertainty principle and list all such elements for some concrete examples. We will prove Young’s inequality for convolutions as an important tool. This work is joint with Jinsong Wu and Chunlan Jiang.

  • Date: 3/7/14
    • No Meeting, Spring Break.

  • Date: 3/14/14
    • Isaac Goldbring, University of Illinois at Chicago
    • Title: Existentially closed II_1 factors
    • Abstract: A II_1 factor M is said to be existentially closed if, roughly speaking, any system of equations involving traces of *-polynomials with coefficients from M that has a solution in an extension of M already has approximate solutions in M. We will make this definition precise using continuous logic and investigate the various operator algebraic properties of existentially closed II_1 factors. In particular, we show that existentially closed II_1 factors are McDuff and have only approximately inner automorphisms. We then discuss the possibility of axiomatizing the class of existentially closed II_1 factors. If time permits, I will mention a recent interesting connecting between existentially closed C^* algebras and a conjecture of Kirchberg. No prior knowledge of model theory will be necessary. Various parts of this work are joint with Ilijas Farah, Bradd Hart, David Sherman, and Thomas Sinclair.

  • Date: 3/21/14 – 3/23/14

  • Date: Thursday, 3/27/14, Departmental Colloquium, 4:10-5:00.

  • Date: Friday, 3/28/14, 4:10-4:50.
    • Chenxu Wen, Vanderbilt University
    • Title: Character rigidity for special linear groups [After Peterson-Thom]
    • Abstract: A character on a group $G$ is a positive definite function $\tau: G \to \mathbb C$ which is invariant under conjugation and normalized so that $\tau(e) = 1$. Via the GNS-construction, characters are naturally related to the unitary representations of the group on finite von Neumann algebras.

      Recently, Bekka proved that for $G = PSL(n; \mathbb Z)$, $n \geq 3$, $G$ has character rigidity. He also noticed that from this fact it follows that the only II$_1$-factor representation for these groups must be the left regular representation. More recently, Peterson and Thom, proved character rigidity for $PSL(2; k)$; where $k$ is a infinite field. They also got some applications for this result to the question of freeness of ergodic actions of those groups. In this talk I will go through the main idea of their proof, and compare it with Bekka’s.

  • Date: Friday, 3/28/14, 5:00-5:40.
    • Sandeepan Parekh, Vanderbilt University
    • Title: Character rigidity for commensurators [After Creutz-Peterson]
    • Abstract: I shall present the Property (T)-half of the character rigidity theorem (due to D. Creutz and J. Peterson) for an irreducible lattice $\Gamma$ commensurated by $\Lambda$ in a product of certain Lie groups. Ie, if $\pi : \Lambda \rightarrow U(M)$ is a finite factor representation, then $\pi : L\Lambda \cong M$ or $\pi(\Gamma)” = M$ is finite dimensional. Under the assumption $\pi(\Lambda)”$ is amenable, a property (T) assumption on the ambient group implies the continuous algebra of $\pi$ is finite index in $M$. On the other hand, a Howe-Moore assumption implies the continuous algebra is just $\mathbb{C}$. Thus $M$ turns out to be finite dimensional.

  • Date: 3/30/14 – 4/5/14

  • Date: Thursday, 4/3/14, Departmental Colloquium, 4:10-5:00.

  • Date: 4/4/14
    • Zhengwei Liu, Vanderbilt University
    • Title: Noncommutative uncertainty principle (part II)
    • Abstract: We will talk about the uncertainty principle of subfactor planar algebras. The noncommutative generalization of two classical uncertainty principle of finite abelian groups are stated and proved. Furthermore we will describe the element which achieves the minimal value of the uncertainty principle and list all such elements for some concrete examples. We will prove Young’s inequality for convolutions as an important tool. This work is joint with Jinsong Wu and Chunlan Jiang.

  • Date: 4/5/14 – 4/6/14

  • Date: 4/11/14
    • Michael Hartglass, University of Iowa
    • Title: $C^{*}$-algebras associated to planar algebras
    • Abstract: In this talk, I will sketch the construction of several canonical $C^*$-algebras (Toeplitz, Cuntz, and free semicircular) associated to a (factor/subfactor) planar algebra $\mathcal{P}_{\bullet}$. Afterwards, I will concentrate on the “free semicircular” case, where results of Pimsner and Germain apply to compute the $KK$-groups of these algebras. I will use this result to state some rather strange properties of the Guionnet-Jones-Shlyakhtenko tower of $C^*$-algebras. This is joint work with Dave Penneys.

  • Date: 4/13/14 – 4/18/14

  • Date: 4/25/14
    • Dave Penneys, University of Toronto
    • Title: Classifying small index subfactors
    • Abstract: I’ll discuss the current state of the small index subfactor classification program. I’ll focus mostly on the newly updated map of small index subfactors (adapted from the Bulletin article of Jones-Morrison-Snyder). Recent progress includes joint work with Liu and Morrison which classifies 1-supertransitive standard invariants without intermediates with index at most $6\frac{1}{5}$. I may also discuss the current state of joint work with Afzaly and Morrison attempting to extend the classification of all standard invariants to index $5\frac{1}{4}$, using orderly generation of principal graph pairs based on McKay’s enumeration by canonical construction paths.

  • Date: 5/2/14 – 5/8/14

  • End of Spring Semester.

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Noncommutative Geometry and Operator Algebras
Department of Mathematics
Vanderbilt University
Stevenson Center 1326
Nashville, TN 37240
U.S.A. Phone: (615) 322-6672
Fax: (615) 343-0215
E-mail: ncgoa[at]vanderbilt[dot]edu