Subfactor Seminar

Fall 2009

---

Organizers: Dietmar Bisch, Richard Burstein, Ionut Chifan, and Jesse Peterson

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

---

• 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.