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
- 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
- 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
- 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.
Past NCGOA and Subfactor seminars
NCGOA home page
VU math department's calendar
Dietmar Bisch's home page
Jesse Peterson's home page