Subfactor Seminar
Fall 2006

Organizer: Dietmar Bisch

Mondays, 4:10-5:30pm in SC 1310

• Date: 8/28/06
  • Speaker: Dietmar Bisch, Vanderbilt University
  • Title: Introduction to II₁ factors and subfactors
  • Abstract: I will give an introduction to von Neumann algebras and the theory of subfactors. This meeting serves also as organizational meeting for the Subfactor Seminar.

• Date: 9/4/06
  • Speaker: Shamindra Ghosh, Vanderbilt University
  • Title: Introduction to planar algebras I
  • Abstract: We define planar algebras, give some examples and explain how they are related to subfactors.

• Date: 9/11/06
  • Speaker: Pinhas Grossman, Vanderbilt University
  • Title: Introduction to planar algebras II
  • Abstract: We continue the introduction to Jones' planar algebra formalism from last week. We will present several examples of planar algebras.

• Date: 9/18/06
  • Speaker: Remus Nicoara, Vanderbilt University
  • Title: Group von Neumann algebras, crossed products and property T
  • Abstract: This lecture will be an introductory lecture covering some of the background material needed for Stefaan Vaes' lecture series.

• Date: 9/22/06 (Friday), Special Subfactor Seminar, Lecture Series by Stefaan Vaes, Lecture I at 3:10pm in SC 1310
  • Speaker: Stefaan Vaes, Katholieke Universiteit Leuven and CNRS Paris
  • Title: Deformation and rigidity results for crossed product II₁ factors I
  • Abstract: I will present an introduction to recent work of Popa on Bernoulli actions of property (T) groups. The main goal is to present the proof of Popa's strong rigidity theorem: the II₁ factors given by the Bernoulli action of an ICC property (T) group essentially remember the group and the action.
    
    Lecture 1. I will explain how unitary conjugacy of subalgebras of a II₁ factor can be shown using bimodules.
    Lecture 2. A strong deformation property of Bernoulli actions, called malleability, is played against property (T) to approach the strong rigidity theorem.
    Lecture 3. The strong rigidity theorem is shown. I will also discuss how the techniques of the first two lectures are used in my recent joint work with Popa, yielding a lot of explicit computations of outer automorphism groups of II₁ factors.

• Date: 9/25/06
  • Speaker: Stefaan Vaes, Katholieke Universiteit Leuven and CNRS Paris
  • Title: Deformation and rigidity results for crossed product II₁ factors II

• Date: 9/26/06 (Tuesday), Special Subfactor Seminar, 4:10pm in SC 1432
  • Speaker: Stefaan Vaes, Katholieke Universiteit Leuven and CNRS Paris
  • Title: Deformation and rigidity results for crossed product II₁ factors III

• Date: 9/27/06 (Wednesday), Topology & Group Theory Seminar, Lecture Series by Andreas Thom, Lecture I at 4:10pm in SC 1310
  • Speaker: Andreas Thom, University of Goettingen
  • Title: L²-invariants for groups and von Neumann algebras
  • Abstract: In recent work, Connes and Shlyakhtenko have defined L²-invariants for tracial algebras, generalizing L²-invariants for groups due to Atiyah-cheeger-Gromov. The study of these L²-invariants combines nicely analytic and homological methods. In this talk, we will introduce the subject and give an account of the known results and open questions that area.

• Date: 9/28/06 (Thursday), RTG Seminar, 1:10pm in SC 1310
  • Speaker: Andreas Thom, University of Goettingen
  • Title: L²-invariants for groups and von Neumann algebras II

• Date: 10/3/06 (Tuesday), NCG Seminar, 4:10pm in SC 1432
  • Speaker: Andreas Thom, University of Goettingen
  • Title: L²-invariants for groups and von Neumann algebras IV

• Date: 10/9/06
  • Speaker: Jesse Peterson, UC Berkeley
  • Title: Deformation/Rigidity Techniques in von Neumann Algebras
  • Abstract: I will discuss the deformation/rigidity techniques introduced by Sorin Popa. These techniques have led to a number of remarkable rigidity type results both in von Neumann Algebras as well as Orbit Equivalence Ergodic Theory. I will highlight some of these results while paying special attention to applications with (amalgamated) free products of von Neumann algebras.

• Date: 10/16/06
  • no meeting, fall break

• Date: 10/23/06
  • Speaker: Adrian Ioana, UCLA
  • Title: Orbit inequivalent actions for groups containing a copy of F₂
  • Abstract: I will prove that if a group H admits a rigid action (e.g. H=F₂), then any group G containing a copy of H has uncountably many non orbit equivalent actions.

• Date: 10/30/06
  • Speaker: Ali Chamseddine, American University of Beirut
  • Title: Computational Methods in the Spectral Action Principle I
  • Abstract: In Einstein's general theory of relativity the gravitational field is geometrical where the symmetry is diffeomorphism invariance and the dynamics is formulated in terms of the fluctuations of the metric. In noncommutative geometry diffeomorphism invariance is replaced with the spectral action principle which states that "The physical action depends only on the spectrum". I review properties of noncommutative geometric spaces based on real spectral triples and explain the computational tools needed in the applications of this principle. As an example I will derive the inner fluctuations of the Dirac operator and show that for dimensions less than or equal to four one gets the sum of a Yang-Mills action and a Chern-Simons action.

• Date: 10/31/06 (Tuesday), NCG Seminar, 4:10pm in SC 1432
  • Speaker: Ali Chamseddine, American University of Beirut
  • Title: Computational Methods in the Spectral Action Principle II

• Date: 11/2/06 (Thursday), Mathematics & Physics Colloquium, 4:00-5:00pm in SC 4327
  • Speaker: Ali Chamseddine, American University of Beirut
  • Title: Hidden Noncommutative Geometric Structure of Space-Time
  • Abstract: The geometry of space-time is reconstructed from the low-energy spectrum defined by the quarks and leptons. I show that there is a hidden noncommutative structure and that the dynamics of the unified geometrical theory is governed by the "Spectral Action Principle".

• Date: 11/6/06
  • Speaker: Pinhas Grossman, Vanderbilt University
  • Title: Indices and angles for supertransitive intermediate subfactors

• Date: 11/13/06
  • Speaker: Akram Aldroubi, Vanderbilt University
  • Title: Slanted matrices, Banach frames, Wiener's lemmas and inverse problems
  • Abstract: Click here for the abstract.

• Date: 11/20/06
  • no meeting, Thanksgiving break

• Date: 11/27/06
  • Speaker: Remus Nicoara, Vanderbilt University
  • Title: Subfactors and Hadamard Matrices
  • Abstract: To any complex Hadamard matrix H one associates a hyperfinite subfactor. The standard invariant of this subfactor captures certain symmetries of H. We present several classification results for Hadamard matrices and discuss some properties of the associated subfactors.

• Date: 11/30/06 (Thursday), Mathematics Colloquium, 4:10-5:00pm in SC 5211
  • Speaker: Sorin Popa, UCLA
  • Title: On the superrigidity of malleable actions
  • Abstract: I will present a series of results showing that measure preserving actions $\Gamma\curvearrowright X$ of countable non-amenable groups $\Gamma$ on a probability space $X$ that satisfy certain {\it malleability} and mixing conditions have very sharp rigidity properties. For instance, any cocycle for $\Gamma \curvearrowright X$ with values in a discrete group can be untwisted on the normalizer of any subgroup $H\subset \Gamma$ with the relative property (T). Same if $H$ is the centralizer of a subgroup $G\subset \Gamma$ on which the action has spectral gap. Bernoulli and Gaussian actions are typical examples of malleable actions. As a consequence it follows that if $\Gamma$ is either Kazhdan, is a product of two infinite groups, or has infinite center, then any Bernoulli action $\Gamma \curvearrowright X=X_0^\Gamma$ is {\it orbit equivalent superrigid}, i.e. if $\Lambda \curvearrowright Y$ is a free ergodic measure preserving action whose orbits coincide with the orbits of $\Gamma \curvearrowright X$ then $\Gamma \simeq \Lambda$ and the actions are conjugate.

• Date: 12/1/06 (Friday), Special Subfactor Seminar, 3:10-4:30pm in SC 1308
  • Speaker: Sorin Popa, UCLA
  • Title: Rigidity phenomena from spectral gap and malleability
  • Abstract: I will explain a new strategy for proving rigidity results for II$_1$ factors, relying on the ``tension'' between a spectral gap condition (used in lieu of property T) and malleability. Applications include: a unique decomposition result for MdDuff factors; a primeness result for factors arising from Bernoulli actions of non-amenable groups; strong rigidity results for isomorphisms between group measure space factors $L^\infty X \rtimes \Gamma \simeq L^\infty Y \rtimes \Lambda$, with $\Gamma$ a product group and $\Lambda \curvearrowright Y$ Bernoulli.

• Date: 12/4/06
  • Speaker: Paramita Das, Vanderbilt University
  • Title: Relative commutants of depth two subfactors
  • Abstract: We show how Planar Algebra can be used to give a simple proof of the part of the Ocneanu-Szymanski theorem which asserts that for a finite index, depth two, irreducible subfactor $N \in M$, the relative commutants $N^{'}\cap M_1$ and $M^{'}\cap M_2$ admit mutually dual Kac algebra structures. In the hyperfinite case, the same techniques also prove the other part, namely that there is an action of $N^{'}\cap M_1$ on $M$ with invariants $N$. This method extends to the general depth two case where the theorem had been generalized by Nikshych and Vainerman to obtain the structures of mutually dual `Weak Hopf C^* algebras'.

Past NCGOA and Subfactor seminars