Subfactor Seminar
Spring 2011

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

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

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• Date: 1/21/11
  • Emily Peters, Massachusetts Institute of Technology
  • Title: Connections and classification of subfactors
  • Abstract:
    The project of classifying subfactors with index less than 5 is nearly
    finished. Many of the final cases are ruled out by proving
    non-existence of connections on principal graphs. I will give some
    background and describe how connections rule out graphs in the family
    “B” of Morrison and Snyder’s classification up to index 5. This is
    joint work with Morrison, Penneys and Snyder.

• Date: Monday 1/24/11, Special subfactor seminar at 4:10pm in SC 1432.
  • Dave Penneys, UC Berkeley
  • Title: Classification of subfactors to index 5: quadratic tangles
  • Abstract:
    The classification of subfactors with index less than 5 is nearly
    complete. In part 1, Morrison and Snyder narrow possible principal
    graphs of such subfactors to certain classes of infinite families called
    “weeds” and “vines.” In recent work with Morrison Peters, and Snyder,
    we eliminate three weeds which have initial triple points. In this talk,
    we will eliminate the weed “C” using techniques derived from Jones’
    work on quadratic tangles. We will not assume any of the material from
    Emily Peters’ talk in this “sequel.”

• Date: 1/28/11
  • Andrew Toms, Purdue University
  • Title: A new dynamic dimension?
  • Abstract:
    Acting on a suggestion of Gromov, Lindenstrauss and Weiss introduced a
    notion of dimension for topological dynamical systems. While effective
    for minimal systems, their definition fails to fit meaningfully with
    covering dimension in the case of a trivial action. Can this be
    repaired? We’ll present a candidate for a positive answer which is
    defined in terms of comparison of Hilbert modules in crossed product
    C*-algebras, and some evidence to support it. We’ll also present some
    work in progress with Phillips and Hines which shows that for systems
    that are zero-dimensional in the sense of Lindenstrauss-Weiss, one at
    least obtains crossed products with some good structural properties.

• Date: 2/4/11
  • Kate Jushenko, Texas A&M University
  • Title: Second moments of unitaries in a II₁-factor
  • Abstract:
    We will consider convex sets of matrices composed of second-order mixed
    moments of n unitaries with respect to finite traces. These sets are of
    interest in connection with Connes’ embedding problem. We overview the
    basic facts about Connes’ embedding problem. In particular, we will
    discuss a theorem of E. Kirchberg which was the main motivation to study
    matrices of second-order moments. We present some properties of these
    sets and descriptions in case of small n. We also discuss a connection
    with the sets of correlation matrices and give some related examples.

• Date: 2/18/11
  • Stavros Garoufalidis, Georgia Institute of Technology
  • Title: The Slope Conjecture
  • Abstract: The Slope Conjecture links the leading term of the
    colored Jones polynomial of a knot to Slopes of incompressible surfaces
    in the knot complement. We will describe the structure of the leading
    term of the Jones polynomial, and the finite set of boundary slopes of a
    knot, and verify the slope conjecture for all alternating knots, torus
    knots, and 2-fusion knots.

• Date: 2/25/11
  • Richard Burstein, Vanderbilt University
  • Title: Stitching planar algebras
  • Abstract: A recent paper of Guionnet, Jones, Shlyakhtenko and
    Zinn-Justin provides a new method of subfactor construction:
    ‘stitching.’ This method produces a new non-irreducible subfactor
    planar algebra from two existing subfactor planar algebras, with the
    square root of the index equal to the sum of the square roots of the two
    original subfactors. I will describe this procedure, and give several
    examples.

• Date: Monday 2/28/11, Special subfactor seminar at 4:10pm in SC 1432.
  • Darren Creutz, UCLA
  • Title: Normal subgroups and rigidity for commensurators
  • Abstract: We present a Normal Subgroup Theorem for (dense)
    commensurators of lattices in arbitrary locally compact groups (not
    necessarily Lie). In particular, any normal subgroup of a (dense)
    commensurator of an (integrable) lattice in a simple topological group
    necessarily contains (up to finite index) the lattice.
    
    The approach involves new rigidity theorems for commensurators both in
    the context of representations and in dynamics, in particular a new
    factor theorem for SAT actions (the natural opposite of
    measure-preserving) more general than those for boundaries.
    
    This is joint work with Yehuda Shalom.

• Date: 3/11/11
  • No Meeting, Spring Break.

• Date: 3/25/11
  • Noah Snyder, Columbia University
  • Title: Applications of Number Theory to Subfactors
  • Abstract: Arithmetic results are crucial to the study of
    representations of finite groups. For example, the dimension of each
    irrep divides the dimension of the group, and character values are
    always cyclotomic integers (that is, sums of roots of unity). Fusion
    categories are quantum analogues of finite groups, so one should hope
    for generalizations of these results to fusion categories. In
    particular, there’s an analogue of the character table called the
    “S-matrix of the double,” and a result of de Boere, Goeree, Coste, and
    Gannon shows that the entries in this table are cyclotomic integers.
    I’ll explain this result and a corollary due to Etingof-Nikshych-Ostrik
    showing that dimensions of objects in fusion categories are cyclotomic
    integers. Since the even part of a finite depth subfactor is a fusion
    category, this result has applications to the study of subfactors. The
    first such application, due to Asaeda-Yasuda, eliminated all but
    finitely many graphs in the “Haagerup family” as possible subfactor
    principal graphs. I’ll outline the proof of a generalization of this
    result to any family of the same form. This is joint work with Frank
    Calegari and Scott Morrison. I’ll conclude with some remarks about
    other interesting arithmetic properties of fusion categories and
    subfactors.
    
    (This talk will not assume a background in number theory beyond the
    definition of an algebraic integer, but will assume some familiarity
    with the relationship between subfactors and fusion categories. For
    people already familiar with subfactor theory, my colloquium talk should
    supply adequate background.)

• Date: 4/1/11
  • Steve Avsec, University of Illinois at Urbana-Champaign
  • Title: q-Gaussians have the CCAP
  • Abstract: In 1997, Bozejko, Speicher, and Kummerer introduced
    q-deformed analogs of the free gaussian algebras of Voiculescu. It was
    subsequently proved by Ricard that the q-gaussians are II_1 factors, and
    Nou proved that the q-gaussians are non-injective. We shall give a
    proof that the q-gaussians have the completely contractive approximation
    property. We shall then discuss connections of this property to
    deformation/rigidity theory of Popa.

• Date: Wednesday 4/6/11, 4:10 – 5:00, in SC 1310, Joint with the Topology & Group Theory Seminar
  • Andreas Thom, Universität Leipzig
  • Title: Finite-dimensional approximation properties of groups
  • Abstract: I want to present various finite-dimensional
    approximation properties of discrete groups and explain applications
    towards some longstanding conjectures. On the other side, I will present
    a group which cannot be approximated by finite subgroups of unitary
    groups with the metric given by the operator norm. The techniques
    provide also new information about the interplay between the group
    structure and the topology of U(n). We will see that word-maps can
    contract the whole group into any given neighborhood of the neutral
    element.

• Date: 4/8/11
  • Pinhas Grossman, Instituto Nacional de Matemática Pura e Aplicada
  • Title: Quantum subgroups of the Haagerup fusion categories
  • Abstract: To a finite depth subfactor there corresponds a Morita
    equivalence between two unitary fusion categories. Given such a
    subfactor, it is natural to ask: what are all fusion categories which
    are Morita equivalent to these two? What are all module categories
    (“quantum subgroups”) over these fusion categories, and what are all the
    Morita equivalences between them? What are all the algebra objects in
    these fusion categories, and what are all the subfactors that realize
    them? And what are the full intermediate subfactor lattices of all such
    subfactors?
    
    We answer these questions for the case of the Haagerup subfactor. In
    particular, we find a new fusion category which is Morita equivalent but
    not isomorphic to the two fusion categories coming from the Haagerup
    subfactor. However, the Grothendieck ring of this fusion category is
    isomorphic to that of the non-symmetric Haagerup category, so the two
    categories cannot be distinguished by fusion rules. We also find several
    new subfactors related to these categories. This is joint work with
    Noah Snyder.

• Date: 4/15/11
  • Ionut Chifan, Vanderbilt University
  • Title: Examples of W*-superrigid groups (after Ioana, Popa, and Vaes)
  • Abstract: In a recent paper
    Ioana, Popa, and Vaes constructed the first examples of groups $\Gamma$
    which are completely remembered by their von Neumann algebra $L\Gamma$
    i.e., whenever $L\Gamma \cong L\Lambda$ then $\Gamma \cong \Lambda$.
    For instance, for every non-amenable group $\Sigma$ consider the
    generalized wreath product $\Gamma= { \mathbb Z_2}^{(I)}\rtimes (\Sigma
    \wr \mathbb Z)$ where $I$ is the quotient $(\Sigma \wr\mathbb Z )/
    \mathbb Z$ on which the group $\Sigma \wr \mathbb Z$ acts by
    multiplication. In these two talks I will present their proof showing
    that these groups are $W^*$-superrigid.

• Date: 4/22/11
  • Ionut Chifan, Vanderbilt University
  • Title: Examples of W*-superrigid groups (after Ioana, Popa, and Vaes)
  • Abstract: In a recent paper
    Ioana, Popa, and Vaes constructed the first examples of groups $\Gamma$
    which are completely remembered by their von Neumann algebra $L\Gamma$
    i.e., whenever $L\Gamma \cong L\Lambda$ then $\Gamma \cong \Lambda$.
    For instance, for every non-amenable group $\Sigma$ consider the
    generalized wreath product $\Gamma= { \mathbb Z_2}^{(I)}\rtimes (\Sigma
    \wr \mathbb Z)$ where $I$ is the quotient $(\Sigma \wr\mathbb Z )/
    \mathbb Z$ on which the group $\Sigma \wr \mathbb Z$ acts by
    multiplication. In these two talks I will present their proof showing
    that these groups are $W^*$-superrigid.

• End of Spring Semester.