• 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.

Past NCGOA and Subfactor seminars