Subfactor Seminar
Fall 2013

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Organizers: Dietmar Bisch, Vaughan Jones, and Jesse Peterson

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

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• Date: 8/23/13, Joint with the Topology & Group Theory Seminar
  • Cary Malkiewich, Stanford University
  • Title: Coassembly Maps, Gauge Groups, and K-Theory
  • Abstract: Calculus of functors is a powerful technique from homotopy theory, which studies computationally difficult constructions by means of “linear approximations.” We will describe a new variant of this calculus, based on the embedding calculus of Weiss and Goodwillie. This theory provides us with a sequence of approximations to the stable gauge group of a principal bundle, in which the linear approximation is the Cohen-Jones string topology spectrum. We will finish with some future applications to algebraic K-theory, a subject which provides a powerful (but difficult to compute) invariant for rings, algebras, and groups.

• Date: 9/6/13
  • Arnaud Brothier, Vanderbilt University
  • Title: Subfactors with prescribed fundamental groups
  • Abstract: In a joint work with Stefaan Vaes we study fundamental groups for subfactors. We consider a large class of subgroups $\mathcal S$ of $\mathbb R_+^\times$ that contains all countable subgroups and some uncountable groups with any Hausdorff dimension between 0 and 1.
    
    Using Popa’s deformation/rigidity theory and Bisch-Haagerup subfactors, we construct a hyperfinite subfactor of index 6 with its fundamental group equal to $G$, for any group $G$ in $\mathcal S$. Those subfactors are build from an ergodic measure preserving transformation and they remember it. This proves in particular that there are unclassifiably many subfactors at index 6 with the same standard invariant.
    
    Furthermore, using those technics we provide an explicit uncountable family of non outer conjugate actions on the hyperfinite II$_1$ factors for any non amenable group.

• Date: 9/13/13
  • Zhengwei Liu, Vanderbilt University
  • Title: Singly generated planar algebras
  • Abstract: Subfactor planar algebras generated by a 2-box are classified by Bisch and Jones, for at most 13 dimensional 3-boxes. We extend the classification to 14 dimensional 3-boxes. We give two proofs. One is based on the skein theory of planar algebras generated by a 2-box. The other is based on a new approach to the complexity of subfactor planar algebras, called the thickness.

• Date: 9/20/13
  • Ionut Chifan, University of Iowa
  • Title: $W^*$-superrigidity for arbitrary actions of central quotients of braid groups
  • Abstract: For any $n > 4$ let $\tilde B_n = B_n/Z(B_n)$ be the quotient of the braid group $B_n$ through its center. We prove that any free ergodic probability measure preserving (pmp) action $\tilde B_n \ca (X, \mu)$ is virtually $W^*$-superrigid: whenever $L^\infty (X,\mu)\rtimes \tilde B_n =L^\infty (Y,\nu)\rtimes \Lambda$, for an arbitrary free ergodic pmp action $\Lambda \ca (Y, \nu)$ it follows that the actions $\tilde B_n \ca X,\Lambda \ca Y$ are virtually conjugate. Moreover, we prove that the same holds if $\tilde B_n$ is replaced with any finite index subgroup of the direct product $\tilde B_{n_1} \times \cdots \times \tilde B_{n_k}$, for some $n_1, \ldots , n_k > 4$. The proof uses a dichotomy theorem of Popa-Vaes for normalizers inside crossed products by free groups in combination with a $OE$-superrigidity theorem of Kida for actions of mapping class groups. This is based on joint work with A. Ioana and Y. Kida.

• Date: 9/27/13
  • Darren Creutz, Vanderbilt University
  • Title: Character Rigidity for Lattices and Commensurators
  • Abstract: Characters on groups (positive definite conjugation-invariant functions) arise naturally both from probability-preserving actions (the measure of the set of fixed points) and unitary representations on finite factors (the trace). I will present joint work with J. Peterson showing the nonexistence of nontrivial characters for irreducible lattices in semisimple groups and for their commensurators. Consequently, any finite factor representation of such a group generates either the left regular representation or a finite-dimensional representation, generalizing our earlier result that every nonatomic probability-preserving action of such groups is essentially free. The key new idea is to use the contractive nature of the Poisson boundary to bring it in operator algebraic setting and along with it the rigidity behavior of lattices in their ambient groups.

• Date: 10/4/13
  • Stavros Garoufalidis, Georgia Tech
  • Title: The 3D index of a cusped hyperbolic 3-manifold
  • Abstract: The 3D index of Dimofte-Gaiotto-Gukov is a partially defined function on the set of ideal triangulations of 3-manifolds with torus boundary, which is partially invariant under 2-3 moves. It turns out that an ideal triangulation has 3D index if and only if it is 1-efficient. Moreover, the 3D index descends to a topological invariant of cusped hyperbolic manifolds. Parts are joint work with Hodgson-Rubinstein-Segerman.

• Date: 10/11/13
  • No Meeting, Fall Break.

• Date: 10/12/13 – 10/13/13

• ECOAS 13: The eleventh east coast operator algebra symposium, at University of Cincinnati.

Date: 10/18/13

• Kamran Reihani, Vanderbilt University
• Title: Reduction to type II in dynamical systems
• Abstract: The talk reports on a frequently used strategy that seems to be useful when some sort of “type-III” phenomena prevent the existence of invariant structures for dynamical systems. The approach is called “reduction to type II” (in analogy with the reduction of type-III factors to type II in the theory of von Neumann algebras). It involves some extension of the dynamical system in such a natural way that the resulting system is large enough to carry the desired invariant structure. We will discuss a few examples in geometry, topology, and measure theory. The reduction process in the measurable case naturally involves Tomita-Takesaki theory, and the computations are based on a joint work with Bill Paschke.

Date: Thursday, 10/24/13, Departmental Colloquium, 4:10-5:00.

• Alice Guionnet, Massachusetts Institute of Technology
• Title:About topological expansions
• Abstract: Maps are connected graphs which are properly embedded into a surface, their genus is the minimal genus of such a surface. Matrix integrals have been shown to be related with the enumeration of maps since the seventies, after the work of ‘t Hooft and Brézin-Itzykson-Parisi and Zuber. This is the so-called topological expansion. Such an expansion has been used in many fields of physics and mathematics. In this talk, we shall describe this correspondance, discuss some applications and some generalisations.

Date: 10/25/13, 2:10-3:30 in SC 1310.

• Uffe Haagerup, University of Copenhagen
• Title: Ultraproducts, QWEP von Neumann Algebras, and the Effros-Maréchal Topology
• Abstract: The talk is based on a joint work with Hiroshi Ando and Carl Winslöw (arXiv:1306.0460). Based on analysis on the Ocneanu/Groh-Raynaud ultraproducts and the Effros-Maréchal topology on the space vN(H) of von Neumann algebras acting on a separable Hilbert space H, we show that for a von Neumann algebra M in vN(H), the following conditions are equivalent:

  (1) M has the Kirhcberg’s quotient weak expectation property (QWEP).
  
  (2) M is in the closure of the set of injective factors on H with respect to the Effros-Maréchal topology.
  
  (3) M admits an embedding i into the Ocneanu ultrapower N^omega of the injective III_1 factor N with a normal faithful conditional expectation from N^omega to M.
  
  (4) For every epsilon > 0, natural number n, and x_1,…,x_n in natural cone P_M^{natural} in the standard form for M, there is a natural number k and a_1,…,a_n in M_k(C)_+, such that | < x_i, x_j > – tr_k(a_ia_j) | is less than epsilon for i,j = 1,…,n, where tr_k is the tracial state on M_k(C).

Date: 10/26/13 – 10/27/13

• DARPA & Shanks Workshop “Quantum Symmetries”, at Vanderbilt University.

Date: 11/8/13

• Alberto Grünbaum, UC Berkeley
• Title: Looking for matrix valued Tau functions

Date: 11/22/13

• Zhengwei Liu, Vanderbilt University
• Title:The subgroup $E_{n+2}$ of quantum $SU(n)$
• Abstract:
• We construct a family of subfactor planar algebras generated by a non-self contragredient 2-box by skein theory. Along this process, we will solve several fundamental problems. The generator and relations are derived from the classification of singly generated planar algebra. Based on HOMFLY we show that these relations are consistent. Then we construct the matrix units of this planar algebra inductively and compute the trace of minimal idempotents. For a discrete series the partition function is positive semidefinite and its kernel is generated by those trace-zero minimal idempotents. The quotient planar algebra by the kernel is a subfactor planar algebra with index $cot^2(\pi/2(n+1))$. Its principal graph is a subset of the Young’s lattice with a dihedral group $D_{n+1}$ symmetry. For each $n$, we obtain a pair of complex conjugate subfactor planar algebras corresponding to a subgroup $E_{n+2}$ of quantum $SU(n)$. When $n=3,4$, they are listed in Ocneanu’s classification result.

  Principal graphs for n=3,4.

Date: 11/29/13

• No Meeting, Thanksgiving Break.

End of Fall Semester.