Fall 2007

**Organizer: Dietmar Bisch**

**Mondays, 4:10-5:30pm in SC 1432**

- Date:
**9/3/07****Organizational meeting**(first 15 minutes or so)- Speaker:
**Alan Wiggins, Vanderbilt University** - Title:
**Applications of intertwining techniques to subfactors of II**_{1}factors - Abstract: Click here for the abstract.

- Date:
**9/8/07-9/9/07 (Saturday, Sunday), Wabash Modern Analysis Seminar, IUPUI, Indianapolis** - Date:
**9/10/07**- Speaker:
**Yves de Cornulier, CNRS Rennes** - Title:
**Lamplighter groups associated to free groups have Haagerup's approximation property** - Abstract:
In this talk we discuss the following recent result (joint work
with Stalder and Valette): Given a finite group H and a free group
F
_{n}, then the wreath product of H with F_{n}has the Haagerup approximation property.

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- Date:
**9/13/07 (Thursday), Mathematics Colloquium, 4:10-5:00pm in SC 5211**- Speaker:
**John Lott, University of Michigan** - Title:
**Long-time behavior of Ricci flow** - Abstract: Perelman proved Thurston's geometrization conjecture using Hamilton's Ricci flow. The first part of the talk will be an introduction to this work. Perelman gave enough information about the long-time behavior of a 3-dimensional Ricci flow to prove the validity of Thurston's geometric decomposition. However, it is not known whether Ricci flow performs the decomposition for you, i.e. whether as time passes one sees the various geometries appearing. I will give some results in this direction.

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- Date:
**9/17/07** - Date:
**9/24/07**- Speaker:
**Alan Wiggins, Vanderbilt University** - Title:
**Applications of intertwining techniques to subfactors of II**_{1}factors, continued

- Speaker:
- Date:
**9/29/07-9/30/07 (Saturday, Sunday), Fifth East Coast Operator Algebras Symposium, Wellesley College, Boston** - Date:
**9/27/07 (Thursday), Mathematics Colloquium, 4:10-5:00pm in SC 5211**- Speaker:
**Carl Sundberg, University of Tennessee, Knoxville** - Title:
**Von Neumann's inequality and model theory for several commuting operators** - Abstract:
Let T be a contraction on a Hilbert space H, i.e. a bounded linear operator
on H whose norm is less than or equal to 1. Von Neumann's inequality says
that if p is an analytic polynomial then the norm of p(T) is bounded by
the supremum of the values of |p(z)| for z in the unit disk. Following
Sz. Nagy-Foias this can be proved by developing a "model-theory" for
contractions, i.e. one shows that every contraction can be modelled as
the restriction to an invariant subspace of a particular nice kind of
operator for which von Neumann's inequality can be easily proved. An
abstract approach to model theory was developed by Agler and this
approach suggests far reaching applications to the study of a variety
of classes of operators.

There are many ways to generalize von Neumann's inequality to several commuting operators, some of which work and some of which don't. We will discuss these generalizations, focusing particularly on one due to Drury in 1978. This generalization is to "row-contractions" of commuting operators. Drury's result was rediscovered and further developed by Arveson, who produced a model theory for row contractions. We discuss Drury and Arveson's work along with recent joint work with Stefan Richter, in which we show how Arveson's model theory can be produced using Agler's approach.

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- Date:
**10/1/07**- Speaker:
**Claus Koestler, University of Illinois at Urbana-Champaign** - Title:
**On noncommutative random sequences from braid group representations** - Abstract:
Recently we have proven a noncommutative version of the extended De
Finetti theorem for infinite sequences of random variables. In contrast to
the classical result, the distributional symmetries of exchangeability and
spreadability are shown to be no longer equivalent in an operator
algebraic framework.

As our joint work with Rolf Gohm shows, spreadable random sequences are induced by braid group representations. We prove that such random sequences lead to triangular towers of von Neumann algebras, such that all cells form commuting squares. Our approach covers all examples as they arise from the Jones fundamental construction for inclusions with small index. Moreover, it is applicable to inclusions with infinite index. This will be illustrated by examples coming from the left regular representation of the braid group and the free group. Our results give strong evidence for the conjecture that there is a braided extension of free probability.

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- Date:
**10/3/07 (Wednesday), Special Subfactor Seminar, 3:10-4:00pm in SC 1310**- Speaker:
**Uffe Haagerup, University of Southern Denmark** - Title:
**Solution of the Effros-Ruan conjecture for bilinear forms on C*-algebras (joint work with Magdalena Musat)** - Abstract:
n 1991 Effros and Ruan conjectured that a certain Grothendieck
type inequality for a bilinear form on a pair of C*-algebras holds if (and
only if) the bilinear form is jointly completely bounded. In 2002 Pisier
and Shlyakhtenko proved that this inequality holds in the more general
setting of operator spaces, provided that the operator spaces in question
are exact, in particular they proved the Effros-Ruan conjecture for pairs
of exact C*-algebras. In a recent joint work with Magdalena Musat we prove
the Effros - Ruan conjecture for general C*-algebras (and with constant
one), i.e. for every jointly completely bounded (jcb) bilinear form u on a
pair of C*-algebras A,B there exists states f1,f2 on A and g1,g2 on B,
such that

|u(a,b)| =< ||u||_jcb (f1(aa*)^ g1(b*b)^ + f2(a*a)^ g2(bb*)^)

While the approach by Pisier and Shlyahktenko relied on free probability theory, our proof uses more classical operator algebra methods, namely Tomita Takesaki theory and special properties of the Powers factors of Type III-lambda, 0 < lambda < 1.

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- Date:
**10/4/07 (Thursday), Mathematics Colloquium, 4:10-5:00pm in SC 5211**- Speaker:
**Uffe Haagerup, University of Southern Denmark** - Title:
**Random matrices and the Ext-invariant for C*-algebras** - Abstract: In this talk I will discuss a surprising connection between two quite different areas of mathematics, namely "random matrices" and "operator algebras" (i.e. C*-algebras and von Neumann algebras), a connection, that has been developed over the last 16 years. In 1991 Voiculescu introduced a random matrix model for a free semicircular system, which has led to the solution of a number of classical problems in von Neumann algebra theory. More recently, Steen Thorbjoernesn and I have developed methods which allowed us to apply random matrices to problems in C*-algebra theory as well. In particular, we proved (Annals of Math. 2005) that the Brown-Douglas-Fillmore Ext-invariant for the reduced C*-algebra C*r(F2) for the free group on two generators is not a group, but only a semi-group, a problem, which had been open since 1978. Further results in this direction have been obtained in collaboration with Hanne Schultz and Steen Thorbjoernsen (Advances in Math. 2006).

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- Date:
**10/8/07**- Speaker:
**Thomas Sinclair, Vanderbilt University** - Title:
**Group cocycles and the ring of affiliated operators (after Peterson and Thom)** - Abstract: see arXiv:0708.4327.

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- Date:
**10/15/07**- Speaker:
**Thomas Sinclair, Vanderbilt University** - Title:
**Group cocycles and the ring of affiliated operators (after Peterson and Thom), continued** - Abstract: see arXiv:0708.4327.

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- Date:
**10/18/07 (Thursday), Mathematics Colloquium, 4:10-5:00pm in SC 5211**- Speaker:
**Vaughan Jones, UC Berkeley** - Title:
**Random matrices and planar algebras** - Abstract: I will discuss a connection between random matrices and subfactors discovered through the planar algebra approach to subfactors and the fact that there is a genus expansion for certain integrals over large matrices with the leading term being of genus zero. This is joint work with Alice Guionnet and Dimitri Shlyakhtenko.

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- Date:
**10/19/07 (Friday), Special Subfactor Seminar, 4:10-6:00pm in SC 1308**- Speaker:
**Vaughan Jones, UC Berkeley** - Title:
**Free probability and subfactors** - Abstract: I will give some of the details of the colloquium talk including a purely diagrammatic construction of a subfactor realising a lambda-lattice in the sense of Popa. This gives another proof of Popa's theorem on the subject.

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- Date:
**10/22/07****no meeting, fall break**

- Date:
**10/26/07 (Friday), Special Subfactor Seminar, 3:10-4:00pm in SC 1310**- Speaker:
**Teodor Banica, Universite Paul Sabatier (Toulouse III)** - Title:
**Hadamard matrices at roots of unity** - Abstract:
An n x n complex Hadamard matrix is known to produce a maximal abelian
subalgebra of the n x n matrices which is
orthogonal to the diagonal matrices. This can be used to construct
a subfactor of the hyperfinite II
_{1}factor with integer index, and hence a planar algebra. I will discuss a number of results on Hadamard matrices having as coefficients roots of unity. This is joint work with Nicoara (arxiv 0610) and Schlenker (arxiv 0707).

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- Date:
**10/29/07** - Date:
**11/5/07**- Speaker:
**Hans Wenzl, UC San Diego** - Title:
**Centralizers for spinor representations** - Abstract: We show that centralizers of tensor products of spinor representations can be conveniently described using Jones' basic construction. We compare this with related duality results coming from quantum statistical mechanics and discuss applications towards constructions of subfactors and reconstructions of tensor categories.

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- Date:
**11/12/07**- Speaker:
**Romain Tessera, Vanderbilt University** - Title:
**On property tau** - Abstract: We will give a survey on property tau and its applications.

- Speaker:
- Date:
**11/19/07****no meeting, Thanksgiving break**

- Date:
**11/26/07**- Speaker:
**Akram Aldroubi, Vanderbilt University** - Title:
**The least squares problem revisited** - Abstract: Given a set of functions, is there an optimal collection of subspaces minimizing the sum of the square of the distances between each function and its closest subspace in the collection?

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- Date:
**12/3/07**- Speaker:
**Shamindra Ghosh, Vanderbilt University** - Title:
**Topological quantum field theories from subfactors I** - Abstract: We will explain how one can construct a rational topological quantum field theory from a finite depth subfactor. Such a theory gives rise to a 3-manifold invariant via a state sum a la Turaev-Viro.

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- Date:
**12/10/07**- Speaker:
**Shamindra Ghosh, Vanderbilt University** - Title:
**Topological quantum field theories from subfactors II** - Abstract: We will explain how one can construct a rational topological quantum field theory from a finite depth subfactor. Such a theory gives rise to a 3-manifold invariant via a state sum a la Turaev-Viro.

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