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
Fn, then the wreath product of H with Fn has
the Haagerup approximation property.
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.
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.
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.
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
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.
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).
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.
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.
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.
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.
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 II1 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).
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.
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.
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?
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.
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.