Spring 2005

**Organizer: Dietmar Bisch**

**Tuesdays, 4:00pm-5:00pm in SC 1432**

- Date:
**1/13/05 (Thursday) Mathematics Colloquium, 4:10pm-5:00pm in SC 1206**- Speaker:
**Sergei Treil, Brown University** - Title:
**Fun Around Corona** - Abstract: See http://www.math.vanderbilt.edu/~colloq

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**1/14/05 (Friday) Mathematics Colloquium, 3:10pm-4:00pm in SC 1206**- Speaker:
**Brett Wick, Brown University** - Title:
**Holomorphic Vector Bundles, Analytic Projections and the Corona Problem** - Abstract: The Corona problem is one of the more important problems in complex analysis and operator theory. Most of the proofs of this theorem rely on the function theory behind the problem. However there is a substantial geometric approach to the problem that provides an elegant answer to the Corona problem. In this talk I will discuss some aspects of the Corona problem and the relationship with holomorphic vector bundles and analytic projections.

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**1/18/05 (Tuesday) Mathematics Colloquium, 4:10pm-5:00pm in SC 1432**- Speaker:
**Jeff Raven, Penn State University** - Title:
**Topological K-homology** - Abstract: In the 1980s Baum & Douglas defined a topological K-homology group using cycles built from vector bundles on Spinc manifolds. One might hope that these groups would agree with the corresponding analytic K-homology, at least for finite CW-complexes; however to get anywhere along these lines one must first show that the topological K-homology groups are actually a homology theory (a result whose proof is missing from the original paper). I will describe one means of proving this, and then show how these arguments can be generalized to the equivariant case.

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**1/20/05 (Thursday) Mathematics Colloquium, 4:10pm-5:00pm in SC 1206**- Speaker:
**Greg Friedman, Yale University** - Title:
**Singular Knots and Stratified Spaces** - Abstract: I will present an overview of some of my work on singular knots and stratified spaces. Singular knots are codimension two embeddings of spheres that may possess singularities - points at which the embedding is not smoothable. I will discuss some of the properties of such knots and how they differ from those of smooth knots, as well as provide some motivation for the mathematical interest in such objects. From there, I will generalize to stratified spaces - spaces that are not quite manifolds, but which possess sets of singularities that can be filtered into manifold strata. I will introduce intersection homology, a useful algebraic tool for studying such spaces, and discuss some of my own work on intersection homology theory.

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**1/25/05 (Tuesday) Mathematics Colloquium, 4:10pm-5:00pm in SC 1432**- Speaker:
**Robert Yuncken, Penn State University** - Title:
**Group C*-algebras and the BGG-resolution** - Abstract: In this talk I will discuss the problem of understanding the representation theory of a locally compact group from the point of view of "noncommutative topology". I will begin by explaining what this means. I will then turn my attention to connected Lie groups, in particular the groups SL(2,C) and SL(3,C), and their discrete subgroups. I will indicate how the differential complex of Bernstein, Gelfand and Gelfand is pertinent to these examples.

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**2/1/05**- Speaker:
**Xiang Fang, University of Alabama** - Title:
**Some Interplay between Operator Theory and Commutative Algebra** - Abstract:
Roughly speaking, we are going to do commutative algebra on
Hilbert spaces. This approach was intended to give a new way to
study Fredholm operators, and Fredholm tuples on Hilbert spaces.
So far there has been quite successful theories for the one and two
variables. As for concrete applications, we will talk about Hilbert
spaces of analytic functions such as the Hardy, Bergman, and
Dirichlet spaces.

The long term goal is develop a Fredholm theory in several variables, or the index theory of an abstract Dirac operator. It is expected to be an interplay of many more areas including algebra, geometry, and topology.

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**2/15/05**- Speaker:
**Marius Dadarlat, Purdue University** - Title:
**Continuous Fields of Cuntz Algebras**

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**2/17/05 (Thursday) Mathematics Colloquium, 4:10pm-5:00pm in SC 1206**- Speaker:
**Liming Ge, University of New Hampshire** - Title:
**Factors and Groups** - Abstract: The connections between (discrete) groups and their group von Neumann algebras will be the focus of the talk. Some of the old and new problems in these two areas are discussed.

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**2/18/05 (Friday) Special RTG Seminar, 3:00-4:00pm, SC 1432**- Speaker:
**Liming Ge, University of New Hampshire** - Title:
**Introduction to Free Probability and Free Entropy** - Abstract: Basic laws, such as the "free central limit theorem", will be explained in the talk. Its connection with random matrices and applications to operator algebras will be discussed.

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**2/22/05**- Speaker:
**Serban Belinschi, Indiana University** - Title:
**Regularity for Free Convolutions** - Abstract: We will discuss analytic subordination for Cauchy transforms of free convolutions of probability measures. An investigation of the boundary behaviour of the subordination functions will yield several regularity results for the free additive convolution of two probabilities.

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**3/8/05****no meeting, spring break**

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**3/15/05 (Joint with the Topology/Group Theory Seminar)**- Speaker:
**John Crisp, Universite de Bourgogne (Dijon)** - Title:
**Quasi-isometric Embeddings in the Pure Braid Group** - Abstract: In this joint work with Bert Wiest (Rennes) we develop a technique for embedding groups quasi-isometrically in a pure braid group, as well as in the group of area preserving diffeomorphisms of the 2-disk (with a suitable metric). I shall describe how this can be done for a large class of right-angled Artin groups. This result can then be used to exhibit quasi-isometric embeddings of a variety of hyperbolic groups in both the braid group and the group of area preserving diffeomorphisms of the disk.

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**3/21/05 (Monday), 4:00pm-5:00pm in SC 1432 (N.B.: Different day this week.)**- Speaker:
**Shamindra Ghosh, University of New Hampshire** - Title:
**Representations of a Planar Algebra** - Abstract:
This talk will describe `representations of a planar algebra' in the sense
of Vaughan Jones and classification of the irreducible ones according to
their weights.

To every representation of a planar algebra, one can associate a power series called `dimension'. Vaughan Jones posed the question: `Is the radius of convergence of the dimension of any irreducible representation of a planar algebra at least as big as the inverse-square of the modulus of the planar algebra?' We answer this in the affirmative for the case of planar algebras with finite depth. Finally, we will list all the irreducible representations of the planar algebra associated to a finite group.

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**3/22/05**- Speaker:
**Denis Osin, Vanderbilt University** - Title:
**Weakly Amenable Groups**

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**3/24/05 (Thursday) Mathematics Colloquium, 4:10pm-5:00pm in SC 1206**- Speaker:
**Craig Evans, UC Berkeley** - Title:
**Games and some Degenerate Nonlinear PDE** - Abstract: I will discuss some simple nonlinear partial differential equations, depending on a parameter p lying between 1 and infinity, and then explain that the most interesting cases occur in the singular limits as p tends to either 1 or infinity. These two limit cases in fact have recently discovered game theoretic interpretations, which I will discuss and connect to other mathematical issues.

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**3/29/05**- Speaker:
**Brian Curtin, University of South Florida** - Title:
**Spin Models for Link Invariants, Bose-Mesner Algebras, and Planar Algebras** - Abstract: We discuss spin models for link invariants and Bose-Mesner algebras and place these objects in the context of planar algebras.

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**4/1/05 (Friday), 3:00-4:00pm, SC 1432**- Speaker:
**Jingbo Xia, SUNY Buffalo** - Title:
**On the Essential Commutant of ${\Cal T}(\text{QC})$** - Abstract: Let ${\Cal T}$(QC) (resp. ${\Cal T}$) be the $C^\ast $-algebra generated by the Toeplitz operators $\{T_\varphi : \varphi \in $ QC$\}$ (resp. $\{T_\varphi : \varphi \in L^\infty \}$) on the Hardy space $H^2$ of the unit circle. A well-known theorem of Davidson asserts that ${\Cal T}$(QC) is the essential commutant of ${\Cal T}$. We show that the essential commutant of ${\Cal T}$(QC) is strictly larger than ${\Cal T}$. Thus the image of ${\Cal T}$ in the Calkin algebra does not satisfy the double commutant relation. We also give a criterion for membership in the essential commutant of ${\Cal T}$(QC).

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**4/5/05 to 4/15/05****Lecture series ``Noncommutative geometry: from Quantum Physics to Motives'' by Alain Connes (IHES, College de France & Vanderbilt) and Matilde Marcolli (Max-Planck Institute Bonn)**- 10 lectures, rooms and times as follows:

4/5/2005 3:00 pm to 5:00 pm STEV 1307 (2 hours)

4/6/2005 4:00 pm to 5:00 pm STEV 1308

4/7/2005 3:00 pm to 4:00 pm STEV 1307

4/8/2005 4:00 pm to 5:00 pm STEV 1307

4/11/2005 4:00 pm to 5:00 pm STEV 1307

4/12/2005 3:00 pm to 4:00 pm STEV 1307

4/13/2005 4:00 pm to 5:00 pm STEV 1308

4/14/2005 3:00 pm to 4:00 pm STEV 1307

- Date:
**4/22/05 (Friday) Special NCGOA Seminar, 4:00pm to 5:00pm in SC 1307.**- Speaker:
**John Roe, Penn State University** - Title:
**The Analytic Structure Set** - Abstract: I will discuss recent work of N. Higson and myself which provides an analytic counterpart (a C*-algebra K-theory group) for the set of manifold structures within a given homotopy type (the 'structure set' of classical surgery theory.) This analytic structure set is also relevant to the classification of positive scalar curvature metrics, and of the 'noncommutative spectral sections' used in the definition of higher eta invariants.

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