Spring 2003

**Organizers: Dietmar Bisch and Guoliang Yu**

**Tuesdays 3:00 pm - 4:30 pm, 1403 Stevenson
Center **

- Date:
**1/28/03 Graduate Student Colloquium, 3:10-4:00 pm in SC 1307**- Speaker:
**Dietmar Bisch, Vanderbilt University** - Title:
**What Evariste Galois might have done 150 years later** - Abstract:
Galois theory is certainly one of the most fascinating
theories in mathematics. If Galois had lived in the 20th century
he might have been an analyst and studied extensions of certain
rings of operators on Hilbert space instead of field extensions.
An operator algebra version of Galois theory was discovered by
Fields Medalist Vaughan Jones in the early 1980's. It led to an
amazingly rich theory which has many surprising applications to
representation theory, low dimensional topology, quantum physics
and quantum information theory. I will try to give some glimpses
of Jones' non-commutative Galois theory, which is nowadays called
the
.**theory of subfactors**

- Speaker:
- Date:
**2/4/03**- Speaker:
**Yuri Bahturin, Memorial University of Newfoundland & Moscow State University (visiting Vanderbilt)** - Title:
**Direct limits of simple Lie algebras** - Abstract: In this talk we will establish a very close connection between an important class of so called diagonal simple locally finite Lie algebras with simple or involutory simple locally finite associative algebras. In particular this expresses diagonal direct limits of finite-dimensional simple algebras in terms of similarly defined limits of finite-dimensional simple or involutory simple associative algebras.

- Speaker:
- Date:
**2/6/03 (Thursday) Mathematics Colloquium at 4:10pm in SC 1431**- Speaker:
**Stanley Chang, Wellesley College** - Title:
**A New Invariant and the Surgery Exact Sequence** - Abstract:
We will construct a ``higher" Hirzebruch-type invariant of compact
manifolds based on the L
^{2}-signature and motivated by the work of Cheeger-Gromov. This invariant is useful in studying the structure set of manifolds whose fundamental group contains torsion. The talk will be geared towards a graduate student audience.

- Speaker:
- Date:
**2/11/03**- Speaker:
**Mark Sapir, Vanderbilt University** - Title:
**Isoperimetric functions of groups, connections between group theory, topology and computer science** - Abstract: This is a joint work with J-C. Birget, Olshanskii and Rips. One of the main results says that the function $f(l)=l^a$, $a>4$, is (O-equivalent to) the isoperimetric function of a group if and only if $a$ is a relatively fast computable number. For example, there exists a manifold with isoperimetric function equivalent to l^(\pi+e). We also construct the first example of an NP-complete group, and characterize the groups with word problem in NP: a finitely generated group has word problem in NP if and only if it is a subgroup of a finitely presented group with polynomial isoperimetric function.

- Speaker:
- Date:
**2/18/03**- Speaker:
**Alexis Alevras, US Naval Academy** - Title:
**Irreversible flows on von Neumann algebras** - Abstract: The study of one-parameter semigroups of endomorphisms of von Neumann algebras was initiated some fifteen years ago by| R.T. Powers as a first step towards an index theory for unbounded derivations. It may be appropriately viewed as the study of the differential operator d/dx in a noncommutative setting. In the talk I will give a self-contained overview of the theory of such semigoups, and explain its connections to continuous tensor products of Hilbert spaces, stochastic processes and to more general quantum dynamical semigroups, with a focus on attempts at classification, up to cocycle conjugacy, of endomorphism semigroups of type I factors.

- Speaker:
- Date:
**2/26/03 (Wednesday) Special Talk, 1:00-2:00pm in SC 1403**- Speaker:
**Michael Burns, UC Berkeley** - Title:
**Infinite Index Subfactors: Extremality and Rotations** - Abstract:
We will generalize the notion of extremality to II
_{1}subfactors of infinite index, in addition defining a weaker notion of approximate extremality. We will connect these properties to the existence of certain rotation operators on the L^{2}spaces of the higher relative commutants of the subfactor.

- Speaker:
- Date:
**2/27/03 (Thursday) Mathematics Colloquium at 4:10pm in SC 1431**- Speaker:
**Michael Burns, UC Berkeley** - Title:
**Planar Operations on Subfactors** - Abstract:
Jones' planar algebra formalism provides the most elegant and powerful
description of the standard invariant of a finite index, extremal
II
_{1}subfactor, allowing the use of diagramatic techniques to prove results in the theory of operator algebras. After reviewing some of the theory of planar algebras, von Neumann algebras and subfactors, we will discuss a number of extensions of the planar algebra results.

- Speaker:
- Date:
**3/4/03****no meeting (spring break)**

- Date:
**3/11/03 (Monday) Mathematics Colloquium at 4:10pm in SC 1431**- Speaker:
**Nick Wright, Vanderbilt University** - Title:
**Coarse Geometry and Scalar Curvature** - Abstract:
For manifolds, one of the most intuitive geometric properties is the curvature. The scalar curvature is dependent on the Riemannian metric however the topology also plays a role in determining whether there are metrics with positive curvature. Coarse geometry studies the large scale structure of a manifold and is a useful tool for analyzing curvature questions.

I will describe the ideas and methods underlying coarse geometry. The relation with curvature is given by a geometric differential operator (the Dirac operator). The index theory for this operator gives various obstructions to positive scalar curvature. I will present some of these obstructions on open manifolds and draw conclusions for general closed manifolds.

- Speaker:
- Date:
**3/13/03 (Thursday) Mathematics Colloquium at 4:10pm in SC 1431**- Speaker:
**Sorin Popa, UCLA** - Title:
**L**^{2}-Betti Numbers and the Fundamental Group of Finite von Neumann Factors - Abstract: Click here to download the pdf file of the abstract.

- Speaker:
- Date:
**3/14/03 (Friday) Special Talk, 3:10-4:30pm in SC 1432**- Speaker:
**Sorin Popa, UCLA** - Title:
**Playing amenability against rigidity to unfold the structure of II**_{1}factors - Abstract:
Click here to download the ps file of the
abstract.

I will explain a general strategy for studying type II$_1$ factors that proved extremely succesful in the last couple of years, and which consists in ``playing amenability against rigidity'', whenever some very weak versions of these properties are met. These opposing properties create enough ``tension'' within the algebra to unfold the entire structure. I will exemplify with three distinct types of situations and results: 1) When the II$_1$ factor $M$ contains Cartan subalgebras $A\subset M$ such that $A\subset M$ satisfies an operator algebra version of the Kazhadan-Margulis relative property (T), while the equivalence relation $\Cal R_{A\subset M}$ satisfies Haagerup's property. 2) When $M=R \rtimes_\sigma G$ with $\sigma$ a Bernoulli shift action and $G$ a group which contains an infinite normal rigid subgroup. 3) When $M$ is a tensor product of ``hyperbolic'' factors (a recent property due to Ozawa).

- Speaker:
- Date:
**3/25/03**- Speaker:
**Ed Saff, Vanderbilt University** - Title:
**Minimal Riesz Energy Points on Manifolds** - Abstract: We discuss asymptotics (as N -> infty) for minimal Riesz s-energy N-point configurations on the union of smooth manifolds. Recent work with Doug Hardin leads to some very general results in the case when s is greater than the Hausdorff dimension of the manifold. Motivation for the investigation is the question of distributing many points on a sphere and best-packing problems.

- Speaker:
- Date:
**3/27/03 (Thursday) Mathematics Colloquium at 4:10pm in SC 1431**- Speaker:
**Zhong-Jin Ruan, University of Illinois at Urbana-Champaign** - Title:
**Operator Spaces: A Natural Non-commutative Quantization of Functional Analysis** - Abstract:
An operator space is a norm closed subspace of bounded operators on some
Hilbert space together with a distinguished ``matrix norm''. Morphisms
between operator spaces are ``completely bounded linear maps''.

Operator space theory is a natural non-commutative quantization of functional analysis (Banach space theory). In this talk, I will first discuss some fundamental results in operator spaces, and then discuss some interesting applications to operator algebras and non-commutative harmonic analysis.

- Speaker:
- Date:
**4/1/03**- Speaker:
**Marius Dadarlat, Purdue University** - Title:
**K-theory and approximations of C*-algebras** - Abstract: The validity of the universal coefficient theorem in KK-theory is shown to be equivalent to an approximation property for residually finite dimensional C*-algebras. We will also discuss approximation properties of representations of amenable residually finite groups.

- Speaker:
- Date:
**4/3/03 (Thursday) Mathematics Colloquium at 4:10pm in SC 1431**- Speaker:
**Zhenghan Wang, Indiana University** - Title:
**Topological Quantum Computation** - Abstract: An equivalent model of quantum computing based on topological quantum field theories has been proposed in the work of Freedman, Kitaev, Larsen and Wang. This new way of looking at quantum computation provides efficient quantum algorithms to approximately compute quantum invariants of links and 3-manifolds, and a possible way to realize a large scale quantum computer. We will start with a general introduction to quantum information science, and then discuss the connection to topology, computer science and condensed matter physics.

- Speaker:
- Date:
**4/14/03 (Monday) Special Talk, 3:00-4:00pm in SC 1404**- Speaker:
**Erik Guentner, University of Hawaii** - Title:
**The Novikov conjecture for linear groups** - Abstract:
A central problem in C*-algebra K-theory, addressed by the
Baum-Connes conjecture, is to compute the K-theory of the
reduced group C*-algebra of a discrete group. The Baum-Connes
conjecture implies the Novikov conjecture on homotopy invariance
of higher signatures.

N. Higson and G. Kasparov proved the Baum-Connes conjecture for groups acting properly and affine isometrically on Hilbert space. Building on their work G. Yu proved the Novikov conjecture for groups that embed uniformly in a Hilbert space.

We will describe results, obtained with Higson and S. Weinberger, proving the Novikov conjecture for subgroups of Lie groups which are not necessarily discrete. This generalizes earlier results of Kasparov. Our results build directly on those mentioned above, with elementary aspects of the theory of valuations playing an important role.

- Speaker:
- Date:
**4/15/03**- Speaker:
**Jingbo Xia, SUNY at Buffalo** - Title:
**Bounded functions of vanishing mean oscillation on compact metric spaces** - Abstract: A well-known theorem of T. Wolff asserts that for every $f \in L^\infty $ on the unit circle $T$, there is a non-trivial $q \in $ QA $=$ VMO$\cap H^\infty $ such that $fq \in $ QC. We consider the situation where $T$ is replaced by a compact metric space $(X,d)$ equipped with a measure $\mu $ satisfying the condition $\mu (B(x,2r)) \leq C\mu (B(x,r))$. We generalize Wolff's theorem to the extend that every function in $L^\infty (X,\mu )$ can be multiplied into VMO$(X,d,\mu )$ in a non-trivial way by a function in VMO$(X,d,\mu )\cap L^\infty (X,\mu )$. Wolff's proof relies on the fact that $T$ has a dyadic decomposition. But since this is not available for $(X,d)$ in general, our approach is completely different. Furthermore, we show that the analyticity requirement for the function $q$ in Wolff's theorem must be dropped if $T$ is replaced by $S^{2n-1}$ with $n \geq 2$. Move precisely, if $n \geq 2$, then there is a $g \in L^\infty (S^{2n-1},\sigma )$, where $\sigma$ is the standard spherical measure on $S^{2n-1}$, such that if $q \in H^\infty (S^{2n-1})$ and if $q$ is not the constant function 0, then $gq$ does not have vanishing mean oscillation on $S^{2n-1}$. The particular $g$ that we construct also serves to show that a famous factorization theorem of S. Axler for $L^\infty $-functions on the unit circle $T$ cannot be generalized to $S^{2n-1}$ when $n \geq 2$. We conclude the talk with an index theorem for Toeplitz operators on $S^{2n-1}$.

- Speaker: