Colloquia, Spring 2003

Mathematics Colloquia, Spring 2003

Thursdays 4:10 pm in 1431 Stevenson Center, unless otherwise noted

Tea at 3:30 pm in 1425 Stevenson Center

January 23, 2003 

Glenn Webb, Vanderbilt University

Mailborne Transmission of Anthrax:
Modeling and Implications

A mathematical model is developed to analyze the transmission of
inhalational anthrax through the postal system by cross-contamination of
The model consists of state vectors describing
the numbers of cross-contaminated letters
generated, the numbers of anthrax spores on these letters,
the numbers of resulting infections in recipients, and
matrices of transition probabilities acting on these vectors.
The model simulates the recent outbreak in the US, and provides a
general framework to investigate the potential impact of possible future

Contact person: Dietmar Bisch

February 5, 2003  

Charles Chui, University of Missouri-St. Louis & Stanford University

Note: Wednesday, 4:10 pm in SC 1431

Image Edge Analysis and Noise Removal

Inspired by the work of Mumford and Shah on problems in computer
vision, the total variational norm is shown to be a natural choice for
the Perona-Malik anisotropic diffusion approach to image edge enhancement,
for which the bilateral filter provides a most effective computational
scheme. We recently extended this to a trilateral filter for removing both
white and impulse noises.

Contact person: Larry Schumaker

February 6, 2003 

Stanley Chang, Wellesley College

A New Invariant and the Surgery
Exact Sequence

We will construct a “higher” Hirzebruch-type invariant of compact
manifolds based on the L2-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.

Contact person: Guoliang Yu

February 13, 2003 

Matthias Hieber, University of Darmstadt (visiting UC Berkeley)

Maximal Lp Regularity for
Parabolic Evolution Equations

Lp properties of solutions for linear parabolic equations have far
consequences for many nonlinear problems, such as free boundary
In this talk we discuss the developments of the so-called
maximal Lp regularity problem in the last year and show how it is
related to heat-kernel bounds, imaginary powers, Fourier multipliers
and the notion of R-boundedness.

Contact person: Gieri Simonett

February 20, 2003 

Neil Robertson, Ohio State University

The Strong Perfect Graph Theorem

This talk concerns the perfect graphs of Claude Berge.
When for all (vertex) induced subgraphs H of a graph G the maximum
clique size of H equals the chromatic number of H then G is said
to be perfect. The simplest examples of perfect graphs are the
2-colorable (bipartite) graphs B and of nonperfect graphs are the
simple circuits C(t) of length t>3. It is easy to see that the
(edge-set) complements of such graphs B and C(t) are also perfect
and not perfect, respectively. Berge, in a study of the Shannon
capacity of graphs, further noted that the line-graphs of
bipartite graphs and their complements are perfect, so that their
capacity is easy to compute, while graphs with the above odd
circuits or their complements as induced subgraphs (these are
called the odd holes or antiholes, respectively, of G) are not
perfect and the capacity is difficult to compute. From this
evidence he made his famous conjecture in 1961 that a graph is
perfect if and only it contains no odd hole or antihole. The
direct corollary, that G is perfect if and only if its complement
is perfect, was proved in 1972 by Lovasz. This established the
Berge conjecture as a premier graph coloring problem. In 1976
Appel and Haken proved the 4-color conjecture for planar graphs,
bringing Berge’s conjecture to the forefront. Strong attacks on
this problem were made by several graph theorists associated with
Berge, Lovasz, Chvatal and Cornuejols over the years. In June
of 2002 this conjecture was proved by another group (Chudnovsky,
Robertson, Seymour and Thomas), using methods of structural
graph theory. This talk will discuss the problem further and
will describe the general methods and the line of proof in an
intuitive way. A 160-page paper covering the proof details is
available on the home page of Robin Thomas. More recent joint
work of Chudnovsky, Cornuejols, Liu, Seymour and Vuskovic has
developed a polynomial-time algorithm to recognize a perfect
graph, not depending on the graph structural decomposition
theorems used to prove the Berge conjecture.

Contact person: Mike Plummer

February 24, 2003  

Edward Swartz, Cornell University

Representations of Matroids

Note: Monday, 4:10 pm in SC 1431

What is the nature of linear independence over fields of different
characteristics? For a specific vector space, what are the possible
geometric point configurations? Matroids, introduced by Whitney in
are a framework for answering these and other questions involving
of independence such as algebraic independence. In the 70’s
of real hyperplane arrangements, the simplex algorithm and directed
were independently and simultaneously led to oriented matroids. This
combinatorial abstraction of linear independence in an ordered field
always be realized by an arrangement of pseudospheres. We now know
if we allow homotopy spheres then all matroids have such a

Contact person: Paul Edelman

February 27, 2003  

Michael Burns, UC Berkeley

Planar Operations on Subfactors

Jones’ planar algebra formalism provides the most elegant and powerful
description of the standard invariant of a finite index, extremal
II1 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.

Contact person: Dietmar Bisch

March 10, 2003 

Nick Wright, Vanderbilt University

Coarse Geometry and Scalar

Note: Monday, 4:10 pm in SC 1431

For manifolds, one of the most intuitive geometric properties is
the curvature. The scalar curvature is dependent on the Riemannian
however the topology also plays a role in determining whether there
metrics with positive curvature. Coarse geometry studies the large
structure of a manifold and is a useful tool for analyzing curvature

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

Contact person: Guoliang Yu

March 11, 2003 

Martin Kassabov, Yale University

Kazhdan Property and Finite

Note: Tuesday, 3:10 pm in SC 1424

In this talk I survey several classical results about
Kazhdan Property T and apply them to two combinatorial
problems involving finite graphs — construction of family
of expanders and working time of product replacement
algorithm in computational group theory.

Kazhdan property T originated from the representation
theory of Lie groups. Shortly after its introduction it was
used by Margulis to construct an explicit example of a
family of expanders. Unfortunately, the expanding constant
of this family was unknown, because all proofs that a group
has a property T were not quantitative, and the expanding
constants of this family of expanders was unknown.

In a resent paper, A. Lubotzky and I. Pak showed that
Kazhdan property T of the group SLn(Z) implies that the
working time of the product replacement algorithm on
k-generated abelian groups is logarithmic in the size of
the groups, but its dependence on k was unknown.

A recent result by Y. Shalom gave an explicit bound of the
Kazhdan constant for the group SLn(Z), which lead to
quantitative bounds for the constants in these two
combinatorial constructions.

Contact person: Mark Sapir

March 13, 2003 

Sorin Popa, UCLA

L2-Betti Numbers and the
Fundamental Group of Finite von Neumann Factors

Click here to download the pdf file of the

The fundamental group F(M)
of a type II1 factor M was introduced by
Murray and von Neumann in 1943 in connection with their notion of
continuous dimension. It measures the extent to which “amplifications”
of M are isomorphic to M (e.g., if the algebra of
2×2 matrices over M is isomorhic to M then 2 is in F(M)}.
It is a puzzling and still poorly understood invariant.

We will present results providing the first examples of factors M
with trivial fundamental group. Thus, if G is the arithmetic group
Z2 \rtimes SL(2, Z) and M=L(G) is the associated
group von Neumann algebra then F(M)={1}. The proof uses in a
crucial way the “weak amenability” of \Gamma=SL(2, Z)
(i.e. Haagerup’s approximation property or equivalently Gromov’s
a-T-menability) and the relative property (T) of Kazhdan-Margulis
of the inclusion Z2 \subset Z2 \rtimes \Gamma.
The combination of these two properties makes it possible to prove
a unique decomposition of M as a cross-product
M = L^\infty(T2) \rtimes \Gamma, thus allowing us to define
l2-Betti number invariants bn(M) from the
l2-Betti numbers bn(R), defined by Gaboriau
in 2001, of the equivalence relation R induced by \Gamma on

Contact persons: Dietmar Bisch and Gennadi

March 20, 2003 

Ralph McKenzie, Vanderbilt University

Defining and Recognizing Structure in General Algebras;
Congruence Lattices are the Key to Deep Results

In the last three decades of the twentieth century,
universal algebra began to realize many of the lofty goals Garrett
Birkhoff had envisioned for it in 1933. Especially notable is the
ability to
formulate and proof deep results about all finite and locally finite
algebraic sysems. Tame congruence theory is an analysis of the possible
ways a clone of operations on a set may be organized relative to a
covering pair of congruences that it admits. Applied to all the covering pairs
of congruences of a finite algebra A, and also those of finite
algebras of functions derived from A, this theory reveals a
wealth of previously unrecognized structural features in finite
algebras, and provides natural and useful new ways of classifying them.
The task of working out the implications and extending the insights
of tame congruence theory has been the dominant theme of research
in general algebra for the past twenty years. Many of the results
discovered with its aid have since been extended by other means
to all algebraic systems (without local finiteness assumptions).

I originated this theory in 1981–84 (with the considerable help of my
then graduate student David Hobby).
In this talk, I will tell the story of how a long-running fascination
with one little problem and several big problems, combined with
stubbornness and luck, led to some big results.

Contact person: Dietmar Bisch

March 27, 2003 

Zhong-Jin Ruan, University of Illinois at Urbana-Champaign

Operator Spaces: A Natural
Non-commutative Quantization of Functional Analysis

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.

Contact person: Guoliang Yu and Dechao

April 3, 2003 

Zhenghan Wang, Indiana University

Topological Quantum Computation

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.

Contact persons: Dietmar Bisch and Bruce

April 14, 2003 

Rostislav Grigorchuk, Texas A&M University

The Ihara Zeta Function of Infinite
Graphs, the KNS Spectral Measure and Integrable Maps

Note: Special Colloquium, Monday, 4:10pm in SC 1431

(Part 1 of talk 4:10-5:00pm, 5 minutes break, Part 2 of talk

We define the Ihara zeta function for Cayley graphs of infinite finitely
generated groups.
We extend the definition of the Ihara zeta function to infinite
graphs which are limits of sequences Xn
of finite k-regular graphs such that Xn+1 covers
We associate to such a graph a measure mu with support
in [-1,1] called the Kesten-von Neumann-Serre spectral measure.
We present a few examples of computation of zeta function and a measure
\mu for Schreier graphs of some fractal groups generated by finite automata.
These computations are closely related to the integrability of
some 2-dimensional mappings which are also in focus of our
considerations. (joint work with A. Zuk (University of Chicago))

Contact person: Mark Sapir

April 22, 2003 

Yuri Bahturin, visiting Vanderbilt University

Note: Tuesday, 4:10-5:00pm in SC 1431

Bicharacters on Hopf Algebras

The notion of skew-symmetric bicharacter on a Hopf algebra is dual to
that of
R-matrix widely used in mathematics and beyond. It appears in the theory of
quantum groups, Yang – Baxter equations, etc. While R-matrices work in the
case of finite-dimensional Hopf algebras, bicharacters are more universal
and work fine in the case of infinite dimensions. One of the basic examples
of bicharacters are the commutation factors on abelian groups used, in
particular, to define so called color Lie superalgebras, a notion generated
in physics few decades ago. In general, bicharacters on a commutative and
cocommutative Hopf algebra H allow one to define Lie structures on the
algebras with the coaction H. On the other hand, Hopf algebras whose
structure includes a fixed skew-symmetric bicharacter form an important
class of cotriangular Hopf algebras, dual to the triangular ones
introduced by Drinfeld. If we want to classify such Lie structures or such Hopf
algebras, we may apply so called Scheunert’s trick, saying that any
bicharacter b(g,h) on a finitely generated abelian group G can be written in
the form b(g,h)=a(g,h)[s(g,h)/s(h,g)], where a(g,h) is either trivial or the
bicharacter defining ordinary Lie superalgebras and s(g,h) is a not
necessarily skew-symmetric bicharacter. In particular, if we deform the
product in a G-graded algebra using s(g,h), then the color commutator defined
by b(g,h) becomes either an ordinary Lie bracket or an ordinary superbracket.
The goal of this talk is to report on most recent result in this area,
including the extension of Scheunert’s trick to arbitrary cocommutative Hopf
algebras of characteristic different from 2 and the classification of
finite-dimensional algebras, which are commutative under a suitable
generalized Lie bracket.

Contact person: Mark Sapir


Colloquium Chair (Spring 2003): Dietmar Bisch

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