Colloquia, Fall 2002

Vanderbilt Mathematics

Colloquia are listed in reverse chronological order. The
top of the list is subject to change, since more colloquia
are still being planned.
All colloquia are held at 4:10 pm
in 1431 Stevenson Center unless otherwise noted.

Our colloquia, as well as our
seminars and other activities, feature
speakers not only from our own department but also from other
departments all over the world.
For further information on activities in the department,
you may also consult our

Thursday, December 5th.
Bruce Richter, of the
Department of Combinatorics and
The University of Waterloo.
(Host: Mark Ellingham)

Thursday, November 7th.
John Roe, of the
Department of Mathematics,
The Pennsylvania State University.
(Host: Guoliang Yu)

Thursday, October 31st.
Ronald G. Douglas, of
Texas A&M University.
Multivariate Operator Theory and Complex Geometry.
In considering the study of multivariate operator theory on Hilbert space, a
module approach is useful in bringing to bear concepts and techniques from
complex and algebraic geometry. In the talk I will demonstrate instances of
such applications to the study of submodules and quotient modules determined
by algebraic objects. The emphasis will be on concrete examples that
illustrate the general results. Holomorphic hermitian bundles along with
curvature and other spectral invariants will be shown to be relevant.
(Host: Guoliang Yu)

Tuesday, October 29th.
Ken-ichi Kawarabayashi, of
Princeton University.
Title TBA.
Abstract TBA.
(Host: Mark Ellingham)

Friday, October 25th.
, of
Yale University and University of California at San Diego.
Lie algebras graded by root systems.

I will talk about a classification project that includes classical Lie
algebras, the Freudenthal-Tits “magic” square and recently discovered
infinite dimensional superconformal algebras.

Thursday, October 24th.
Yiannis Vourtsanis, of
Vanderbilt University.
The product operation on structures and its role in developments in

Abstract TBA.

Thursday, October 17th.
Guihua Gong, of
University of Puerto Rico.
C*-algebras and classification.

A C*-algebra is a closed selfadjoint subalgebra of
the algebra of all bounded linear operators on a Hilbert
space. C*-algebras can be considered as noncommutative topological
spaces and have significant applications geometry, topology
and physics.

In this talk, we will survey some recent developement on the
classification of C*-algebras.

Thursday, October 10th.
Cornelia Drutu, of
University of Lille-1.
Quasi-isometry invariants and asymptotic cones.
Finitely generated groups G become geometric objects when
endowed with the word metric (depending on the generating set): the
distance between a and b from G is the length of
the shortest word representing a-1b.

If G acts “nicely” on a metric space (X,d) (for instance
if X is the universal cover of a compact manifold M,
G is the fundamental group of M) then G as a metric
space is quasi-isometric to (X,d) (a quasi-isometry is a
bilipschitz map up to an additive constant).This justifies the interest
in the study of groups up to quasi-isometry.

In this talk we shall discuss some quasi-isometry invariants of groups.
One of the tools in this study is the asymptotic cone of a metric space.
For a metric space (X,d), an asymptotic cone represents “an image of
the space seen from infinitely far away”. We shall present some
relations between geometric properties of asymptotic cones and the
behavior of quasi-isometry invariants.

Thursday, October 3rd.
Manny Knill,
Alamos National Laboratory
Algebraic Methods for Quantum Noise Control.
To be useful, a model of information processing (computation,
communication) needs to be robustly realizable using physical devices.
The fundamental theorem of quantum information processing (QIP) is that
QIP can in principle be realized in the presence of constant physical
error rates. The unifying idea underneath most approaches for
controlling quantum errors is that of a subsystem, defined as a tensor
factor of a subspace of a Hilbert space. There is a close relationship
between subsystems and properties of matrix algebras.

I will start with a hands-on introduction to quantum computing. After a
short explanation of “everything you need to know about robust
realization of information”, I will describe a few of the ways in which
elementary algebra and representation theory are contributing toward
understanding and using noisy quantum systems.

Thursday, September 19th.
Goulnara Arjantseva, of
Université de Genève.
Graphs, subgroups, and group actions.
Let G be a group and A a finite set of generators for G.
One can associate a directed A-labelled graph Gamma to every
subgroup H of G. It turns out that this graph carries the essential
information on the structure of H. For a free group G the idea behind
this association has been developed by J.Stallings in an algebraic
topology terminology. For every finitely generated group G an approach
proposed by A. Olshanskii and the author is applied.

In this talk we review some known results where such an approach via
graphs is used. Then using our graph technique, we give a sufficient
condition for a finitely generated subgroup of a word hyperbolic group G
to be free and quasiconvex. Finally, we generalize this result to groups
acting by isometries on a delta-hyperbolic space.

Thursday, September 12th.
Rob Donnelly, of the

Department of Mathematics & Statistics
Murray State University.

Semisimple Lie algebras acting on partially ordered sets.
The study of groups acting on sets has many well-known “enumerative”
consequences. As an example, “Burnside’s” Theorem can be used to
address the question: How many different bracelets can one make using
six beads if there are three choices for the color of each bead? In
this talk, we explore some of the combinatorial and algebraic
consequences of a certain kind of action of “semisimple” Lie algebras
on partially ordered sets. In one direction, this notion leads to
answers to certain “extremal” cominatorics questions, of which a
simple example is: What is the maximum possible size of a collection
of pairwise “incomparable” subsets of an n-element set? In
another direction, this idea has led to new explicit constructions of
many families of irreducible representations of semisimple Lie
algebras (as well as a new interpretation of the existing
Gelfand-Tsetlin explicit constructions for sl(n), the simple
Lie algebra consisting of traceless n x n matrices). Bases for
representations so constructed enjoy several distiguishing “extremal”
and combinatorial properties: For example, the partially ordered sets
which are the most useful for this approach are “modular” lattices.
The cumulative evidence of these many examples suggests the following
question: Can each irreducible representation of a semisimple Lie
algebra be constructed from a modular lattice?
(Host: Paul Edelman)

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(Consistent archiving began in Fall 1998.)

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