Colloquia, Fall 2001

Vanderbilt Mathematics
Fall 2001

Colloquia are listed in reverse chronological order. The
top of the list is subject to change, since more colloquia
are still being planned. 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. You may also want to consult our
and past

Wednesday, December 19th.
Chunlan Jiang, of
the Chinese Academy of Science, Hei Bei University of Technology
and the University of Puerto Rico.
Structure of bounded linear operators.
The Jordan standard form theorem is a fundamental
theorem in matrix theory. It reveals the structure of bounded
linear operators on a finite dimensional space. For the
operators on an infinite dimensional Hilbert space, such a
structure theorem does not exist. We will prove an
approximated version of such a theorem: Every bounded
linear operator on separable Hilbert space can be
approximated by operators which are the direct sums of
strongly irreducible operators. Here strongly
irreducible operators are the counterpart of Jordan block
in case of infinite dimensional Hilbert space.

Thursday, November 29th.
Andrzej Kisielewicz, of the
University of Wroclaw, Poland
Vanderbilt University
Randomness Properties of Self-Complementary Symmetric Graphs.
There are a few results in the literature showing that
Paley graphs behave in many ways like random graphs.
In fact, this family of graphs is the most widely used example of
the so called concrete random graphs. The topic is
especially attractive, because it combines results from different
areas of mathematics: combinatorics, probability, algebra,
logic, and number theory! In my recent paper with W. Peisert,
we extended the results in question onto the family of all
self-complementary symmetric graphs, thus showing that the reason
why Paley graphs behave in such a way is their
highest degree of symmetry.
(Host: Ralph McKenzie.)

Tuesday, November 27th.

Charles Little
, of
Massey University,
New Zealand
and the
Georgia Institute of Technology.
The Pfaffian Property of Graphs.
Given a graph G and an orientation for G one may
assign plus or minus signs to the 1-factors of G according to
a procedure described by Kasteleyn. Kasteleyn’s goal was to enumerate
the 1-factors, and for this purpose he wanted the orientation to be
chosen so that each 1-factor had the same sign. A graph is said to be
Pfaffian if it has such an orientation. Pfaffian bipartite graphs
have been characterised in terms of forbidden subgraphs, but a general
characterisation of Pfaffian graphs remains elusive. This talk will
discuss some recent results.
(Hosts: Mark Ellingham and Mike Plummer.)

Thursday, November 15th. No colloquium.
DOUBLE FEATURE in Biomathematics

Please refer to the

Analysis and Biomathematics Seminar
for further information.

Thursday, November 8th.
Atsushi Uchiyama, of
Tohoku University, Japan and the
University of North Carolina at Charlotte.
An Application of Xia’s Methods for p-Hyponormal Operators to
p-Quasihyponormal Operators.

Theory of hyponormal operators and their generalization,
p-hyponormal operators has been in the center of attention
in operatory theory for the last four decades. For the non-normal
operator, its right “absolute value” is different than its
left “absolute value”. If the pth power of its right
“absolute value” is greater than or equal to to pth
power of its left “absolute value”, then this operator is
p-hyponormal. Several mathematicians, such as Xia and
Cho-Ito, established the Putnam type inequality which gives the
estimate of the norm of the difference between these two
“absolute values”, that measures the non-normality of the
p-hyponormal operatory by an integral on the spectrum. We study
a more general class of operators, p-quasihyponormal operators.
If the pth power of the right “absolute value” of an operator is
greater than or equal to the pth power of its left “absolute
value” on its range, then this operator is p-quasihyponormal.
A Putnam type inequality is also established for this class of
operators. It is preferable that the audience know some elementary
knowledge of what is a Hilbert space, a linear operator and its adjoint,
but it is not required. All the necessary terminology will be
(Host: Daoxing Xia.)

Tuesday, October 30th.
Georgij M. Kobelkov, of

Moscow M. V. Lomonosov University
Parabolic Approximations of the Navier-Stokes Equations.
Two types of parabolic approximations of the nonstationary
Navier-Stokes equations are considered.
Comparison and qualitative analysis of these two approaches are
carried out. Symmetrization of the artificial compressibility
method is proposed and studied. The talk is accessible
to graduate students.
(Hosts: Alexander Ol’shanskii and Maxim Ol’shanskii.)

Thursday, October 25th.
Victor Guba, of
Vologda State Pedagogical University

Vanderbilt University
Some Problems on the R. Thompson’s Group F.
The group F was discovered by Richard J. Thompson in the 1960s
during his work in Lambda-calculus. Since that, it was
rediscovered at the end of 1970s by some homotopy theory
specialists. This group seems to be interesting from many
points of view. First of all, it appears in many different
branches of mathematics such as mathematical logic, algebra,
the theory of functions, homotopy theory, homology theory,
probability theory, and others. Secondly, this group has many
specific properties. It is finitely presented, it has several
nice representations (e.g., it can be represented by
piecewise-linear functions). The word and the conjugacy
problems for it are solvable. There are no free non-abelian
subgroups in F. However, F does not satisfy any nontrivial
group law. We are going to discuss several properties of F,
including some very recent ones (such as the problem about
isoperimetric functions for this group). Also we will
discuss some open problems about this group.
(Host: Mark Sapir.)

Thursday, October 18th.
Jaroslav Jezek, of
Charles University, Prague
and Vanderbilt University.
Topics in Universal Algebra.
We will start with a characterization of the algebras that
are isomorphic to the factor of a subdirectly irreducible
algebra by its monolith. (It is possible to give the
characterization in the case when the similarity type
is not too poor; surprisingly, for poor similarity types
the question remains an open problem.) A survey of some
other recent and not-so-recent results and open problems
will follow.
(Host: Ralph McKenzie.)

Tuesday, October 16th.
Maxim Olshanskii, of

Moscow M. V. Lomonosov University

Vanderbilt University
Finite Element Methods in Fluid Dynamics: Stability and
Iterative Solvers.

After refreshing basic concepts of the Galerkin method for PDE
we consider some particular problems from fluid dynamics: The
convection-diffusion, the Stokes, and the incompressible
Navier-Stokes. The application of finite element method gives
rise to non-symmetric, non-definite and nonlinear systems of
algebraic equations, respectively. Our concern is iterative
solvers for these systems and a quality of approximate solutions.
This quality depends on stability properties of the Galerkin
method. Possible ways of enhancing the stability are discussed.

Tuesday, October 9th.
Zuowei Shen, of
National University of Singapore
Wavelet Frames: Theory and Constructions.
This talk is about wavelet frames.
The redundant representation offered by wavelet frames has already
been put to good use for signal denoising, and is currently explored for
image compression. We restrict our attention to wavelet frames
constructed via multiresolution analysis, because this guarantees
the existence of fast implementation algorithms.
We shall explore the `power of redundancy’ to establish general
principles and specific algorithms for constructing framelets and
tight framelets. In particular, we shall give several systematic
constructions of spline and pseudo-spline tight frames and symmetric
bi-frames with short supports and high approximation orders. Several
explicit examples are discussed.

This talk is based on joint work with
Ingrid Daubechies, Bin Han and Amos Ron.
(Host: Doug Hardin.)

Friday, October 5th.
Congming Li, of
University of Colorado, Boulder
Analysis of PDEs and the Problem of Prescribing Gaussian
and Scalar Curvature.

In the study of nonlinear partial differential equations (PDEs) of
elliptic type, the equations with the so-called “critical exponent” is
one of the central problems. Many problems from physical sciences and
differential geometry are of this type. Prescribing Gaussian and
scalar curvature is one of them. The main difficulty in dealing with this
type of problems is the lack of estimates (or compactness) on the
solutions. Very often, a small perturbation leads to dramatical
change of the set of solutions. We will discuss several methods
designed to solve this type of
problems and their applications to differential geometry.
(Host: Guoliang Yu.)

Thursday, October 4th.
Rene Sperb, of
the ETH Zürich.
Some Estimates for the Logistic Population with Diffusion.
A population is considered whose growth rate is governed by
Verhulst’s quadratic law in a habitat surrounded by a hostile
environment. The influence of diffusion versus the rate constant is
incorporated in a single parameter which plays a crucial role. The
main issues are sharp estimates for the total number of species which
can coexist.
(Host: Philip Crooke.)

Thursday, September 20th.
Shigui Ruan, of
Dalhousie University
and Vanderbilt University.
Homoclinic Bifurcations and Chaos.
Consider a dynamical system at an equilibrium, if the variational
matrix has at least one eigenvalue with positive real part and at
least one eigenvalue with negative real part, then the equilibrium
is a saddle (and unstable). If the one dimensional unstable manifold
joins the stable manifold to form a homoclinic orbit, what are the
dynamics of the system near the homoclinic orbit? In this talk, we
will see that in two dimensions there is a periodic orbit bifurcated from the
homoclinic orbit. However, in three dimensions we can have either a homoclinic
bifurcation leading to a periodic orbit or a Smale Horseshoe Map
leading to chaos. Examples in FitzHugh-Nagumo model and couples
three oscillators model will be given. Generalization to higher
dimensional systems and infinite dimensional dynamical systems will
be discussed. (The talk is accessible to seniors and graduates).
(Host: Glenn Webb.)

Tuesday, September 11th.
Pierre Magal, of the

Université du Havre
Mutation, Selection, and Recombination in a Model of Phenotype Evolution.
A model of phenotype evolution incorporating mutation, selection,
and recombination is investigated. The model consists of a partial
differential equation for population density with respect to a
continuous variable representing phenotype diversity. Mutation
is modeled by diffusion, selection is modeled by differential
phenotype fitness, and genetic recombination is modeled by an
averaging process. It is proved that if the recombination process is
sufficiently weak, then there is a unique global asymptotically
stable attractor.
(Host: Glenn Webb.)

Tuesday, September 4th.
Herbert Amann, of the
Universität Zürich.
Navier-Stokes Equations in Spaces of Low Regularity.
(Host: Gieri Simonett.)

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(Consistent archiving began in Fall 1998.)

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