# Vanderbilt MathematicsColloquiaSpring 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
weekly
calendar
and past calendars.
at the
web page maintained by this year’s Colloquium Chairperson
.

June 11.
Suhrit K. Dey, of
Eastern Illinois University.
Biomechanics of Lymphocytes with Applications for Prevention /
Cure of Breast Cancer.

Lymphocytes are white cells, which fight infections and cancer. T cells
are lymphocytes, which attack all antigens very aggressively. Thymus is
a lymphoid organ which produces a hormone called thymosin which help T –
cells to proliferate and while staying in the thymus, T – cells become
trained soldiers to defend the body against the invasion of cancer.
Thymus is located inside the chest cavity above the breast and in the
space between the lungs. —
Three mathematical models have been developed to represent Surveillance
of lymphocytes, Rate of mobilization of lymphocytes and the defense
mechanism of lymphocytes. The third model deals with activator-inhibitor
response system. Cancer cells are activators. They activate the
lymphocytes to respond as inhibitors. This model consists of two coupled
nonlinear partial differential equations, which were solved numerically.
If the activator prevails, cancer spreads and if the inhibitor prevails
the immune system overpowers cancer. The models are all one-dimensional
based on the assumption that the cancer site is very near to at least
one lympnode. Physiologically this scenario is applicable to breast
cancer.

June 7.
Joachim Escher,
of the
University of Hannover.
Analytic Solutions for a Stefan Problem with Gibbs-Thomson Correction.
Stefan problems are widely used to model the freezing/melting process of
water/ice. In this talk a general existence and uniqueness result of
classical solutions for a class of Stefan problems with Gibbs-Thomson
correction in arbitrary space dimensions is provided. In addition, it
will show that the moving boundary depends analytically on the temporal
and spatial variables. — Of crucial importance for the analysis is the
property of maximal Lp-regularity for the linearized problem,
which is based on the Dore-Venni theorem.

April 19.
Paul Baum, of
Pennsylvania State University.
K theory for group C* algebras.
Several issues in representation theory and geometry-topology
can be unified by studying the K-theory of group C* algebras.
P.Baum and A.Connes have conjectured a formula for this K-theory.
This talk states the conjecture and indicates how it is related
to various questions. The talk is intended for a general mathematical
audience, and basic definitions (C* algebra , K theory) will
be carefully stated.

April 18.
Roger Horn, of the
University of Utah.
Equalities and Inequalities for Matrix Eigenvalues and Singular Values.
Many classical inequalities for matrix eigenvalues and singular values can
be understood in the context of simple counting arguments involving
subspace intersections. These arguments often have the added benefit of
identifying cases of equality. We illustrate these ideas by discussing
the Weyl inequalities for the eigenvalues of a sum of two Hermitian
matrices, and the Cauchy interlacing inequalities for a bordered Hermitian
matrix.

April 13.
Richard Laver,
of the
Large cardinals and their implications in classical mathematical
areas.

This will be an expository talk. We will review some of the basic
concepts of set theory, and state some “large cardinal” axioms—axioms which
assert the existence of infinite cardinals having properties which make them,
roughly speaking, so large that their existence cannot be proved by ordinary
mathematical methods. The question arises as to what applications the
existence of such cardinals might have to classical mathematics; we’ll
discuss a couple of examples, one about projections of Borel sets and
one about finite algebras in an area related to knot theory.
[In this last example, the large cardinal assumption is not known to
be necessary.]

April 12.
Daniel E. Gonsor,
of
The Boeing Company,
Seattle, Washington.
Three Problems from Industrial Mathematics.
Most talks on mathematics in industry present difficult problems
for which mathematics played an important role in deriving an effective
solution. In this talk we will take a different approach and look at
three routine problems that arose in the context of everyday work at The
Boeing Company. In each case the (then) current solution method
produced unsatisfactory results. The first problem involves data
fitting, the second calculating a geodesic, and the third calculating
arc length. We will analyze each solution method, determine the
source(s) of the deficiency, and propose a better solution. The reason
for choosing these particular problems is threefold. First, each problem
is representative of an approach or mentality that one often encounters
in industrial mathematics. In the case of data fitting it is the
insistence on interpolation, for the geodesic example it is the reliance
on intuition and visual verification, and for the arc length example it
is the disregard of fundamental hypothesis. The second reason for
choosing these examples is that the mathematics involved is fairly
elementary, and therefore will be accessible to undergraduate math
majors. The last reason is that the fundamental deficiency in each
example can be traced to a lack of mathematical expertise.

April 11.
Karin Goosen,
of the
University of Stellenbosch,
Republic of South Africa.
Interpolatory Subdivision and Wavelets on an interval.
We consider a method of adapting the Dubuc-Deslauriers subdivision
scheme to accommodate sequences of finite length, in a way which ensures
convergence of the adapted scheme and the existence of an associated
refinable function. Then with an appropriate definition of a
interpolation wavelet, we obtain decomposition and reconstruction
algorithms. Illustrations of the theory are provided.

March 29.
Eric Weber,
of
Texas A&M.
Translation Invariant Wavelets.
All wavelets can be associated to a multiresolution like
structure, i.e. an increasing sequence of subspaces of L2. We
consider the interaction of a wavelet and the translation operator in
terms of which of the subspaces in this multiresolution like structure are
invariant under the translation operator. This action defines the notion
of the translation invariance property of order n. In this talk we
shall characterize such wavelets and show that they exist. We shall
also discuss how these special types of wavelets might lead to techniques
for edge detection in images.

March 27.
Misha Kapovich,
of the University of Utah.
Singularities of representation varieties of finitely generated
groups.

Given a finitely-generated group G and an
algebraic Lie group G (for instance, SL(2)) the space of representations
Hom(G,G) itself has structure of an algebraic
variety. In this talk I will outline several “universality” theorems for
the singularities of Hom(G,G), whose main
message is that these singularities could be as bad as one wishes. This
applies to such classes of finitely generated groups as Coxeter groups,
Artin groups, fundamental groups of 3-manifolds and discrete groups of
isometries of the hyperbolic 3-space. This is a joint work with John
Millson.

March 19.
Kirby Baker, of
UCLA.
Unavoidable patterns in long strings of symbols.
Thue showed that using an alphabet of three symbols it is
possible to construct an infinite string with no block that is
immediately repeated. We say that such a string avoids the
pattern xx . On the other hand some patterns, such as
xyxzxyx, are unavoidable no matter how many symbols are
in the alphabet; in other words, there are always blocks
X, Y, Z so that XYXZXYX occur consecutively. There are
intriguing questions as to which patterns can be avoided
using what sizes of alphabets.

March 15.
Wai Shing Tang,
of the
National University of Singapore.
A Hilbert space approach to wavelets.
In this talk, we first review the concept of multiresolutions of
L2(R), as introduced by Meyer and Mallat in the mid 1980’s,
and show how an orthonormal wavelet can be obtained from a
multiresolution. Next we describe a connection between the existence of
wavelets and Robertson’s result on wandering subspaces for unitary
operators on Hilbert spaces. Finally, we give a brief summary of some
recent work of the speaker and his collaborators on the approach of
wavelets in Hilbert spaces.

March 13.
Alexander Kostochka,
University of Illinois.
Equitable colorings of graphs. In many applications of graph
colorings color classes should not be large. A good model for such
applications is the notion of equitable coloring — a proper
coloring where the difference between the sizes of any two color classes
is at most one. Hajnal and Szemerédi proved that for every
1
and
k³D+1,
each graph with maximum degree at
most
D
admits equitable coloring with k colors. The aim of the
talk is to survey recent progress in studying equitable colorings and to
prove a conjecture on equitable colorings of outerplanar graphs. We also
will discuss an analog of equitable coloring for list colorings.

March 5.
Dmitry Kozlov, of the
Royal
Institute of Technology, Stockholm, Sweden
.
Stratifications indexed by partitions and combinatorial models for
homology.

In this talk I will discuss several connections between various
combinatorial objects (partitions, partially ordered sets, labeled forests,
matchings) and algebraic invariants (homology groups, Betti numbers) of
certain
topological spaces.

More specifically, I shall consider several topological spaces equipped with
stratifications indexed by integer partitions. In each case I consider the
problem of studying homology groups of strata. I shall describe how to
construct various models for computing these groups and present the following
applications:

1. determining the homology of resonance-free orbit arrangements (with
the help of general lexicographic shellability), thereby settling a
conjecture of Bjorner for this special case;
2. a combinatorial reproof of Arnol’d theorem regarding the rational
homology of the space of monic complex polynomials with at least q roots
of multiplicity k;
3. a counterexample to a conjecture by Sundaram and Welker;
4. a computation of the homology groups of the space of hyperbolic
polynomials with at least q roots of multiplicity k.

March 1.
Bjarne Toft,
University of Southern Denmark
& Vanderbilt University.
Julius Petersen – the man, the myth, the legend.
The emergence of Danish mathematics at the end of the nineteenth century and
the dominant role of geometry in Danish research is closely linked to the work
of two mathematicians, Zeuthen and Petersen. Both are still remembered for
their geometry, but Zeuthen also as a historian of mathematics, and Petersen
for his graph theory.

Petersen did pioneering work in a number of fields, including cryptography and
economics, but both his graph theory and some of the other brilliant pieces
went unnoticed or met with outright rejection in his own time. This is not to
imply that his life was one of disappointment – far from it! He was the
embodiment of the best sense of humor and the most vigorous joy in life.

A biography of Petersen by Jesper Lutzen, Gert Sabidussi and Bjarne Toft has
been published in Discrete Mathematics Volume 100. The talk is based on that
paper.

(Related
web page)

February 22.
Nigel Higson,
of Penn State.
Group C*-Algebras and Topology.
I will give a survey of some current work at the interface
of C*-algebra theory and the topology of manifolds. The central
problem here is the Baum-Connes conjecture, which has
implications not only for topology but also for harmonic analysis.
The conjecture will be the focus of the lecture.

February 21.
Don Hong,
of
East Tennessee State University.
Multilevel structure of bivariate spline spaces over
triangulations.

In this talk, we’ll investigate multilevel structure of bivariate spline
spaces. Wavelet decomposition method, multigrid technique in finite
element, and subdivision scheme in splines and approximation theory
actually reach the same goal by different routes. Wavelet theory
provides very efficient algorithms in decomposition and reconstruction
by using the so-called wavelets, which are actually locally supported
basis functions like “little waves”. We’ll present some recent results
on multivariate splines and wavelet-type basis construction for
bivariate spline spaces over triangulations.

February 20.
Anthony
Hilton
, of University
, UK.
Some coloring theorems and conjectures.
There is a well known conjecture that if a regular graph of even order has
degree greater than half the order then it is the union of edge-disjoint
1-factors.
With David Cariolaro, I have shown that this is true if 3/4 is substituted
for 1/2.
I shall sketch the proof of this result, and also discuss its relationship
with two other well known conjectures, the
Overfull Conjecture and
the Conformability Conjecture.

Let us call the conjecture above about regular graphs the
Regular Graph Conjecture.
Let us now explain what the other two conjectures say. —
The chromatic index of a graph is the least number of colours needed to
colour the edges of the graph so that no two edges with the same colour
are incident with the same vertex.
By Vizing’s Theorem the chromatic index equals either the maximum degree
(in which case the graph is called Class 1) or the maximum degree plus 1
(in which case the graph is Class 2).
A graph is Overfull if it is of odd order and the number of edges is
greater than half the product of the maximum degree and
(the order lass one).
The Overfull Conjecture is that if the maximum degree of a graph
is greater than one third the order, then the graph is Class 2 if and only
if it contains an overfull subgraph of the same maximum degree.
The Regular Graph Conjecture follows from the Overfull Conjecture.

The total chromatic number of a graph is the least number of colours needed
to colour the vertices and the edges of the graph so that no two incident
or adjacent elements get the same colour.
An old conjecture of Behzad and Vizing (independently) is that the total
chromatic number of a graph equals either (1 or 2) plus the maximum
degree.
In the first case the graph is called Type 1 and in the second Type 2.
A vertex colouring of a graph with a number of colours equal to 1 plus the
maximum degree is called Conformable if the number of colour classes
having parity different from that of the order is at most the deficiency
of the graph, where the Deficiency is defined to be the product of
(the maximum degree and the order) less twice the number of edges.
(The deficiency measures the amount by which the graph fails to be
regular.)
A graph is called conformable if it has a conformable vertex colouring.
The Conformability Conjecture is that a graph with the property that
the maximum degree is greater than twice the order is Type 2 if and only if
it has a subgraph of the same maximum degree which is either nonconformable
or is a complete graph of odd order with one edge subdivided.

February 15.
Friedrich Wehrung, of the
Université de Caen, France.
Congruence lattices of lattices: a survey.
The Congruence Lattice Problem, that asks whether every distributive
algebraic lattice is isomorphic to the congruence lattice of a lattice,
is, despite many attemps, still unsolved. I present the most recent
results say essentially that one cannot solve the problem by using
lattices with permutable congruences. The positive results imply that all
known representation theorems can be done with relatively complemented
lattices with zero.

February 8.
Ivo Dinov, Neurology,
UCLA School of Medicine.
Mathematical and Computational Challenges in Brain
Mapping and Neuroimaging.

The incredibly complex (yet robust, efficient and
elegant) functional, anatomical and bio-physiological
organization of the brain provides a rich source for
developing interesting mathematical and computational
models. Following an introduction to the goals of brain
mapping research and the variety of brain-data acquisition
methods we will describe a number of problems and obstacles
researchers in this field encounter. Among the most needed
algorithms and data filters are models for: Stereotactic
data registration (alignment); Cortical surface modeling;
Tissue segmentation; Skull stripping and feature
extraction; Construction of population specific brain
atlases; Measures of temporal/developmental changes and
variability; Statistical assessment of structural or
functional differences. We will devote most of our
attention to the problems of signal representation and
quantitative evaluation of different image registration
methods. MRI (magnetic resonance imaging) and PET (positron
emission tomography) data from elderly normal controls and
dementia patients will be used to illustrate the
functionality and disadvantages of a variety of
mathematical techniques.

January 18.
Edward Saff,
of the University of South Florida.
Distributing Many Points on a Sphere.
The problem of distributing a large number of points
uniformly over the surface of a sphere arises in
many practical and theoretical situations. We discuss
generating such points by optimization with respect
to a generalized energy criterion. Our interest is
primarily with the asymptotic behavior of these
optimal spherical configurations of N points as
N tends to infinity. Methods for generating “near
optimal” points will also be discussed along with
several challenging open problems.

January 15.
Gui
Quian Chen
, of Northwestern University.
Hyperbolic Conservation laws and Divergence-Measure Vector Fields.
In this talk we first describe some connection between
hyperbolic conservation laws and divergence-measure vector fields.
Then we introduce a theory of divergence-measure vector fields,
including the Gauss-Green formula and the normal traces, and discuss
its applications to solving various nonlinear problems in partial
differential equations.

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