Colloquia — Spring 2005
Friday, April 15, 2005
| Title |
Topological Degree Theory and Nonlinear Operator Equations
in Banach Spaces |
| Speaker |
A. Kartsatos |
| Time |
TBA |
| Place |
TBA |
Abstract
We study problems of invariance of domain and eigenvalues for nonlinear operator
equations (*) Tx + Cx = 0, where T, C are
from a Banach space X to its dual
space X* with T maximal monotone. Invariance of domain
refers to the property that the image of a relatively open set in a mapping's
domain is an open set in the range space. We show how to extend the famous
Schauder invariance of domain theorem, involving injective compact displacements
of the identity I + C,
to operators T + C,
where T is, possibly, densely defined. We also show how to obtain
eigenvalues q for operator equations Tx + C(q,x)
= 0, where T is maximal monotone and C is demicontinuous,
bounded, and of type (S+) w.r.t. the variable x. We use
the Leray-Schauder degree theory when C is compact (reducing the
problem (*) to a problem of the type (I+C)x =
0), and the Browder degree theory for C demicontinuous,
bounded and of type (S+) w.r.t. x.
Friday, April 8, 2005
| Title |
Bayes Bandwidth Selection in Kernel Density Estimation with
Censored Data |
| Speaker |
W. J. Padgett
Distinguished Professor Emeritus
Department of Statistics
University of South Carolina
and Visiting Professor
Mathematical Sciences
Clemson University |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Professor George Yanev |
Abstract
Problems with right-censored data arise frequently in survival analysis
and reliability applications, and estimation of the lifetime density function
is often of interest. Two inherent problems in kernel density estimation
for lifetime data are the “spillover” at
the origin and the selection of the smoothing parameter (or bandwidth)
values to use in computing the density estimate. To address these issues,
we propose the
use of asymmetric kernels with a Bayesian approach to bandwidth selection.
In particular, the inverse Gaussian density function is used here as the kernel,
although other asymmetric densities such as the lognormal can be considered.
The (local) Bayes bandwidth obtained is exact for any sample size, only
depends on the prior parameters, and can be easily calculated from the
censored data. Strong pointwise consistency of the density estimator is
proven, and it is also shown that meaningful bandwidths with the same rates
of convergence as for the classical asymptotically optimal bandwidths can
be obtained for suitable choices of the prior parameters. (Joint work with K.B.
Kulasekera, Clemson University.)
Friday, April 1, 2005
| Title |
Old problems — new solutions in Knot Theory |
| Speaker |
Prof. Jozef Przytycki
Mathematics Department
George Washington University |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Professor M. Saito |
Abstract
In its long history, the knot theory abounds with elementary open problems.
One of them, recently solved, was the Montesinos-Nakanishi 3-move conjecture.
We start by discussing the history of the problem and the story of its solution.
But the story does not end with the solution.
The Nakanishi's 4-move conjecture still remains open. We will discuss this
and many other related elementary problems. Maybe you can solve one of them.
Monday, March 28, 2005
| Title |
Effective methods for resolving singularities in the plane
with applications to analysis |
| Speaker |
Dr. Michael Greenblatt |
| Time |
3:00-4:00 p.m. |
| Place |
ENB 108 |
| Note |
Speaker is a candidate for the faculty position in Analysis. |
Abstract
We describe an effective method for locally resolving the zero set of a real-analytic
function f(x,y). The method is geometric and
involves doing a finite sequence of transformations taking a point (x,y)
to a point (x, y - g(x^(1/N)))
for appropriate real-analytic functions g, where N is
an integer.
After these transformations,
a branch of the zero set of f(x,y) will be
(locally) given by {(x,y): x > 0, y =
0}or {(x,y): x < 0, y = 0}.
This method has applications to oscillatory integral operators, as well as
to the determination of the largest e > 0 for which the integral
of |f|^(-e) is finite near
a given zero of f(x,y).
Friday, March 25, 2005
| Title |
Fast-growing polynomials, best approximation on the complex
plain and matrix preconditioning |
| Speaker |
Dr. Maurice Hasson
Applied Mathematics Program
University of Arizona |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 120 |
| Note |
Speaker is a candidate for the faculty position in Analysis. |
Abstract
We will review the theory of “fast-growing” polynomials and show
how to construct them using the exterior conformal mapping. We will expand
analytic functions in series of fast growing polynomials and show how to use
these expansions to construct the (near) best uniform approximation of these
analytic functions on a given curve in the complex plane.
We will show how the near best approximation is used for the purpose of matrix
preconditioning. Numerical experiments will then be presented.
Friday, March 25, 2005
| Title |
Homological Algebra Methods in Graph Theory |
| Speaker |
Prof. Yongwu Rong
Mathematics Department
George Washington University |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Professor M. Saito |
Abstract
In 1999, M. Khovanov introduced a graded homology theory for knots, and proved
their graded Euler characteristic is the Jones polynomial. These homology groups
turn out to be surprisingly strong invariants and have sparked much attention
in low dimensional topology. In this talk, we introduce an analogous homology
theory for graphs, whose graded Euler characteristic is the chromatic polynomial.
Most results are joint work with Laure Helme-Guizon.
Friday, March 11, 2005
| Title |
Cherny's conjecture and the road coloring problem |
| Speaker |
Jarkko Kari |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Professor N. Jonoska |
Abstract
A directed graph G = (V,E) is called k-admissible
if all vertices have the same outdegree k. Let us color the edges
using k colors
in such a way that in every vertex the outgoing edges have distinct colors.
Any sequence w of colors specifies a vertex transformation fw :
V
V where fw(v) is the unique vertex reached from vertex v by following
the edges colored by letters of w. Word w is called synchronizing
if fw is a constant function, that is, if one reaches the same
vertex regardless of the starting position in the graph. The coloring
of G is called synchronized if a synchronizing word w exists. We
investigate two old open synchronization problems:
- The road-coloring problem asks which graphs have synchronized colorings.
It is conjectured that a synchronized coloring exists for all strongly
connected graphs that are not periodic. A graph is periodic if
some number m > 1 divides the lengths of all cycles.
- Suppose we have a synchronized coloring of a graph with n vertices.
The Cerny conjecture states that a synchronizing word of length
at most (n-1)2 must exist.
We prove these conjectures in the special case that the graph is Eulerian,
that is, all indegrees of all vertices are also the same constant
k. This is an interesting special case as such graphs seem difficult
to synchronize due to the lack of vertices with large numbers (>k) of incoming edges.
Friday, March 4, 2005
| Title |
TBA |
| Speaker |
Prof. V. Andrievski |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Professor B. Shekhtman |
Friday, March 4, 2005
| Title |
Spaces with maximal projection constants |
| Speaker |
Prof. Grzegorz Lewicki
Jagiellonian University, Poland/
University of Northern Iowa |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Professor L. Skrzypek |
Abstract
Let for n ∈ N, Bn denotes
the Banach-Mazur compactum, i.e., the set of all n-dimensional,
real Banach spaces equipped with the Banach-Mazur distance. Let Sn denote
a subset of Bn consisting of all symmetric, n-dimensional,
real Banach spaces.
Consider for any n ∈ N a function
λn : Sn R
defined by
$$
λn(X) = λ(X, l∞),
$$
where λ(X, l∞) denotes the norm of minimal
projection from l∞ onto X. The aim of this talk
is to present a construction of n-dimensional, real, symmetric spaces Xn for
which λn (Xn)
is large.
In particular, we show that
$$
liminfn λ(Xn)/\sqrt{n} > (2-\sqrt{2/π})-1,
$$
which disproves a conjecture of H. Koenig.
Also some open problems will be indicated.
Friday, February 25, 2005
| Title |
Zeros of the Riemann zeta function: Computations and implications |
| Speaker |
Andrew Odlyzko
Digital Technology Center
University of Minnesota |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 141 |
| Sponsor |
Department of Mathematics/College of Arts & Sciences |
Abstract
The Riemann Hypothesis is now left as the most famous unsolved problem in
mathematics. Extensive computations of zeros have been used not only to provide
evidence for its truth, but also for the truth of deeper conjectures that predict
fine scale statistics on the distribution of zeros of various zeta functions.
These conjectures connect number theory with physics, and are regarded by many
as the most promising avenue towards a proof of the Riemann Hypothesis. However,
as is often true in mathematics, numerical data is subject to a variety of
interpretations, and it is possible to argue that the numerical evidence we
have gathered so far is misleading. Whatever the truth may be, the computational
exploration of zeros of zeta functions is flourishing, and through projects
such as the ZetaGrid is drawing many amateurs into contact with higher mathematics.
Friday, February 18, 2005
| Title |
The Color of My Hat: An Introduction to Error-Correcting Codes |
| Speaker |
Vera Pless
University of Illinois at Chicago |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Professor X.-D. Hou |
Abstract
Error-correcting codes are widely used to correct errors in either the transmission or storage
of information. A specific error-correcting code provides the high fidelity
on compact discs. Because of their demonstrated practical usefulness, electrical
engineers started studying these codes about fifty years ago. Now they
are studied by engineers, mathematicians and computer scientists and a
wide theory has been developed with many connections to mathematical topics.
I will give all the basic definitions with examples and main problems in
error-correcting codes. We will discuss syndrome decoding and perfect codes.
We will then use this to determine “what color is my hat.”
Friday, February 11, 2005
| Title |
Topologized Graphs and S1 Spaces |
| Speaker |
Antoine Vella
University of Waterloo/
Technical University of Denmark |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Professor N. Jonoska |
Abstract
We present the novel model of “topologized graphs,” in which a
(possibly finite) graph is a toplogical space. We discuss the role of
topological ideas in extending well-known results about cycle spaces from finite
to infinite (topologized) graphs. We show how (non-Hausdorff) graph-theoretic
paths and trees, respectively, can be unified with the (Hausdorff) orderable and
dendritic spaces of general topology and how our “orderable” spaces
are naturally topologized graphs. We show how an attractive relaxation of the
T1 axiom emerges naturally in different ways. Some results
are similar to those of Whyburn (topology, 1968), Ward and others (partial-order
characterizations, 1970s) and Diestel and Kühn (cycle spaces, 2004) in more
general and unified settings. This is joint work with Bruce Richter.
Friday, January 28, 2005
| Title |
How Jean Pierre Serre revolutionised Algebraic Topology |
| Speaker |
Peter Hilton
Distinguished Professor Emeritus
State University of New York, Binghamton |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Professor M. Elhamdadi |
Abstract
The great French mathematician Jean Pierre Serre worked in homotopy theory
during the decade 1951-1960 and completely revolutionized the subject. I was
fortunate to know J. P. Serre well during that period, and I will reminisce
about the unique experience of working with him. At the end of the talk I will
describe some of Serre's key ideas.
Wednesday, January 26, 2005
| Title |
Studies of some low order quadrilateral nonconforming finite
elements |
| Speaker |
Professor Zhongci Shi
Academy of Mathematics and System Sciences
Chinese Academy of Sciences
Beijing, China |
| Time |
3:00-4:00 p.m. |
| Place |
ENB 108 |
| Sponsor |
Professor Y. You |
Abstract
In practice, quadrilateral mesh is much flexible in the finite element approximation
for a curved domain than rectangular and equally well suitable as triangular.
However, the existing results of convergence and superconvergence properties
of nonconforming elements over rectangular meshes can hardly be extended directly
to quadrilaterals.
In this talk, we study some low order nonconforming finite elements. The convergence
and superconvergence over general quadrilateral meshes are discussed.
Friday, January 21, 2005
| Title |
Classical polynomials with non-classical parameters |
| Speaker |
Professor Ramon Orive
University of La Laguna
Spain |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Professor E. Rakhmanov |
Abstract
In this talk we analyze some recent results on classical families of polynomials
(Jacobi, Laguerre) when the parameters take non-classical values. In particular,
we study the asymptotics of Jacobi polynomials with varying parameters. To
this end, a Riemann-Hilbert approach is used.
Friday, January 14, 2005
| Title |
On the space of pairs of immersions of the 2-disk to the
plane with common boundary circle |
| Speaker |
Dr. Minoru Yamamoto
Hokkaido University
Japan |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Professor M. Saito |
Abstract
In 1970's, Eliashberg classified pairs of immersions of the 2-disk to the
plane up to the regular homotopy. He used the homotopy principle and proved
that there are precisely two regular homotopy classes of such pairs.
In this talk, we classify such pairs up to the regular homotopy by using another
method. Comparing to Eliashberg's method, our method is combinatorial.
Monday, January 10, 2005
| Title |
Integrable semi-discretizations of two model equations for
shallow water waves |
| Speaker |
Prof. Xing-Biao HU
Academy of Mathematics and Systems Sciences
Academia Sinica
Taiwan |
| Time |
2:30-3:30 p.m. |
| Place |
CPR 118 |
| Sponsor |
Professor W.-X. Ma |
Abstract
Integrable semi-discretizations of two model equations for shallow water waves
are investigated. As a result, one integrable differential-difference version
for the AKNS equation and three integrable differential-difference versions
for the Hirota-Satsuma equation are found. These four differential-difference
versions are transformed into bilinear forms. Bäcklund transformations, soliton
solutions and Lax pairs for these differential-difference equations are presented.