Colloquia — Spring 2003
Friday, May 2, 2003
| Title |
A Statistical Method for Identifying Informative Genes in
Microarrays |
| Speaker |
Dr. James Yang |
| Time |
2:00-3:00 p.m. |
| Place |
PHY 109 |
| Sponsor |
Professor A. Rao |
| Note |
Speaker is a candidate for Asst. Prof. in Biostatistics.
|
Abstract
DNA microarrays can be used to monitor thousands of gene expressions in a single
experiment. Statistical analysis on microarray data provides genetics researchers
a scientific approach to answering research questions. In this talk, a cost-effective
method of making microarrays and reading microarray data will be presented.
Statistical methods to solve the following three primary methodological problems
in microarray data analysis are proposed: (1) identify differentially expressed
genes; (2) estimate the expression difference; and (3) determine the sample
size.
This talk provides a comprehensive review of statistical methods for identifying
differentially expressed genes in two-condition microarray experiments. Following
this review, a new method is proposed to select informative genes. Simulation
experiments and statistical analysis on real data were conducted to compare
the proposed method with commonly used methods. The results indicate that the
proposed gene selection method did better than commonly used methods.
To estimate the gene expression differences under different conditions, a new
method has been developed in this study. The estimator is proved to be consistent.
This study investigates a practically important yet relatively unexplored issue:
sample size determination. A new statistical method is developed and compared
with two existing methods.
Monday, April 28, 2003
| Title |
Virus Evolution: Micro-scale Epidemic Simulations |
| Speaker |
Dr. Prasith Baccam
Los Alamos |
| Time |
2:00-3:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Professor A. Rao |
| Note |
Speaker is a candidate for Asst. Prof. in Biostatistics.
|
Abstract
This talk consists of two separate parts — virus evolution and epidemiology.
In the first part, I will discuss the role of viral viration in virus persistence
and disease pathogenesis. We focus on the evolution of a regulatory protein
for a horse lentivirus and how it evolves over the course of infection. We use
phylogenetic and non-hierarchial clustering methods to tease out the dynamics
of the viral quasispecies over time and conjecture on how it affects disease
pathogenesis. In the second part of the talk, I will describe a complex agent-based
epidemiological simulation which we created to test medical interventions in
hopes of effectively managing the outbreak. Case studies using the simulation
will be presented. Real-world data is analyzed and its implication on the simulation
will be discussed.
Friday, April 25, 2003
| Title |
On Coarse Approach to the Novikov Conjecture |
| Speaker |
Professor Alexander Dranishnikov
University of Florida |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Boris Shekhtman |
Abstract
The Novikov Conjecture about homotopy invariance of higher signatures of
manifolds has been under very active investigation during last three decades. It
was discovered that it has very close relation to many different areas of
mathematics such as Group Theory, Differential Geometry, Functional Analysis.
During last five years it was found that the Novikov Conjecture has relation to an
abstract Dimension Theory as well to Computer Science, namely to building
communication networks.
Monday, April 21, 2003
| Title |
A Bayesian Approach for Estimating Antiviral Efficacy in HIV
Dynamic Models |
| Speaker |
Dr. Yangxin Huang
Harvard School of Public Health |
| Time |
2:00-3:00 p.m. |
| Place |
CHE 201 |
| Sponsor |
Professor A. Rao |
| Note |
Speaker is a candidate for Asst. Prof. in Biostatistics.
|
Abstract
The study of HIV dynamics is one of the most important developments in recent
AIDS research. It has led to a new understanding of the pathogenesis of HIV
infection. Although important findings in HIV dynamics have been published in
prestigious scientific journals, the statistical methods (nonlinear least
squares, for example) for parameter estimation and model-fitting used in those
papers appear surprisingly crude and have not been studied in more detail. In
this talk, a viral dynamic model is developed to evaluate the effect of
pharmacokinetic variation, drug resistance and adherence on antiviral response.
In the context of this model describing HIV infection, we investigate a Bayesian
modeling approach under a nonlinear hierarchical model framework. In particular,
our modeling strategy allows us to estimate time-varying antiviral efficacy of a
regimen during the whole course of treatment period by incorporating the
information of drug exposure and drug sensitivity. Both simulation and real
clinical data examples are given to illustrate the proposed approach. The
Bayesian approach involves assumptions of probability distributions for model
parameters prior to an analysis being performed, allowing the fitting of complex
models and enabling analysis of all of the model parameters, and has great
potential to be used in many aspects of viral dynamics modeling. It is suggested
that Bayesian approach for estimating parameters in HIV dynamic models is more
flexible and powerful than the nonlinear least squares method.
Friday, April 18, 2003
| Title |
Pseudo-Orthogonality, Maximal Orthogonality and Generalized
Inverses |
| Speaker |
Dr. Michael Rieck
Drake University |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Edwin Clark |
Abstract
Vector spaces over finite fields equipped with a notion of “orthogonal
vectors” do not generally admit an orthogonal complement for a given
subspace. The weaker notion of a “pseudo-orthogonal complement” and
the even weaker notion of a “maximally orthogonal complement”
sometimes provide satisfactory substitutes, and are guaranteed to exist. A number
of related definitions for these will be presented. Several constructive
approaches for obtaining pseudo-orthogonal complements will also be given. It will
be seen that these provide for the construction of a certain kind of generalized
inverse to a given linear transformation. Finally, the nature of the vectors that
can occur in such a complement will be identified.
Friday, March 28, 2003
| Title |
Introduction to Solitons |
| Speaker |
Professor David Kaup
University of Central Florida |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Wen-Xiu Ma |
Abstract
Solitons are nonlinear pulses that have applications all the way from hydrodynamics
through optics to plasma physics. We will describe the basic features of soliton
theory, and some of their applications.
Friday, March 21, 2003
| Title |
Viscosity Solutions and Absolute Minimizers on the Heisenberg
Group |
| Speaker |
Dr. Thomas Bieske
Department of Mathematics
University of Michigan |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Yuncheng You |
| Note |
Speaker is a candidate for Asst. Prof. in Analysis.
|
Abstract
In this talk, we begin by defining the concept of viscosity solutions to a
class of fully nonlinear equations in the Heisenberg group, which is the simplest
non-abelian Lie group. A Heisenberg maximum principle and comparison principle
follow. In particular, existence-uniqueness of (viscosity) infinite harmonic
functions on a domain with given boundary data is shown. Finally, absolute
minimizers are shown to be (viscosity) infinite harmonic; hence, they are
unique.
Friday, March 21, 2003
| Title |
Homogenized and Concentrated Limits in Visual Transduction |
| Speaker |
Professor Emmanuele DiBenedetto
Vanderbilt University |
| Time |
3:00-4:00 p.m. |
| Place |
TBA |
| Sponsor |
Professor Yuncheng You |
Abstract
We compute the homogenized-concentrated limit of solutions of a system of heat
equations set in a layered almost disconnected cylindrical structure and with
non-linear variational data.
The problem arises from Visual Transduction, i.e., the process by which signals
generated by photons in the rod outer segments of the rod of vertebrates, are
transformed into electrical pulses that generate vision.
The main mathematical significance is in (a) some compactness equi-Hölder
continuity estimates for solutions in such a layered domain and (b) an application
of the Kirzbraun-Pucci Extension Theorem for function with a concave modulus
of continuity.
We present also results of some numerical simulations.
Wednesday, March 19, 2003
| Title |
Tensor Products and p-Summing Operators With Hilbertian
Domain |
| Speaker |
Dr. Qingying Bu
Mississippi State University |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Boris Shekhtman |
| Note |
Speaker is a candidate for Asst. Prof. in Analysis.
|
Abstract
In the first part, we will give sequential representations of the projective
and injective tensor products of Lp[0,1] with
a Banach space X. Then by using their sequential representations
we will discuss several geometric properties that can be lifted from X
to the tensor products of Lp[0,1] with X.
In the second part, by using the sequential representations in the first part,
we will discuss p-summing operators with Hilbertian domain, and then
in Banach lattice case, we give a positive answer to Pisier's conjecture about
Banach spaces verifying Grothendieck's theorem.
Wednesday, March 19, 2003
| Title |
On the New Primality Test |
| Speaker |
Professor Michael Pohst
Tech University of Berlin |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Joseph Liang |
Abstract
TBA
Monday, March 17, 2003
| Title |
On the Distribution of the Zeros of an Entire Function of
an Exponential Type |
| Speaker |
Dr. Dimiter Dryanov
University of Montreal |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Yuncheng You |
| Note |
Speaker is a candidate for Asst. Prof. in Analysis.
|
Abstract
An entire function f is said to be of exponential type σ >
0 if for every ε > 0, there exists a constant c(ε)
such that
$$
|f(z)|\le c(\epsilon)e^{(\sigma+\epsilon)|z|}\quad (z\in\mathbb{C}).
$$
We obtain Bernstein's type interpolation formulae for exact recovery of an
entire function of exponential type. These formulae can be used to accelerate
the convergence of sampling series. We generalize a theorem of R. J. Duffin and
A. C. Schaeffer about the distribution of the zeros of a real entire function
of exponential type. By using this generalization we extend a result of L. Hörmander
on local behavior of an entire function of exponential type. Typical for our
consideration is the following theorem:
Let f be an entire function of exponential type σ > 0 and
let f(x) = o(x) as |x|\to\pm\infty.
If f vanishes at the origin and f is bounded by a constant
M at the extrema of sin σ z, then |f(x)| ≤
M| sin σ x| for all $x\in(-\pi/2\sigma,\pi/2\sigma)$.
Equality holds at any point $x\in(-\pi/2\sigma,0)\cup(0,\pi/2\sigma)$ if and
only if $f(z)\equiv e^{i\gamma}\sin\sigma z$ for some real γ.
In the process of our study we introduce a notion for a Chebyshev function
of exponential type. Some other results, one being an analog of a result of
M. Riesz about trigonometric polynomial whose zeros are real and simple are
proved.
Monday, March 17, 2003
| Title |
How to Prove That a Projection is Unique Minimal, or Why the
Smoothness, Trace Duality and Chalmers-Metcalf Operator Come in Handy |
| Speaker |
Dr. Leslaw Skrzypek
Jagiellonian University
Poland |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Boris Shekhtman |
| Note |
Speaker is a candidate for Asst. Prof. in Analysis.
|
Abstract
After a brief introduction to projections we will try to shed light on the
question from the title.
First, as a dramatic evidence of a gap between norm-one projections and those
of norm greater then 1 we will present an especially easy proof of the
Cohen-Sullivan theorem (norm-one projections are unique minimal in smooth
spaces).
Then the essential role of the smoothness of a considered space will be
justified by a consideration of L1 spaces. Next we will
provide a general framework to handle the problem of the uniqueness of minimal
projections. As an example we will outline the proof that a projection onto a
symmetric subspace is unique minimal. A notion of a trace, trace duality and the
Banach-Alaoglu as well as Krein-Millman theorem will be exploited. The above will
result in the uniqueness of the Fourier projection onto Rademacher functions.
Additionally an approach to classical Fourier projections (associated with a group
of characters on a unit disc) as well as Walsh projections (associated with a
group of characters on [0,1]) will be provided and some corresponding open
problems in classical Fourier analysis stated. Other directions for further
development will be mentioned.
Friday, March 7, 2003
| Title |
Classical and Relativistic Compressible Flows |
| Speaker |
Dr. Ronghua Pan
Department of Mathematics
University of Michigan |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Yuncheng You |
| Note |
Speaker is a candidate for Asst. Prof. in Analysis.
|
Abstract
I will present some of our recent progresses in the theories of nonlinear hyperbolic
systems for both classical and relativistic compressible flows. For classical
compressible flows, my emphasis is our recent breakthrough for the long time
behavior of the solutions to one-dimensional isentropic compressible flows through
porous media. The central conjecture in this field has been proven under physical
conditions. For relativistic compressible flows, I shall report my joint work
with Joel Smoller in singularity formation for relativistic Euler equations
in four-dimensional Minkowski spacetime. One of our results shows that any nontrivial
smooth solutions of relativistic Euler equations with compact support initial
data blows up in finite time.
Friday, March 7, 2003
| Title |
Impulsive Stochastic Evolution Inclusions on Infinite Dimensional
Spaces and Their Control |
| Speaker |
Dr. N. U. Ahmed
Department of Mathematics
University of Ottawa |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Yuncheng You |
Abstract
We consider a general class of stochastic evolution inclusions on infinite
dimensional spaces.
Here A is the infinitesimal generator of a C0
-semigroup of bounded linear operators in a Hilbert space, β ∈
BVloc(R), F a single-valued nonlinear
operator and μ is a vector measure, C is a multivalued map and
W is the cylindrical Brownian motion on a separable Hilbert space.
Our major concern here, for this class of systems, is the question of existence
of solutions and their regularity properties. We present some results in this
area from a recent paper of the author (Diff. Incl. Contr. & Optim. 22 (2002),
125-149).
In a more recent paper, optimal control problems for this class of systems
have been also considered which may be briefly described.
Study of this class of systems is motivated mainly by recent interest in impulsive
systems, impulsive controls and the so-called uncertain systems. A couple of
examples from engineering problems will be presented for illustration. The paper
is concluded with some comments on open problems in the area.
Monday, March 3, 2003
| Title |
Convex Bodies and the Fourier Transform |
| Speaker |
Dr. Artem Zvavitch
Department of Mathematics
University of Missouri |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Boris Shekhtman |
| Note |
Speaker is a candidate for Asst. Prof. in Analysis.
|
Abstract
The study of geometric properties of bodies using information about sections
and projections of these bodies has important applications to many areas of
mathematics and science. A new approach to projections and sections of convex
bodies, based on methods of Fourier analysis, has recently been developed. The
idea is to express certain geometric properties of bodies in terms of the Fourier
transform and then apply methods of harmonic analysis to solve geometric problems.
The crucial role in the Fourier approach to sections belongs to a certain formula
connecting the volume of sections with the Fourier transform of powers of the
Minkowski functional. In this talk we present an analog of this formula for
the case of projections, which expresses the volume of projections in terms
of the Fourier transform of the curvature function. Using this formula we study
the extremal projections of lp-balls and present a new, Fourier analytic,
solution of the Shephard problem, asking whether bodies with smaller hyperplane
projections necessarily have smaller volume.
Friday, February 28, 2003
| Title |
Combinatorial Group Theory and Discrete Tiling Problems |
| Speaker |
Dr. Michael Reid
Department of Mathematics
University of Arizona |
| Time |
4:15-5:15 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Edwin Clark |
| Note |
Speaker is a candidate for Asst. Prof. in Algebra.
|
Abstract
In 1990, Conway and Lagarias published a method of analyzing discrete tiling
problems by using finitely presented groups. Their method has been successfully
used to understand a handful of tiling problems, both in their original article,
and by subsequent authors.
Many computational questions about general finitely presented groups are not
solvable by any algorithm, and those that are, are often difficult.
My work shows that the finitely presented groups that typically arise can often
be understood, at least to some degree. I have found numerous new cases of tiling
problems in which I can apply the Conway-Lagarias method successfully, and they
should exist in abundance, in light of my technique.
I also consider tiling restrictions that cannot be detected by the Conway-Lagarias
technique, some open problems and conjectures, as well as some further directions
for future research.
This talk will be accessible to a fairly general mathematical audience.
Friday, February 28, 2003
| Title |
General Position Properties Which Characterize Low-Dimensional
Manifolds |
| Speaker |
Professor Dusan Repovs
University of Ljubljana
Slovenia |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Masahiko Saito |
Abstract
We shall present a historical survey of the geometric topology of generalized
manifolds, i.e., ENR homology manifolds, from their early beginnings in the
early 1930's to the present day, concentrating on those geometric properties
of these spaces which are particular for dimensions 3 and 4, in comparison with
the generalized (n > 4)-manifolds.
In the second part of the talk we shall present the current state of the main
two problems concerning this class of spaces — the Resolution problem (the
work of Bestvina-Daverman-Venema-Walsh, Bryant-Lacher, Brin-McMillan, Lacher-Repovs,
Thickstun, and others) and the General position problem (the work of Bing, Brahm,
Lambert-Sher, Daverman-Eaton, Lacher-Repovs, Daverman-Thickstun, Daverman-Repovs,
Brahm, and others). We shall list open problems and related conjectures.
Monday, February 24, 2003
| Title |
The Great Impactor |
| Speaker |
Professor Larry Frederick
University of Virginia |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Professor Carol Williams |
| Note |
This colloquium is joint with the Physics Department. |
Abstract
A rather non-descript star, as far as stars go, has proven to be of more than
passing interest. Although first observed in about 1855 it was not studied in
depth until Vyssotsky obtained its spectrum at the McCormick Observatory in
1943 under WWII blackout conditions. He made a quick set of parallax observations
and found that it was a nearby star. A second study was undertaken by Osvalds
and Alden in 1950 and this study is a follow-up on those two. In the meantime,
Vyssotsky convinced other observers to obtain high-dispersion spectra. Those
observations indicated the star was traveling directly at the Solar System.
Assuming an impact radius for the Solar System of two light years, the epoch,
circumstances, and consequences of this impact will be discussed.
Friday, February 21, 2003
| Title |
Noncommutative Geometry: Old and New |
| Speaker |
Professor Masoud Khalkhali
The University of Western Ontario
London, Ontario
CANADA |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Mohamed Elhamdadi |
Abstract
I will give an overview of noncommutative geometry a' la Alain Connes, starting
from very general ideas in mathematics relating algebra and geometry. I will
give, in elementary terms, various examples of things that came to be known
as noncommutative spaces and their corresponding topological invariants like
K-theory and cyclic homology. This talk will be non-technical and elementary
and should be understandable by graduate students.
Wednesday, February 19, 2003
| Title |
Construction of Convolutional Codes |
| Speaker |
Dr. Heide Gleusing-Luerssen
Fachbereich Mathematik
Universitat Oldenburg
GERMANY |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Joseph Liang |
| Note |
Speaker is a candidate for Asst. Prof. in Algebra. |
Abstract
Coding theory is concerned with reliability of data transmission. The two most
important types of error-correcting codes used in engineering practice are block
codes and convolutional codes. While block codes are subspaces of
Fn, where F denotes a finite field,
convolutional codes can be regarded as direct summands of the module
F[z]n. In both cases, the
error-correcting capability of the code is described by the distance. We will
discuss two methods of constructing convolutional codes, one of them leads to
codes with optimal distance, the other one to the subclass of cyclic
convolutional codes.
Friday, February 14, 2003
| Title |
Enumeration of Random Walks |
| Speaker |
Dr. Alec Mihailovs
Department of Mathematics
Shepherd College
Shepherdstown, WV |
| Time |
4:15-5:15 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Edwin Clark |
| Note |
Speaker is a candidate for Asst. Prof. in Algebra.
|
Abstract
We'll talk about counting of random walks on Cartesian products, bi-products,
symmetric and exterior powers and bi-powers, Schur operations, coverings and
semi-coverings of weighted graphs. This has various combinatorial and representation-theoretical
applications. In particular, we'll discuss the case of nonnegative parts of
weight lattices of semi-simple Lie groups and algebras. This gives formulas
for the decompositions of tensor powers of irreducible representations. Because
the tensor invariants may be parametrized by wave graphs, this also enumerates
the corresponding wave graphs.
Friday, February 14, 2003
| Title |
Tiling Groups and Random Sampling |
| Speaker |
Professor Ivan Rapaport
Universidad de Chile
Santiago, CHILE |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor Natasha Jonoska |
Abstract
We apply tiling groups and height functions to tilings of regions in the plane
by Wang tiles, which are squares of colored boundaries where the colors of shared
edges must match. We define a set of tiles as unambiguous if it contains all
tiles equivalent to the identity in its tiling group. For all but one set of
unambiguous tiles with two colors, we give efficient algorithms that tell whether
a given region with colored boundary is tileable, show how to sample random
tilings, and how to calculate the number of local moves or "flips"
required to transform one tiling into another. We also analyze the lattice structure
of the set of tilings, and study several examples with three and four colors
as well (joint work with Cris Moore and Eric Remila).
Friday, January 17, 2003
| Title |
Malfatti-Steiner Problem |
| Speaker |
Professor Sam Sakmar
Department of Physics |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 118 |
Abstract
It is often said that Morley's theorem is the most beautiful theorem of the
20th century. One can arguably say that Malfatti-Steiner problem is the most
beautiful problem of the 19th century. Even today it has aspects which need to
be understood. Malfatti gave only an analytic solution. It was Steiner's genial
insight that lead to the solution. Actually he did not give the proof of his
conjecture, but indicated the path by which the problem could be solved.
Following his path the problem was finally solved, but the solution is terribly
complicated.
Here we give a relatively “simpler” construction. As will be seen,
even this “simpler” one is very complex. We will dissect the solution
to its basic constituents and with the help of a large number of progressive
slides prove all the points needed to complete the proof, starting with the
simplest one to the final complex construction making sure that no obscure points
are left.
Monday, January 13, 2003
| Title |
Brownian Motion in Matrix Groups |
| Speaker |
Professor Gyula Pap
University of Debrecen
Hungary |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 108 |
| Sponsor |
Professor A. Mukherjea |
Abstract
The entries of a Brownian motion with values in certain matrix groups satisfy
some stochastic differential equation (SDE). In simple cases (as the affine
group, the Heisenberg group, SO(2) or the motion group of the plane), this SDE
can be solved explicitely. For the moment functions, the SDE implies a (deterministic)
ordinary differential equation. Solving it, in some cases one can prove uniqueness
of embedding of measures into a Brownian motion. Applying this SDE approach,
one can also derive explicit formula for the Fourier transform of a Brownian
motion in some groups, which helps to answer certain probabilistic questions.
It is also interesting to classify Brownian motions in a given matrix group
concerning the support and absolute continuity or singularity of their distributions.
Friday, January 10, 2003
| Title |
New General Inequalities and Their Applications |
| Speaker |
Professor Arcadii Grinshpan |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 118 |
| Sponsor |
Professor M. Ismail |
Abstract
We will present inequalities for arbitrary complex vectors and binomial weights.
In addition to the theory of functions and inequalities, the result may be of
interest in such fields as approximation theory, matrix theory, discrete mathematics,
and probability/statistics. New inequalites link the classical Cauchy-Schwarz
inequality (a trivial case) with Euler's integrals (the gamma and beta functions).
Applications include multiparameter combinatorial, exponential, and integral
inequalities as well as a family of positive definite matrices and kernels.