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Mathematics & Statistics

# Colloquia — Spring 2003

## Friday, May 2, 2003

Title
Speaker
Time
Place
Note

A Statistical Method for Identifying Informative Genes in Microarrays
James Yang
2:00pm-3:00pm
PHY 109
A. N. V. Rao
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
Speaker

Time
Place
Note

Virus Evolution: Micro-scale Epidemic Simulations
Prasith Baccam
Los Alamos National Laboratory
2:00pm-3:00pm
PHY 120
A. N. V. Rao
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
Speaker

Time
Place

On Coarse Approach to the Novikov Conjecture
Alexander Dranishnikov
University of Florida
3:00pm-4:00pm
PHY 118
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
Speaker

Time
Place
Note

A Bayesian Approach for Estimating Antiviral Efficacy in HIV Dynamic Models
Yangxin Huang
Harvard School of Public Health
2:00pm-3:00pm
CHE 201
A. N. V. Rao
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
Speaker

Time
Place

Pseudo-Orthogonality, Maximal Orthogonality and Generalized Inverses
Michael Rieck
Drake University
3:00pm-4:00pm
PHY 118
W. 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
Speaker

Time
Place

Introduction to Solitons
David Kaup
University of Central Florida
3:00pm-4:00pm
PHY 118
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
Speaker

Time
Place
Note

Viscosity Solutions and Absolute Minimizers on the Heisenberg Group
Thomas Bieske
University of Michigan
4:00pm-5:00pm
PHY 118
Yuncheng You
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
Speaker

Time
Place

Homogenized and Concentrated Limits in Visual Transduction
Emmanuele DiBenedetto
Vanderbilt University
3:00pm-4:00pm
TBA
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
Speaker

Time
Place
Note

Tensor Products and $$p$$-Summing Operators With Hilbertian Domain
Qingying Bu
Mississippi State University
4:00pm-5:00pm
PHY 118
Boris Shekhtman
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 $$L^p[0,1]$$ 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 $$L^p[0,1]$$ 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.

Title
Speaker

Time
Place

On the New Primality Test
Michael Pohst
Tech University of Berlin
3:00pm-4:00pm
PHY 118
Joseph Liang

## Monday, March 17, 2003

Title
Speaker

Time
Place
Note

On the Distribution of the Zeros of an Entire Function of an Exponential Type
Dimiter Dryanov
University of Montreal
4:00pm-5:00pm
PHY 118
Yuncheng You
Speaker is a candidate for Asst. Prof. in Analysis.

Abstract

An entire function $$f$$ is said to be of exponential type $$\sigma > 0$$ if for every $$\epsilon > 0$$, there exists a constant $$c(\epsilon)$$ 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 $$\sigma > 0$$ 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 $$\sin\sigma z$$, then $$|f(x)|\le M| \sin\sigma x|$$ 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 $$\gamma$$.

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

Speaker

Time
Place
Note

How to Prove That a Projection is Unique Minimal, or Why the Smoothness, Trace Duality and Chalmers-Metcalf Operator Come in Handy
Lesław Skrzypek
Jagiellonian University
Poland
3:00pm-4:00pm
PHY 118
Boris Shekhtman
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 than $$1$$ 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 $$L_1$$ 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
Speaker

Time
Place
Note

Classical and Relativistic Compressible Flows
Ronghua Pan
University of Michigan
4:00pm-5:00pm
PHY 118
Yuncheng You
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.

Title
Speaker

Time
Place

Impulsive Stochastic Evolution Inclusions on Infinite Dimensional Spaces and Their Control
N. U. Ahmed
University of Ottawa
3:00pm-4:00pm
PHY 118
Yuncheng You

Abstract

We consider a general class of stochastic evolution inclusions on infinite dimensional spaces.

Here $$A$$ is the infinitesimal generator of a $$C_0$$-semigroup of bounded linear operators in a Hilbert space, $$\beta\in BV^{\operatorname{loc}}(R)$$, $$F$$ a single-valued nonlinear operator and $$\mu$$ 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 (Impulsive perturbation of $$C_0$$-semigroups and stochastic evolution inclusions, Discuss. Math. Differ. Incl. Control 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
Speaker

Time
Place
Note

Convex Bodies and the Fourier Transform
Artem Zvavitch
University of Missouri
4:00pm-5:00pm
PHY 118
Boris Shekhtman
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 $$l_p$$-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
Speaker

Time
Place
Note

Combinatorial Group Theory and Discrete Tiling Problems
Michael Reid
University of Arizona
4:15pm-5:15pm
PHY 118
W. Edwin Clark
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
Speaker

Time
Place

General Position Properties Which Characterize Low-Dimensional Manifolds
Dusan Repovs
University of Ljubljana
Slovenia
3:00pm-4:00pm
PHY 118
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
Speaker

Time
Place
Note

The Great Impactor
Larry Frederick
University of Virginia
3:00pm-4:00pm
PHY 130
Carol Williams
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
Speaker

Time
Place

Noncommutative Geometry: Old and New
Masoud Khalkhali
University of Western Ontario
London, Ontario
3:00pm-4:00pm
PHY 118

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
Speaker

Time
Place
Note

Construction of Convolutional Codes
Fachbereich Mathematik
Universitat Oldenburg
GERMANY
4:00pm-5:00pm
PHY 118
Joseph Liang
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 $$F^n$$, 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
Speaker

Time
Place
Note

Enumeration of Random Walks
Alec Mihailovs
Shepherd College
Shepherdstown, WV
4:15pm-5:15pm
PHY 118
W. Edwin Clark
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.

Title
Speaker

Time
Place

Tiling Groups and Random Sampling
Ivan Rapaport
Santiago, CHILE
3:00pm-4:00pm
PHY 118
Nataša 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
Speaker

Time
Place

Malfatti-Steiner Problem
Sam Sakmar
Department of Physics
3:00pm-4:00pm
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
Speaker

Time
Place

Brownian Motion in Matrix Groups
Gyula Pap
University of Debrecen
Hungary
3:00pm-4:00pm
PHY 108
Arunava 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, $$\mathrm{SO}(2)$$ 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.

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