Mathematics Department Colloquium (2003-2004)

Speaker: Alexander Pankov (Department of Mathematics, William and Mary) 

Title: Infinite chains of oscillators 
Abstract: Infinite one dimensional chains of oscillators are the simplest mechanical systems with infinite number of degrees of freedom. On the other hand, they model many physical phenomena, like oscillations of atomic chains, harmonic and anharmonic crystals, dislocations (Frenkel-Kontorova model), wave transmission in optical systems, etc. In this talk we consider linear and nonlinear chains of oscillators and discuss some rigorous mathematical results and a number of open problems. 
 

Speaker: Qing Xiang (University of Delaware) 

Title: The invariant factors of the incidence matrices of points and linear subspaces in $PG(n,q)$ and $AG(n,q)$ 
Abstract: Let $V$ be an $(n+1)$-dimensional vector space over ${\rm GF}(q)$, where $q=p^t$, $p$ is a prime. Let $A_{1,r}^n(q)$ be the (0,1)-incidence matrix with rows and columns respectively indexed by the $r$- and $1$-dimensional subspaces of $V$, and with $(X,Y)$-entry equal to one if and only if the 1-dimensional subspace $Y$ is contained in the $r$-dimensional subspace $X$. The $p$-rank of $A_{1,r}^n(q)$ was computed by Smith and Hamada in late 1960s. In this talk, we explain how to determine the Smith normal form of $A_{1,r}^n(q)$. The techniques we used are from number theory ($p$-adic estimates of multiplicative character sums, Jacobi sums) and group representation theory (permutation representations of general linear groups on projective spaces). (This talk is based on joint work with David B. Chandler and Peter Sin) 
 

Speaker: Professor Zhitao Zhang (Institute of Mathematics, Chinese Academy of Sciences, Beijing, China) 

Title: Cone theory and critical point theory 
Abstract: In this talk, we introduce the main methods of nonlinear functional analysis, especially the partial order methods and variational methods. Moreover, we use cone theory to critical point theory, and give some applications to elliptic boundary problems with jumping nonlinearities. 
 

Speaker: Professor Joachim Rosenthal (Notre Dame University) 

Title: Pole placement problems, inverse eigenvalue problems and quantum Schubert calculus 
Abstract: Several prominent problems originating in linear algebra and in control theory are through their nature actually problems in algebraic geometry. Examples include a large class of so called matrix extension problems studied in the linear algebra literature as well as most pole placement problems studied in the linear systems theory literature. In this talk we will explain those different links and we will report on several recent results obtained by geometric techniques. We will show how many of those problems have an interpretation in terms of Schubert calculus and that some of the more prominent problems require computations in the quantum ring of the Grassmannian. The presented results represent joint work with Bill Helton, Meeyoung Kim, M.S. Ravi, Frank Sottile and Alex Wang. 
 

Speaker: Wing Suet Li (Georgia Tech.) 

Title: Inequalities for eigenvalues of sums in a von Neumann algebra 
Abstract: Consider two self-adjoint operators A and B on a finite-dimensional Hilbert space H. Let $\{\lda_j(A)\}$, $\{\lda_j(B)\}$, and $\{\lda_j(A+B)\}$ be sequences of eigenvalues of A,B, and A+B counting multiplicity, arranged in decreasing order. In 1912, while working on problems in PDE, H. Weyl raised the following question: what are all the inequalities that $\{\lda_j(A)\}$, $\{\lda_j(B)\}$, and $\{\lda_j(A+B)\}$ must satisfy? This problem was only solved completely by A. Klyachko in 1996. Since Klyachko's break through, a great deal of activities have been generated, led by Buch, Fulton, Klyachko, Knotson, Tao, Woodward, and others. (Two excellent articles on the subject appeared in the AMS Bulletin and the AMS Notices: W. Fulton, Eigenvalues, invariant factors, hightest weights, and Schuber calculus, AMS Bulletin 37 (2000), pp. 209-249, and A. Knutson and T. Tao, Honeycombs and Sums of Hermitian Matrices, AMS Notices 48 (2001), pp. 175 - 186.) I will talk about analogus of Weyl's question for self-adjoint elements in von Neumann algebras with finite traces. This has the potential to lead to some new theories on symmetric functions, infinite dimensional representation theory, and infinite dimensional algebraic geometry. 
 

Speaker: Xuefeng Wang (Tulane University) 

Title: Stability and metastability of patterns in a nonlocal model for phase transitions 
Abstract: When a pattern persists for a long time but eventually disappears, it is said to be metastable; when it never disappears, it is said to be stable. Of concern is the non-local Allen-Cahn PDE, where the usual Laplacian is replaced by a convolution operator. It could be used as a model for long-range interactions between different states in material. As in the case of Allen-Cahn, a typical initial value quickly evolves into a spatial pattern; on the other hand, the diffusion effect of the convolution operator is not as strong as that of Laplacian. This sometimes leads to the stability of patterns, which does not occur in Allen-Cahn. We also show that the life span of patterns depends on the decay rate of the kernel of the convolution. 
 

Speaker: Pavel Bleher (Indiana University-Purdue University Indianapolis) 

Title: Random matrix models with external source and multiple orthogonal polynomials 
Abstract:  We consider the unitary random matrix ensemble with an external source A, which is a matrix with only two eigenvalues, a and -a, with equal multiplicities. The correlation functions of eigenvalues in this ensemble have a determinantal form, and we express the correspondingreproducing kernel in terms of multiple orthogonal polynomials. We derive a Christoffel-Darboux type formula for the reproducing kernel and we show that it can be also expressed in terms of a solution to a 3x3 matrix Riemann-Hilbert problem. Our main result consists in an asymptotic analysis of the Riemann-Hilbert problem for the Gaussian ensemble of random matrices with the external source A (the Brezin-Hikami model). The model exhibits a critical point, a=1, and we develop an asymptotic analysis of the model in the cases a>1, a<1. The asymptotic analysis of the Riemann-Hilbert problem is based on the steepest descent method of Deift and Zhou. We prove that the local eigenvalue correlations are expressed in terms of the sine kernel in the bulk and in terms of the Airy kernel at the edge of the spectrum. 
 

Speaker: Alexander Koldobsky (University of Missouri, Columbia) 

Title: Methods of Fourier analysis in convex geometry. 
Abstract: A new Fourier analytic approach to sections and projections of convex bodies has recently been developed. The idea is to express different parameters of a convex body in terms of the Fourier transform and then use methods of harmonic analysis to solve geometric problems. This approach has led to several results including a complete analytic solution to the Busemann-Petty problem asking whether symmetric convex bodies with smaller areas of central hyperplane sections necessarily have smaller volume. In the talk we present the main ideas and several applications of this approach. 
 

Speaker: Vladimir Rabinovich (Instituto Politecnico Nacional, Mexico) 

Title: Essential spectrum of perturbed pseudodifferential operators 
Abstract: The aim of the talk is to present a new approach to the study of essential spectrum of the Schrodinger, Klein-Gordon and Dirac operators. We include these operators in a class of pseudodifferential operators perturbed by non-smooth potentials. For an operator under consideration we introduce a family of limit operators, and prove that the essential spectrum of the original operator is the union of the spectra of limit operators. Because the limit operators have, as a rule, simpler structure than the original operator, this approach provides a powerful tool for investigation of essential spectrum of differential and pseudodifferential operators. 
 

October 20, Monday,  4:00pm in Millington 117 

Speaker: Lisa M. McShane (Biometric Research Branch, National Cancer Institute) 

Title: Controlling the Number of False Discoveries: Application to High-Dimensional Genomic Data 

Abstract: Researchers conducting gene expression microarray experiments often are 
interested in identifying genes that are differentially expressed between 
two groups of specimens.  A straightforward approach to the identification 
of such "differentially expressed" genes is to perform a univariate 
analysis of group mean differences for each gene, and then identify those 
genes that are most statistically significant.  However, with the large 
number of genes typically represented on a microarray, using nominal 
significance levels (unadjusted for the multiple comparisons) will lead to 
the identification of many genes that truly are not differentially 
expressed, "false discoveries."  A reasonable strategy in many situations 
is to allow a small number of false discoveries, or a small proportion of 
the identified genes to be false discoveries.  Although previous work has 
considered control for the expected proportion of false discoveries 
(commonly known as the false discovery rate), we show that these methods 
may be inadequate.  We propose two stepwise permutation-based procedures 
to control with specified confidence the actual number of false 
discoveries and approximately the actual proportion of false discoveries. 
Limited simulation studies demonstrate substantial gain in sensitivity to 
detect truly differentially expressed genes even when allowing as few as 
one or two false discoveries.  We apply these new methods to analyze a 
microarray dataset consisting of measurements on approximately 9000 genes 
in paired tumor specimens, collected both before and after chemotherapy on 
20 breast cancer patients. The methods described are broadly applicable to 
the problem of identifying which variables of any large set of measured 
variables differ between pre-specified groups. 

This is joint work with Edward L. Korn, James F. Troendle, and Richard 
Simon. 

Speaker: Peter Braxton and Sarah Grinnell (Northrop Grumman Information Technology, TASC) 

Title: Cost and Risk Analysis Applications: Cool Math in the Real World 

Abstract: Cost and risk analysis provides developers of complex systems the tools for making decisions about the allocation of scarce resources under uncertainty.  College alumni return for an annual presentation on some of the latest applications of probability and statistics, simulation, and other analytical techniques in this growing field.  Vignettes show young analysts doing innovative and important work in a business environment.  Of interest to students of mathematics, operations research, economics, and finance. 

Speaker: Leslie Hogben, Department of Mathematics, Iowa State University 

Title: Matrix Completion Problems for Various Classes of P-Matrices
 

Abstract:A P-matrix is a real square matrix all of whose principal minors are positive. The positive definite matrices can be described as symmetric P-matrices. A partial matrix is a square array in which some entries are specified and the remaining entries are free to be chosen. A partial matrix is a partial P-matrix if every fully specified principal submatrix is a P-matrix. A completion of a partial matrix is a choice of values for the specified entries, resulting in a conventional matrix. A matrix completion problem for the class of P-matrices asks: Does a partial P-matrix have a completion to an P-matrix? The analogous question can be asked for subclasses of P-matrices, such as positive P-matrices. For partial matrices with a given pattern of specified entries this problem can be studied by graphs and digraphs. This talk will provide an introduction to current techniques for matrix completion problems and present recent advances in the subject. 
 

Speaker: Martine Reurings (Collge of William and Mary, formerly Vrije Universiteit Amsterdam) 

Title: A nonlinear matrix equation connected to interpolation theory 

Speaker: Alexander Pankov (Department of Mathematics, College of William and Mary) 

Title: Mathematical problems of photonic crystals 
 

Abstract: A photonic crystal is an optical medium of (close to) spatially periodic 
structure. In this talk we present basic mathematical results that 
concern linear theory of photonic cristals. We also discuss a recent 
result from nonlinear theory which shows the existence of gap solitons in 
certain two dimentional photonic crystals. 

Speaker: Robert Reams (Department of Mathematics, College of William and Mary) 

Title: Semidefiniteness without Hermiticity 

Speaker: Kevin A. Mitchell, Department of Physics, The College of William and Mary 

Title: Chaos-Induced Pulse Trains 
 

Abstract: The ionization of hydrogen in external fields is a simple example of chaotic escape and a convenient testing ground for classical and semi-classical methods in open dynamical systems.  We analyze this process using the general theory of phase space transport and "homoclinic tangles".  We find that ionization occurs as a train of discrete pulses, rather than a decaying exponential.  These pulses result from fractal structures inherent to the classical dynamics, which exhibit what we call "epistrophic self-similarity": repeated self-similar structure as well as variable structure at all levels of resolution.  We explain how the train of ejected pulses carries the imprint of this fractal structure.  A key aspect of our analysis is a new mathematical approach to describing homoclinic tangles, called "homotopic lobe dynamics". 

Speaker: Valerie Girardin (Department of Mathematics, University of Caen, France) 

Title: Entropy and Markov processes: from Boltzman and Shannon to the latest developments 
Abstract: The concept of entropy is the basis of information theory. It has been introduced in the field of probability first by Boltzman in statistical mechanics and then by Shannon for studying communication systems. Among a given family of processes, selecting the process with the maximum entropy is equivalent to adding the least of information possible to the considered problem or to choosing the process which can be realized in the most numerous number of ways. Entropy and Markov processes are linked since the first version of the asymptotic equirepartition property stated by Shannon in 1948 for ergodic discrete time Markov processes. Many extensions for different classes of processes have been made ever since, up to the definition of entropy for semi-Markov processes last year. 

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