Mathematics Department Colloquium (2005-2006)

Speaker: Vaidyanathan Ramaswami, AT&T Labs 

Title: Ensuring emergency service access in the presence of congestion induced by internet dial up calls 
Abstract: In this talk, we demonstrate how we used probabilistic models to identify the root cause of circuit congestion in certain parts of the AT&T network that resulted in unavailability of dial tone. That identification required significant modeling and analysis based on large amounts of data and was the result of many counter-intuitive insights derived from mathematical models. The work not only addressed a very serious problem that could be life threatening in certain cases, but also resulted in a set of five patent filings (three granted already) and a saving of about $10 million per year to AT&T on an ongoing basis. This effort which serves as an example of the power of applied mathematics and operations research techniques in solving important real world problems was selected as a finalist for the prestigious Wagner Prize of INFORMS in the year 2004.

April 14, Friday,  3pm in  Jones 131 

Speaker: Alexander Pankov, College of William and Mary 

Title: Stability of factorization of matrix polynomials: a glance from the outside 
Abstract: In this talk we discuss applications of elementary multidimensional complex analysis and algebraic geometry to the stability problem for factorization of matrix poynomials.

March 31, Friday,  3pm in  Jones 131 

Speaker: Jiabao Su, Capital Normal University, Beijing, China (currently visiting Utah State University) 

Title: Solutions to Semilinear Elliptic Equations via Morse Theorys 
Abstract: This talk will concern with the main idea of applying infinite dimensional Morse Theory in finding solutions to semilinear elliptic boundary value problems.

March 27, Monday,  3pm in  Jones 131 

Speaker: Tin-Yau Tam, Auburn University 

Title: Yamamoto's Theorem on the eigenvalues and singular values of a matrix 
Abstract: The result of Yamamoto asserts that limits of m-th roots of the s-numbers of the matrix X^m are the eigenvalues of the matrix X, arranged non-increasingly.We discuss an extension in the context of real semisimple Lie group. It is a joint work with Huajun Huang.

March 24, Friday,  3pm in  Jones 131 

Speaker: Thomas Wanner, George Mason University 

Title: Computational Homology and the Evolution of Complex Patterns 
Abstract: Many partial differential equation models arising in applications generate complex patterns evolving with time which are hard to quantify due to the lack of any underlying regular structure. Such models often include some element of stochasticity which leads to variations in the detail structure of the patterns and forces one to concentrate on rougher common geometric features. From a mathematical point of view, algebraic topology suggests itself as a natural quantification tool. In this talk I will present some recent results for both the deterministic and the stochastic Cahn-Hilliard equation, both of which describe phase separation in alloys. In this situation one is interested in the geometry of time-varying sub-level sets of a function. I will present theoretical results on the pattern formation and dynamics, show how computational homology can be used to quantify the geometry of the patterns, and will assess the accuracy of the homology computations using probabilistic methods.

Talks in Fall 2005
Munchies are served 15 minutes beforehand in Jones 131 

Speaker: Alexander Pankov, the College of William and Mary 

Title: Gap solitons in periodic discrete nonlinear Schroedinger equation 
Abstract: In the past decade localized solutions of the discrete nonlinear Schroedinger equation (DNLS) become a topic of intense research. Much of this work concerns the standard constant coefficient DNLS. Certainly, the DNLS with periodic coefficients is not less important because it is related to many problems with spatial inhomogeneities. In the physics literature the simplest case of period 2 has been considered recently. While the spectrum of constant coefficient stationary discrete Schroedinger operator consists of a single closed interval, in the spectrum of periodic operator finite gaps typically open up. The corresponding DNLS may have spatially localized standing wave solutions with carrier frequency in such a gap (gap solitons). In this talk we discuss rigorous results on existence of gap solitons.

November 18, Friday,  3pm in  Jones 131 

Speaker: John L Spouge, the National Center for Biotechnology Information (NCBI) 

Title: Importance sampling of marginal distributions and gapped alignment of random sequences 
Abstract: Sequence databases are indispensable in modern molecular biology. As an example, biologists use the BLAST program at NCBI over the web more than once every second. In its essence, the program gives an alignment score, which evaluates how closely a query sequence can be aligned with each database sequence. Biologists then infer the function of their query sequence by looking at high scoring database sequences with known functions. Historically, BLAST distinguished itself as the first database retrieval program to assign p-values to its retrieval scores. The corresponding statistical distributions must be computed off-line, however, restricting the scoring parameters available to BLAST users. Accordingly, a lot of recent effort has been aimed at on-line computation of BLAST p-values. No methods are known that compute on-line to the accuracy the BLAST program requires. I present my group's work, which over the past two years has reduced the simulation times from about two days to about two seconds. After a gentle overview of the biological relevance of the problem, the talk focuses on our two main mathematical tools: loose analogies between gapped sequence alignment and Markov additive processes, and a theorem about marginal distributions in importance sampling.

November 11, Friday,  3pm in  Jones 131 

Speaker: Sebastian Schreiber 

Title: The evolution of dispersal in patchy environments 
Abstract: Plants and animals often live in landscapes where environmental conditions vary from patch to patch. Since the fecundity and survivorship of an individual depends on these conditions, an organism may decrease or increase its fitness by dispersing across the environment. To better understand the evolution of dispersal, one can analyze difference equation models of competing populations living in a landscape consisting of k patches. For competing asexual populations whose likelihoods of dispersing is patch independent, Steve Kirkland, Chi-Kwong Li, and I have proven that the slower dispersing population displaces the faster dispersing population. Since this prediction relies on the absence of environmental fluctuations, genetics, species interactions, and stage structure, I will discuss how including these factors can significantly alter this prediction. For instance, asynchronous population dynamics promotes faster dispersers displacing slower dispersers, while species interactions can promote coexistence of faster and slower dispersers. To fully understand how these factors influence the evolution of dispersal requires conquering several challenging problems in matrix theory and dynamical systems.

November 4, Friday,  3pm in  Jones 131 

Speaker: Shulim Kaliman, Department of Mathematics, University of Miami, Coral Gables 

Title: Actions of C-star and C-plus on affine algebraic varieties 
Abstract: We study the problem of equivalence of algebraic actions of a group G on an affine algebraic variety X (two such actions on X are equivalent if they are conjugate in the group of automorphisms of X). As an example of such a problem one can ask when a G-action on C^n is linear in a suitable polynomial coordinate system on C^n. There were several recent breakthroughs in this area, including a proof of the Linearization Conjecture by Koras and Russell. This conjecture states that any algebraic C-star action on C^3 is linear after conjugation. We discuss such results and their relation to the study of the so called exotic algebraic structures on C^n (i.e., affine algebraic varieties diffeomorphic to R^2n but not isomorphic to C^n). We assume no prior knowledge of the subject.

October 21, Friday,  3pm in  Jones 131 

Speaker: Maribel Bueno, the College of William and Mary 

Title: A forward stable algorithm for computing Jacobi matrices.
Abstract: Algorithms for computing Jacobi matrices are very important in different areas of mathematics and mathematical physics. A monic Jacobi matrix is a tridiagonal matrix which contains the parameters of the three-term recurrence relation satisfied by the sequence of monic polynomials orthogonal with respect to a measure. Christoffel transformation with shift $\alpha$ transforms the monic Jacobi matrix associated with a measure $d\mu$ into the monic Jacobi matrix associated with $(x-\alpha)d\mu$. However the existing algorithms to compute Christoffel transformation with shift $\alpha$ are not stable. We propose a new algorithm and prove its forward stability. This means that the obtained forward errors are of similar magnitude to those produced by a backward stable algorithm. We provide a condition number that allows us to estimate forward errors in O(n) flops. Moreover, we show that the new algorithm yields smaller forward errors than the previous ones.

October 14, Friday,  3pm in  Jones 131 

Speaker: Nicholas Loer, the College of William and Mary 

Title: An Introduction to q,t-Parking Functions.
Abstract: This talk discusses the combinatorial properties of parking functions and their generalizations (the so-called "q,t-parking functions"). Parking functions were originally introduced by Konheim and Weiss in their study of hashing protocols. Generalized parking functions arose in an ongoing study of diagonal harmonics modules, symmetric functions, and Macdonald polynomials by Mark Haiman, Jim Haglund, the speaker, et al. This talk will focus almost entirely on the combinatorial aspects of this investigation. Thus the talk will be totally elementary and accessible to a wide general audience; no particular knowledge of algebraic combinatorics or representation theory will be assumed.

October 6, Thursday,  5pm in  Jones 131 

Speaker: Mark Tomforde, the College of William and Mary 

Title: C*-algebras of directed graphs
Abstract: C*-algebras are important objects in Functional Analysis that have numerous applications. In the past 7 years, a method for constructing a C*-algebra from a directed graph has been described. In this talk, I will explain this method and also discuss how the structure of a C*-algebra created in this way is encoded in the directed graph. Because of this, many complicated questions about the C*-algebra can be translated into (often easier to answer) questions about the graph. This talk will be accessible to a wide audience.

September 30, Friday,  3pm in  Jones 131 

Speaker: Robert Michael Lewis, the College of William and Mary 

Title: Direct search algorithms for constrained optimization
Abstract: We will discuss recent progress in direct search methods for constrained optimization. Direct search methods are algorithms that do not require either the exact or approximate derivatives of the objective and constraints. Nevertheless, as we discuss, for a particular class of these algorithms---generating set search (GSS)---it is possible to prove convergence results comparable to those for derivative--based algorithms. We discuss GSS algorithms for bound constrained, general linearly constrained, and nonlinearly constrained problems. We show that the key to understanding these algorithms is the stationarity properties of a particular subsequence of the iterates generated by the algorithm.

September 28, Wednesday  4pm in  Small 113 

Speaker: Jinchuan Hou, 2005 W&M Freeman Visiting Fellow, President, Shanxi Teachers' University 

Title: Changes of Higher Education in China 
Abstract: The higher education of China has changed dramatically in the last decade. We will discuss some aspects of the changes such as the movement of merging of higher education institutes, rapid increase of enrollments, reformation of higher education, degree and postgraduate education, and challenges faced by Chinese educators.

September 23, Friday,  3pm in  Jones 131 

Speaker: Professor Junping Shi, the College of William and Mary 

Title:Bistablility in chemical reaction and predator-prey systems
Abstract: Bistability describes the phenomenon of multiple attracting regions in a dynamical system, and it has been observed in a wide range of natural phenomena. I will introduce my recent mathematical work on bistability in reaction-diffusion systems. First we consider an autocatalytic chemical reaction. If the spatial domain has dimension higher than 2 and the "order" of the reaction is high enough, then it is known that the system has a family of non-trivial steady states. We prove that each of these steady states is a "hair-trigger" for two types of long time behavior: if the initial value is below the steady state, then the solution of the system converges to a rest state of the system as time goes to infinity and so extinction occurs; if the initial value is above the steady state, then a wave front is developed and so we have the spread of "flame". This a joint work with Xuefeng Wang of Tulane University. Secondly I will consider a diffusive prepador-prey system of ecology. Existence of multiple positive steady states and global bifurcation branch are examined as well as related dynamical behavior. It is found that while the predator population is not far from a constant level, the prey population could extinguish, persist or blow up depending on the initial population distributions, the various parameters in the system, and the heterogeneous environment. In particular, we examine a situation where the Allee effect is caused by the spatial heterogeneity of the environment. If time allows, a diffusive predator-prey system with a protection zone for prey will also be discussed. This is a joint work with Yihong Du of University of New England, Australia.

August 29, Monday,  3pm in  Jones 131 

Speaker: Professor Jan van Mill (Vrije Universiteit, Amsterdam) 

Title: Polishable ideals and complete Erdös space 
Abstract: In solution to a problem posed by L. G. Oversteegen we present a simple and useful topological characterization of complete Erdos space E_c. Here E_c is the set of all vectors in Hilbert space all whose coordinates are irrational. One of the applications states that whenever I is a Polishable F_sigma-ideal on omega then I with the Polish topology is homeomorphic to either omega, 2^omega, omega x 2^omega, or E_c. This result answers a question that was posed by S. Todorcevic.