Colloquium: 2002-2003

Fast explicit methods for stiff ODEs September 13
Alexei Medovikov, Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia

Abstract
Most of known explicit Runge-Kutta methods has small stability domains -- their time steps are bounded by a very restrictive CFL condition, and thus standard explicit methods become useless for stiff ODEs. We propose explicit embedded integration schemes with large stability domains. The construction of these methods is achieved in two steps: first, we compute stability polynomials of a given degree with optimal stability domains, i.e., possessing a Chebyshev alternation; second, we realize a corresponding explicit Runge-Kutta method with the help of the theory of Runge-Kutta composition methods.

Spring 2003

Reed Solomon Codes: An Application of Abstract and Linear Algebra January 9
Ivelisse Rubio, University of Puerto Rico at Humacao

Abstract
Error control codes are used to protect digital information from errors that can occur during transmission or storage. Reed-Solomon codes are a class of codes that have been widely used in applications such as satellite transmission and compact disk storage. Their structure as well as encoding and decoding algorithms derived from them can be understood with an elementary knowledge of linear and abstract algebra.

In this talk I will introduce codes in general and present Reed-Solomon codes from a polynomial ideal point of view. We will see how concepts and methods from linear and abstract algebra can be used to describe the structure of these codes and to give encoding and decoding methods. I will also introduce special generating sets for ideals of polynomials in several variables called "Groebner bases" and I will discuss how they can be used to generalize some of the methods used in Reed-Solomon codes to encode and decode multidimensional Reed-Solomon codes. Some of these generalizations are still open problems.

EM travelling Waves and Surface Waves in Photonic Crystal Slabs January 23
Stefanos Venakides, Duke University

Abstract
We analyze EM wave propagation and scattering in photonic crystal slabs, i.e. dielectric structures that are periodic in y and finite in x. They are made with a material of constant dielectric coefficient embedded in a matrix of another dielectric coeffcient. We study scattering for polarized incident exponential fields of form exp(iKx+i(m+k)y) where m is an integer, k is the Bloch wavenumber (real in our analysis), and K is generally complex so that the free dispersion relation K²+(m+k)²=w² holds, w being the complex frequency. Developing a boundary integral/boundary element method for periodic structures, we probe the scattering dispersion relation w=W(k), that holds exactly when the system of linear integral equations that desribes the scattering acquires a nontrivial nullspace. When the frequency w is real, the null functions correspond to self-sustained fields that exist in the absence of any incident field and are travelling wave states. We refer to them as surface waves when most of the field is concentrated at the end of the crystal.

We focus on resonant cases where w has a small negative imaginary part. Although mathematiclly they correspond to unphysical fields that grow exponentially as x tends to infinity, they cause anomalous behavior of the scattering states at nearby frequencies at which narrow regions of large resonant fields, and near-perfect transmission or reflection occur. Both the resonant phenomena and the transmission/ reflection anomalies are absent from scattering states at nearby frequency values when the imaginary part of the resonant frequency is zero.

Communicating Mathematics to the Masses (without mangling it)   January 30
Sara Robinson, Science Writer

Abstract
As a mathematician-turned-mathematics writer, I will draw upon my experience with the New York Times to discuss the perils of effectively and accurately communicating mathematics to the general public and what mathemeticians can do to help.

See samples of New York Times articles written by Sara Robinson

An extension of Hurwitz's zeta function to polyhedral cones in Rⁿ   February 6
Sinai Robins, Temple University

Abstract
We define an extension of Hurwitz's zeta function, where the summation index extends to a polyhedral cone in Rn. We find that it has a functional relation, analogous to the one that Hurwitz found in 1 dimension. The tools are analytic, involving variations of Lipschitz and Poisson summation in higher dimensions.

Cappell-Shaneson's 4-dimensional s-cobordism   February13
Selman Akbulut, Michigan State University

Abstract
`Stein-cork decomposition' and `Lefschetz fibrations' are our main techniques to study smooth structures of 4-manifolds, but in practice every example requires its own bag of tricks. We will demonstrate this on Cappell-Shaneson's celebrated example: In 1987 they had proposed a possible counterexample to 4-dimensional Poincare Conjecture by constructing a possible nonstandard s-cobordism from S³ to itself. We show that this is no counterexample, i.e. the cobordism is the standard product. By construction, this standard cobordism is the 8-fold covering space of a strange s-cobordism H from the quaternionic 3-manifold Q to itself. Potentially H could still be a nonstandard fake s-cobordism. We reduce the trviality of H to a question about the 3-twist spun trefoil knot in S⁴, and also relate this to a question about a Fintushel-Stern knot surgery.

Numerical methods for compressible multimaterial flows   February20
Smadar Karni, University of Michigan

Abstract
Multimaterial flows are of great interest in a wide variety of physical problems, ranging from studying the dynamics and stability of interfaces, through mixing processes, the dynamics of bubbles, to liquid suspensions and bubbly flows. Different types of flows call for different assumptions, and lead to flow models which raise computational issues of different flavor.

Stratified flows dominated by propagating material fronts are often described by one-velocity one-pressure models. Unexpectedly, numerical methods for these models have proved difficult, often producing material fronts contaminated by nonphysical oscillations. In dispersed flows, such as liquid suspensions, tracking individual interfaces is not of interest for the macroscopic flow description. A common practice is to average the equations, yielding models that are inherently nonconservative due to momentum and energy exchange terms between the phases. They require closure relations which are not available from first physical principles, and even when motivated by physical considerations yield controversial results. Most notoriously, assuming a single (equilibrium) pressure for all species, leads to loss of time-hyperbolicity of the governing equations, often referred to as the ill-posedness of the multiphase flow equations.

The talk will discuss the numerical issues raised by multimaterial flow computations, will present strategies in the design of suitable numerical algorithms and a host of numerical results.

An analogue of Abel's theorem   February 27
Herb Clemens, Ohio State University

Abstract
Abel's classical theorem giving the condition that formal sums of points on a Riemann surfaces may be the zero (polar) set of a rational function `almost' has an analogue for sums of curves on a three-dimensional Kaehler manifold. The role of complex line bundles in the modern interpretation of Abel's theorem is played by `quaternionic line bundles.'

The octic equation and beyond   March 13
Scott Crass, CSULB

Abstract
The requirement for solving a polynomial is a means of breaking its symmetry, which in the case of the octic, is that of the symmetric group S₈. Its eight-dimensional linear permutation representation restricts to a six-dimensional projective action. A mapping of complex projective 6-space with this S₈ symmetry can provide the requisite symmetry-breaking tool.

The talk will describe some of the S₈ geometry in CP⁶ as well as a S₈-symmetric rational map with special geometric and dynamical properties. An explicit algorithm that uses this map to solve a general octic will run in real time. A concluding discussion will treat the generality of this approach.

On the distance function to the boundary and the singular set of solutions of some Hamilton Jacobi equations   March 20
Louis Nirenberg, Courant Institute

Clifford Lectures   March 27

Constrained Dynamics, Conservation Laws, and Control   April 3
Dmitry Zenkov, North Carolina State University

Abstract
This talk will overview some modern trends in dynamics and control of constrained systems. The behavior of such systems is often counterintuitive. For example, in the absence of external dissipation such systems conserve energy but nonetheless can exhibit asymptotically stable relative equilibria. Another interesting behavior which does not occur in unconstrained systems is that symmetries do not always lead to momentum conservation laws as in the classical Noether theorem. Instead, the momentum satisfies a dynamic momentum equation. Conditions for conservation of some of the components of the momentum will be discussed. Usability of both the presence and lack of conservation laws in control will then be covered.

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Mathematics Department Tulane University 6823 St. Charles Ave New Orleans, LA 70118 phone: (504) 865-5727 fax: (504) 865-5063

Last Updated: April 15, 2005