SCU Mathematics Colloquium Series Schedule

Winter 2004

Unless noted otherwise, talks will be at 4:00pm Tuesdays. Room TBA. There will be refreshments before all talks in O'Connor 31 starting around 3:45pm.

**Jan. 6th**John Towers, Mira Costa CollegeTitle: Entropy and Uniqueness for Conservation Laws with Discontinuous Coefficients

Conservation laws having spacially dependent coefficients that are discontinuous arise in several areas of science and industry. The discontinuities in the coefficients generally cause the solutions to be discontinuous, and there may be more than one unique solution. I will discuss additional conditions, so called entropy conditions, that guarantee uniqueness. It has recently become apparent that there is more than one reasonable way to prescribe these entrophy conditions. I will try to explain some of the differences between the two different entropy theories.

**Jan. 13th**Tom Banchoff, Brown UniversityTitle: Characteristic Classes in Multivariable Calculus Classes

Some of the most appealing theorems in differential topolgy already show up in the first course in multivariable calculus, although often in disguised forms. We will visit the parity lemma, the winding number, the Morse critical point theorem, the non-embeddability of the Klein bottle, the ham sandwich theorem, and the Whitney duality theorem as part of an Internet based "paperless" two-variable calculus course.

**(Thursday) Jan. 29th**Thomas A. Henzinger, University of California, BerkeleyTitle: The Symbolic Approach to Hybrid Systems

A hybrid system is a dynamical system whose state has both a discrete component, which is updated in a sequence of steps, and a continuous component, which evolves over time. Hybrid systems are a usefull modeling tool in a variety of situations, including control, robotics, curcuits, biology, and finance. We survey a computational approach to the design and analysis of hybrid systems which is based on the symbolic discretization of continuous state changes. On the theoretical side, we classify infinite, hybrid state spaces as to which finite, discrete abstractions they admit. This classification enables us to apply concepts and results from concurrency theory

**Feb. 3rd**Ed Schaefer, Santa Clara UniversityTitle: Rational Periodic Points of Period 5 with Respect to a Rational Quadratic Polynomial

Plug 5/4 into the polynomial f = x^2 - 29/16 and you get -1/4. Plug -1/4 in and you get -7/4. Plug -7/4 in and you get 5/4 again. So we say that 5/4 is a rational periodic point with respect to the polynomial x^2 - 29/16 with period 3. For periods n = 1, 2 and 3, there are infinitely many pairs of a rational quadratic polynomial and a rational periodic point of period n. In 1996, Morton showed that there are no such pairs for period (exactly) 4 using a result on modular curves. The period 5 case can not be resolved using modular curves and required new techniques developed by Victor Flynn, Bjorn Poonen and the speaker. We will present these techniques in the talk. The period 6 case is still unresolved.

**Feb. 10th**Andrew J. Bernoff, Harvey Mudd CollegeTitle: Stability and Dynamics of Self-similarity in Evolution Equations

Similarity methods have been used to derive special solutions for a broad variety of physical problems in the past few decades. In this talk I will discuss a methodology for studying linear stability for self-similar blow-up and pinch-off. I will present three problems: a simple ODE model, the Sivashinsky equation which arises in solidification, and the pinch-off of a solid filament due to the action of surface diffusion. The goal is to show that self-similar phenomena can be studied using many of the now familiar ideas that have arisen in the study of dynamical systems. In particular, I will discuss rescaling methods, linearization and the role of symmetries in the context of self-similarity. I will demonstrate that the symmetries in the problem give rise to "anomalous" positive eigenvalues associated with the rescaling symmetries as opposed to instability, and show how this stability analysis can identify a unique stable (and observable) solution from a countable infinity of similarity solutions.

**Feb. 17th**Brian Granger, Santa Clara UniversityTitle: Nonseperable Partial Differential Equations in Atomic and Molecular Physics

In physics, differential equations are the primary tool used to encode physical systems into a mathematical language. In atomic and molecular physics one usually knows the differential equation without any ambiguity. However, these differential equations often have high dimensionality and are not seperable, making them extremely difficult to solve exactly. Inthis talk I will describe a number of experimentally accessible systems from atomic and molecular physics whose differential equations have these properties.

**Mar. 9th**Esmond Ng, Berkeley National LabTitle: Scientific Computing in the Berkeley Lab

Scientific computing and computational sciences are becoming the third pillar of science, complementing the traditional theoretical and experimental sciences. In this talk, we will discuss the infrastructure that is needed to support scientific computing research and describe some of the computational science activities at Berkeley Laboratory.

**Mar. 16th**Gene Golub, Stanford UniversityTitle: Matrices, Moments and Quadrature

Given an $n\times n$ symmetric, positive definite matrix $A$ and a real vector $u$, it is desirable to estimate and bound the quadratic form $\frac{u' F(A) u}{u'u}$ where $F$ is a differentiable function. This problem arises in estimating errors of linear systems, computing a parameter in a least squares problem with a quadratic constraint and bounding elements of the inverse of a matrix. Some of these estimates are useful in cross validation and statistical computations; we shall illustrate their use. We describe a method based on the theory of moments and numerical quadrature for estimating the quadratic form. A basic tool is the Lanczos algorithm which can be used for computing the recursive relationship for orthogonal polynomials. Generalizations to other forms will be described.

**Mar. 30th**Jesus De Loera, UC DavisTitle: Counting Lattice Points in Polyhedra: Algorithms and Applications

A wide variety of problems in pure and applied mathematics involve the problem of counting the number of lattice points inside a region in space. Applications range from the very pure (Number theory, Multiplicities and Kostant's partition function in representation theory, Ehrhart polynomials in combinatorics) to the very applied (cryptography, integer programming, statistical contingency tables).

If you have a disability and require a reasonable accommodation, please call Aaron Diaz aaron@turing.scu.edu 1-408-554-6811 or 1-800-735-2929 (TTY - California Relay).

[SCU Homepage | Mathematics/Computer Science Homepage] *Last Updated: 7 March 2005*Maintainer: webmaster@math.scu.edu