## Proposed projects for the 2003 Math REU Program

### Project #: Math2003-01

**Partial Differential Equations**

Mentor: Avy Soffer, Department of Mathematics

Co-mentor: Pieter Blue, graduate student

Dynamical wave phenomena requires the understanding of the large time behavior of partial differential equations. Quantum Mechanics is one such example. Optical and laser systems are of similar nature. Such analysis poses a special challenge both to theorists and computational analysis. The study of various aspects of the nonlinear Schrödinger equation by analytic and numerical methods is proposed. The topic and its nature will be determined according to the interests and experience of the student.

Prerequisites: Students should know linear algebra and differential equations. Also very useful are: programming, graphics, quantum mechanics, and perhaps complex analysis, advanced calculus/real analysis.

### Project #: Math2003-02

**Lie theory and wireless antennae networks**Mentors: Chris Woodward and Shawn Robinson, Department of Mathematics

We will look at some possible approaches to the problem of packing in the unitary group via Lie theory, and some applications to wireless communication.

Prerequisites: Good knowledge of linear algebra and programming skills.

### Project #: Math2003-03

**Root systems and spherical functions**Mentor Siddhartha Sahi, Department of Mathematics

Co-mentor: Aaron Lauve, graduate student

The student will get a quick introduction to the combinatorics of root systems and associated polynomials such as characters and spherical functions. These functions have applications to many different areas of mathematics and physics. There are extensive character tables in the literature, but not so for spherical functions. Using a recently discovered formula, we will concentrate our efforts on writing a program to compute spherical functions for various groups. Hopefully the end-result will be a table of such spherical functions in an accessible format - perhaps a CD/DVD that can be used by other researchers.

Prerequisites: The student should have good mathematical maturity and programming skills.

### Project #: Math2003-04

**Spectral Theory of Random Matrices and Applications**Mentor: Michael Kiessling, Department of Mathematics

Random matrices were originally invented in statistics (who is surprised?), but eventually showed up in many other parts of mathematics and (mathematical) physics, in particular number theory, completely integrable systems, quantum mechanics, statistical mechanics.

In a nutshell, if the matrix is large, its spectral properties not only become deterministic through a law of large numbers (LLN), the LLN seems to belong to one of a handful of so-called universality classes. Thus many seemingly disjoint problems in science and mathematics appear to be linked. This project will introduce the student to this intriguing subject. After a review of the basic concepts and early main results, we will turn to some of the open questions that are in the focus of interest of current research.

Prerequisites: Good basic knowledge in mathematics and physics and a good deal of curiosity.

### Project #: Math2003-05

**Quantum behavior of time dependent systems; ionization problems**Mentor: Ovidiu Costin, Department of Mathematics

Co-mentor: Rodica Costin, Department of Mathematics

The students will work on describing the long time behavior of a quantum particle initially in a bound state and subject to an external quasiperiodic forcing with amplitude which is not necessarily small. This type of problems has applications to the question of ionization of atoms in microwave or laser fields of relatively large amplitude, a problem which cannot be solved by usual perturbation theory. There is a large amount of literature on the subject but we are only now developing a relatively general mathematical theory.

### Project #: Math2003-06

**Connection constants in differential systems**Mentor: Ovidiu Costin, Department of Mathematics

Co-mentor: Rodica Costin, Department of Mathematics

This problem has a long history, and is illustrated by the question of determining the behavior of a system as t→∞ when data is given as t→-∞. There are obviously many problems in which this question is relevant. The project aims at developing a new methodology to attack this issue in settings when solutions cannot be expressed in any usable closed form.