## Proposed projects for the 2004 Math REU Program

### Project #: Math2004-01

**Partial Differential Equations**Mentor:

**Avy Soffer,**Department of Mathematics,

Co-mentor:

Dynamical wave phenomena requires the understanding of the large time behaviour 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 #: Math2004-02

**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.

Prerequisites:

### Project #: Math2004-03

**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.

Prerequisites:

### Project #: Math2004-04

**Automated Proofs of Combinatorial Identities**Mentor:

**Andrew Sills,**Department of Mathematics,

Co-mentor:

A geometric series is a series a(0) + a(1) + a(2) + . . . in which the ratio a(k+1) / a(k) of consecutive terms is constant for all k = 0, 1, 2, 3, . . . . A hypergeometric series (first studied by Gauss) is a series in which the ratio of consecutive terms is a rational function, thus the term "hypergeometric." Since the time of Gauss, many identities involving hypergeometric series were discovered. In the course of studying such series, it became standard to look for, by ad hoc methods, recurrence relations satisfied by the series. A major advance was made in the 1940's by Sister Mary Celine Fasenmyer. Sister Celine discovered an algorithm which finds a recurrence relation satisfied by a given hypergeometric term. In the early 1990's Herbert Wilf (University of Pennsylvania) and Doron Zeilberger (Rutgers University) as a part of what is now known as the "WZ theory" (named in honor of two famous complex variables) found a much faster algorithm which accomplishes much the same goal as Sister Celine algorithms. Furthermore, Zeilberger has implemented his algorithms in Maple, while Peter Paule and Axel Riese of the Research Institute for Symbolic Computation (RISC) in Linz, Austria have done the same for Mathematica. As a result of this work, it is possible to prove (not just verify for a large number of cases, but actually prove) a large class of hypergeometric identities in a completely automatic fashion via the computer.

Despite the fact that Wilf and Zeilberger have proved that such computer generated proofs are possible (and in fact have done so in many cases), due to limitations of computer speed and memory, many interesting identities remain to this day without a WZ-style proof.

The project I propose is to work towards obtaining more automated proofs of identities. In the process we may need to implement specialized versions of the WZ and/or Sister Celine algorithms in order to exploit symmetries which are not present in the general case.

Prerequisites: Enough mathematical background to be comfortable with the manipulation of discrete functions and a basic knowledge of computer programming. Experience with Maple and/or Mathematica would be helpful.

References: (Note: If you only have time to consult one reference, number 1 below is by far the most important.)

1. M. Petrovsek, H.S. Wilf, D. Zeilberger, A=B, A.K. Peters publishing, Wellesley, MA, 1996. (Also available for free download here.)

2. A. V. Sills, Finite Rogers-Ramanujan Type Identities, Electronic J. Combin. 10(1)(2003), #R13, pp. 1-122.

3. H.S. Wilf, D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc., 3(1990), 147-158.

4. D. Zeilberger, A fast algorithm for proving terminating hypergeometric function identities, Discrete Math., 80(1990), 207-211.

5. D. Zeilberger, The method of creative telescoping. J. Symbolic Comput. 11(1991): 195-204.

### Project #: Math2004-05

**Applications of Independent Components Analysis and subspace projection methods in brain mapping and genomics**

Mentor: **Stanley Dunn,** Department of Biomedical Engineering,

Co-mentor:

Description to be announced soon

Prerequisites:

### Project #: Math2004-06

**Looking at Geometric Algebra as a unifying tool for the many theories and models of medical imaging systems**

Mentor: **Stanley Dunn,** Department of Biomedical Engineering,

Co-mentor:

Description to be announced soon

Prerequisites:

### Project #: Math2004-07

**Representation Theory**Mentor:

**Christopher Woodward,**Department of Mathematics,

Co-mentor:

Description to be announced soon

Prerequisites: