Mentor: **Avy Soffer,** Department of Mathematics,
soffer@math.rutgers.edu

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.

Mentor: **Ovidiu Costin,** Department of Mathematics,
costin@math.rutgers.edu

Co-mentor:
**Rodica Costin,** Department of Mathematics,
rcostin@math.rutgers.edu

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:

Mentor: **Ovidiu Costin,** Department of Mathematics,
costin@math.rutgers.edu

Co-mentor:
**Rodica Costin,** Department of Mathematics,
rcostin@math.rutgers.edu

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:

Mentor: **Andrew Sills,** Department of Mathematics,
asills@math.rutgers.edu

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 consective 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.

Mentor: **Stanley Dunn,** Department
of Biomedical Engineering,
smd@occlusal.rutgers.edu

Co-mentor:

Description to be announced soon

Prerequisites:

Mentor: **Stanley Dunn,** Department
of Biomedical Engineering,
smd@occlusal.rutgers.edu

Co-mentor:

Description to be announced soon

Prerequisites:

Mentor: **Christopher Woodward,** Department
of Mathematics,
ctw@math.rutgers.edu

Co-mentor:

Description to be announced soon

Prerequisites:

Send your comments to
the Webmaster.