## Projects for the 2005 Math REU Program

### Project #: Math2005-01

**Partial Differential Equations**Mentor:

**A**

**vy Soffer**

**,**Department of Mathematics, soffer@math.rutgers.edu

Co-mentor:

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 #: Math2005-02

**Mathematical Physics**Mentor:

**Joel Lebowitz,**Department of Mathematics, lebowitz@math.rutgers.edu

### Project #: Math2005-03

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.

### Project #: Math2005-04 and 05

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

### Project #: Math2005-06

**A∞ algebras and categories**Mentor:

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

Co-mentor:

An A

_{∞}algebra is a vector space with a collection of operations that satisfy associativity constraints. These first arose in topology, and more recently symplectic geometry and mathematical physics. We will try to classify A

_{∞}structures satisfying some simple constraints that arise in symplectic geometry/mathematical physics.