## Proposed projects for the 2002 VIGRE REU Program

### Project #: VIGRE2002-01

**Matrix Analysis**Mentor: Peter Landweber, Department of Mathematics

Co-mentor: Brian Lins, VIGRE graduate student

There is a lot more to linear algebra than one usually learns. Matrix analysis is devoted to that part of the subject which concerns inequalities and makes use of several other analytic techniques. There are several fine texts in the area, with the title Matrix Analysis: one by Horn and Johnson (available in paperback) is thorough and especially well-written, and a more recent one by Bhatia is more challenging with many substantial exercises and problems. In addition, a recent text by Lax gives a superb account of the essentials of linear algebra needed for matrix analysis. The aim of the project will be to become familiar with this area, and to concentrate on questions concerning eigenvalues of sums of Hermitian matrices which have been studied with great success in recent years.

Prerequisites: Linear algebra (ideally, an introductory course, followed by a second course on linear algebra, or by a course on abstract algebra), and advanced calculus (or introductory real analysis).

References:

1. R. Bhatia, Linear algebra to quantum cohomology: the story of Alfred Horn's inequalities, American Mathematical Monthly 108 (April 2001), 289-318.

2. R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics 169, Springer-Verlag, 1997.

3. R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1985 (paperback).

4. P. Lax, Linear Algebra, Wiley, 1997.

### Project #: VIGRE2002-0

**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 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 SchrÃ¶dinger 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 #: VIGRE2002-03

**Representation Theory**Mentors: Friedrich Knop, Department of Mathematics, and Shawn Robinson, Department of Mathematics

Co-mentors: Bill Cuckler and Vince Vatter VIGRE graduate students

The group of invertible n x n matrices whose entries are complex numbers has certain important subgroups, called Lie groups. Representations of these groups are central to both mathematics and physics, and a wide variety of tools are used to study them. We will focus on the algebraic and combinatorial aspects of the theory. In particular we aim to implement a recent combinatorial model on a computer, and use this to test existing conjectures and search for new ones.

Prerequisites: linear algebra (necessary), abstract algebra (preferred), interest and/or experience in java or c++ programming (preferred).

### Project #: VIGRE2002-04

**Symplectic Geometry**Mentor: Matthew Leingang, Department of Mathematics

Co-mentor: Alex Zarechnak, VIGRE graduate student

Symplectic manifolds are manifolds equipped with a skew-symmetric pairing between tangent vectors. One can deem half of the coordinates in such a manifold to denote "position" and a complementary half "momentum," and in so doing practice a kind of Hamiltonian mechanics on these manifolds. Of particular interest is the case when a Lie group of symmetries acts on the manifold in a nice way; then we can reduce the total number of coordinates and arrive at a simpler phase space. Symplectic manifolds and various generalizations of them are also applicable to string theory.

Prerequisites: at the very least, advanced calculus. Preferably, differentiable manifolds, intermediate mechanics, linear algebra.

References: Introduction to Mechanics and Symmetry (J. E. Marsden and T. S. Ratiu), Symplectic Geometry (D. MacDuff and D. Salomon).

### Project #: VIGRE2002-05

**Statistical Mechanics**Mentor: Carlo Lancellotti, Department of Mathematics

Self-consistent wave-particle systems play an important role in many physical problems, e.g. in plasma physics. In this kind of systems the dynamics of a large number of charged particles is determined by the collective electro-magnetic field that the particles themselves generate. An important and interesting example of this situation is given by the so-called Bernstein-Greene-Kruskal (BGK) nonlinear solutions to the Vlasov-Poisson kinetic equations for a collisionless plasma. Especially interesting are solitonic BGK solutions and their superpositions. The properties of these solutions can be studied both analytically and numerically.