## Projects for the 2008 Math REU Program

### Project #: Math 2008-01

**Equivariant cohomology of homogeneous spaces**

Mentors: Anders Buch, Department of Mathematics, and Siddhartha Sahi, Department of Mathematics,

The goal of the project is to find formulas for the Schubert calculus of homogeneous spaces. Specifically, we hope to obtain a positive combinatorial formula (also called a Littlewood-Richardson rule) for the multiplicative structure constants for the equivariant cohomology of Lagrangian Grassmannians.

### Project #: Math 2008-02

**Representation theory of affine Lie algebras**

Mentors: William Cook, Department of Mathematics, and Yi-Zhi Huang, Department of Mathematics,

We will study how modules for affine Lie algebras are related to each other by investigating the corresponding vertex operator algebras and representations of these vertex operator algebras. Our goal is to understand the algebraic structure on the direct sum of all irreducible modules for an affine Lie algebra. In particular, we would like to determine the classes of modules which are related by suitable operations.

### Project #: Math 2008-03

**Ultraproducts of finite symmetric groups**

Mentors: Paul Ellis, Department of Mathematics, and Scott Schneider, graduate student,

We shall investigate the extent to which the algebraic structure of ultraproducts of finite symmetric groups depends upon the choice of the ultrafilter. In particular, we shall attempt to compute the number of non-isomorphic groups which arise in this fashion. There is a strong possibility that the answer to this question is independent of the ZFC axioms of set theory.

### Project #: Math 2008-04

**Polynomial equations over 3 by 3 matrices**

Mentor: Robert Wilson, Department of Mathematics,

project description (a pdf file)

### Project #: Math 2008-05

**Polynomial equations over matrices - prescribed sets of roots**

Mentor: Robert Wilson, Department of Mathematics,

project description (a pdf file)

### Project #: Math 2008-06

**Moduli spaces of points on the complex plane**

Mentor: Christopher Woodward, Department of Mathematics,

Co-mentor: Sikimeti Mau, graduate student,

The project concentrates on developing an understanding of toric singularities of these moduli spaces. It will start out by reading a little about toric varieties, from Fulton's book, which can be used as an introduction to algebraic geometry.