# REU

## Projects for the 2008 Math REU Program

### Project #: Math 2008-01

Equivariant cohomology of homogeneous spaces
Mentors: Anders Buch, Department of Mathematics, asbuch@math.rutgers.edu and Siddhartha Sahi, Department of Mathematics, sahi@math.rutgers.edu
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, wjcook@math.rutgers.edu and Yi-Zhi Huang, Department of Mathematics, yzhuang@math.rutgers.edu
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, prellis@math.rutgers.edu and Scott Schneider, graduate student, scottsch@math.rutgers.edu
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, rwilson@math.rutgers.edu
project description (a pdf file)

### Project #: Math 2008-05

Polynomial equations over matrices - prescribed sets of roots
Mentor: Robert Wilson, Department of Mathematics, rwilson@math.rutgers.edu
project description (a pdf file)

### Project #: Math 2008-06

Moduli spaces of points on the complex plane
Mentor: Christopher Woodward, Department of Mathematics, ctw@math.rutgers.edu
Co-mentor: Sikimeti Mau, graduate student, sikimeti@math.rutgers.edu
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.