## Projects for the 2006 Math REU Program

### Project #: Math 2006-01

**Dispersive wave equations of mathematical physics -- analysis and numerics**

Mentor: Avy Soffer, Department of Mathematics,

Co-mentor: Chris Stucchio, graduate student

### Project #: Math 2006-02

**Polytopes related to symplectic geometry and homotopy theory**

Mentor: Christopher Woodward, Department of Mathematics,

### Project #: Math 2006-03

**Polynomial equations over matrices**

Mentor: Robert Wilson, Department of Mathematics,

The quadratic equation X2 + AX + B = 0, where the coefficients A,B and the unknown X are all 2 by 2 matrices over the complex numbers, has some surprising properties. For example, it is possible to choose matrices A and B so that this equation has exactly 0, 1, 2, 3, 4, 5, or 6 solutions, but if there are more than 6 solutions there must be infinitely many solutions. This result is described at http://sites.math.rutgers.edu/~rwilson/polynomial_equations.pdf

It would be interesting to know the corresponding result for an equation of degree m where the coefficients and unknowns are n by n matrices over the complex numbers. A procedure for describing all solutions of such an equation is known and it is known that it is possible to choose coefficients so that the number of solutions is exactly equal to the binomial coefficient "mn choose n". It may well be that any number of solutions between 0 and "mn choose n" is possible but that no larger finite number of solutions is possible, but this has not been proved. The proposed project is to find all possible numbers of solutions for some collection of values of m and n. (It would be nice to do this for all m and n, but that might be too hard. One would probably start with the case of a cubic equation in 2 by 2 matrices, and then proceed to other cases if possible.) This problem can be investigated using standard tools of linear algebra (in particular, rational canonical form).

### Project #: Math 2006-04

**Eulerian Graph Representation for siRNA Sequence Structure Applications**

Mentor: Stanley Dunn, Department of Biomedical Engineering,

RNA interference (RNAi) is a process by which gene expression is suppressed. Since RNAi is a powerful tool for determining gene function and preventing disease, much work has been done to understand the important role that short interfering RNA (siRNA) plays in RNAi. Characteristic sequence, thermodynamic, and structural properties of functional siRNAs have been identified by researchers. We have considered the Eulerian Graph Model of siRNA, introduced by Pancoska et al, in hopes that it would provide valuable insight into the level of functionality of a potential siRNA sequence. Using this graph structure, we have identified siRNA properties, captured in the graphs, that discriminate functional and non-functional siRNAs. The aim of this project is to consider the problems of existence and uniqueness of the Eulerian graphs for functional siRNA sequences.

### Project #: Math 2006-05

**An algebraic study of quantum walks**

Mentor: Mario Szegedy, Department of Computer Science,

### Project #: Math 2006-06

**Set Theory and Discrete Mathematics**

Mentors: Simon Thomas, Department of Mathematics, and János Komlós, Department of Mathematics,

Co-mentor: Paul Ellis, graduate student,