Professor Rodgers' Groups
We will look at some of the graphs associated with finite incidence geometries, and consider the existence of vertex sets having nice combinatorial properties. Often these graphs arise from considering a vector space over a finite field, and using a quadratic form to obtain a collection of distinguished subspaces. The graphs obtained in this way have very strong structural properties. The vertex sets we will be interested in can be described in terms of the eigenspaces of the adjacency matrix of the graph.
Students should have some familiarity working with permutation groups, and a strong background in linear algebra. Some programming experience is a plus.
Professor Tran's Groups
We will analyze the structure of complex zeros of a sequence of polynomials generated by a recurrence relation (any given polynomial in the sequence is given as a combination of previous ones in the sequence). In various cases, the zeros of such polynomials lie on a fixed curve in the complex plane. When this curve is the real axis, we obtain a sequence of hyperbolic polynomials which are the subject of many recent studies.
Students working on this project should be familiar with sequences and series of functions, generating functions, and the basic structure of complex numbers. A good understanding of the Cauchy’s integral formula will be a plus.