Master's Thesis
Radicals of Extended Smash Products of Group-Graded Rings

Abstract:
In this thesis we take a group-graded ring, R, that has unity, and show two equivalent conditions for it to be graded. Then, using the dual of the group ring, k[G], denoted k[G]*, and a subalgebra of it called P_G, we form a new ring called the smash product, written R#P_G. We do this for both finite and infinite groups. In the infinite case, the smash product has no unity, so we adjoin a 1 using a method similar to the Dorroh extension.
Next we introduce the ideas of radicals of rings with some examples. Talking about graded rings, we introduce graded radicals. With this theory, we are able to characterize certain radicals of the smash product. Saorin gave a characterization of the smash product with a 1 adjoined for the Jacobson radical. Here we will do so for all hereditary radicals, thus making Saorin's result a corollary.

Graduate Seminar
Operators on Hilbert Space

Abstract:
We will study bounded linear operators on an infinite dimensional Hilbert space, with examples. In particular, we will focus on the case of the shift operator.

Graduate Seminar
Trilinear Coordinates

Abstract:
We will introduce the notion of using three numbers to represent points in the plane. Using these new coordinates we will determine the area of a triangle and demonstrate that the altitudes of a triangle are concurrent.

Graduate Seminar
The Cauchy Integral Formula

Abstract:
We use Clifford algebras to generalize the Cauchy Integral Formula to compact n-dimensional manifolds.

Graduate Seminar
Path Integrals and the Free Particle

Abstract:
We will look at the formulation of a path integral and use path integrals to calculate the probability amplitude for the free particle.

Seminar
Introduction to Topological K-Theory

Abstract:
Algebraic topology is concerned with functors from topological spaces to groups. The three most important examples are homology (simplicies and triangulations), homotopy (fundamental groups), and K-theory.
In this talk, by studying certain operations on vector bundles over topological spaces, we define K-theory and give some basic results.

Seminar
Towards a Gauss-Bonnet Theorem for Non-Compact Surfaces

Abstract:
An overview of my thesis topic will be presented. In order to generalize the Gauss-Bonnet theorem to non-compact surfaces an analog of integration must be developed. This is accomplished via the heat equation. To make the problem more tractable, the hyperbolic plane is considered, followed by quotients of H^2 by groups of isometries. Examples will be given to illustrate various ideas.

CMS Winter Meeting 2002 / Inter-Campus Seminar Day, 2003
Computing some nonstandard Betti numbers

Abstract:
In previous work, we established an abstract generalized Gauss-Bonnet theorem for surfaces. We discuss the computation of our nonstandard Betti numbers in the theorem, at least for the special case of Riemann surfaces of constant curvature. By the uniformization theorem we may as well consider hyperbolic space modulo a discrete subgroup of isometries and we discuss the case of hyperbolic space in detail.