Master's and Doctoral Defenses

Resolutions of Ore Extensions

Resolutions provide important tools for studying the homology and deformation theory of non-commutative algebras, so one desires convenient, finite resolutions to work with. We construct resolutions for noncommutative PBW algebras that arise iteratively from setting quantum or skew commutators to lower order terms, like Weyl algebras, shift algebras, skew polynomial rings, many universal enveloping algebras and others. To do so, we extend a quantum summarization map to a chain map between resolutions, allowing us to take advantage of the twisted product resolution for twisted tensor product algebras.

Marker Sets And Their Use To Show The Hyperfiniteness of 2^{Z^n}

In this talk I will show two proofs that for each natural number n the free part of the shift action on 2^{Z^n} is hyperfinite and I will show it also admits a continuous 4 coloring. To do this, I will present the theory of marker sets and their uses in these arguments. Types of marker sets considered are Slaman-Steel markers, basic clopen markers, marker regions, and orthogonal marker regions.

A Tree-based Discrete Time Model for Jump Diffusion Option Valuation

Quantum Group Actions and Hopf Algebras

The Subgroup Structure of GL(2, p) for p a prime

Hidden Markov Models with Applications

Option Pricing Under New Classes of Jump-Diffusion Processes