Undergraduate Research and Expository Writing

Operator Algebras

C*-Algebras Generated by Weakly Quasi-Lattice Ordered Groups

In 1992 Alexandru Nica published a highly influential paper examining the C*-algebras generated by a semigroup he calls a quasi-lattice ordered group. Of particular interest to Nica were the C*-algebras generated by the Toeplitz representation of a quasi-lattice order, and a universally defined C*-algebra he calls C*(G, P). Due to the concreteness of the first C*-algebra and the universal properties of the second, Nica was interested in finding sufficient conditions to determine when these two algebras were isomorphic. He called quasi-lattice orders that satisfied this isomorphism condition amenable. Finding techniques that establish amenability is now a topic at the cutting edge of research in the study of C*-algebras of semigroups. In this dissertation, we follow the outline set out by Nica's paper to study the C*-algebras generated by weakly quasi-lattice ordered groups. Most of the proofs throughout this dissertation required a substantial amount of original work to fill in details omitted by Nica, as he often only offered a proof outline.

Low Dimensional Topology

Khovanov Homology of Knots and Links

Classical knot theory is the study of embeddings of the one-sphere into Euclidean three-space or the three-sphere. Knot theorists aim to completely classify links into disjoint equivalence classes and, failing that, to develop tools which can distinguish two distinct links. Due to the difficulty of studying links directly in three-space, knot theorists tend to study projections of the link onto a plane with crossing information included. Such a projection of a link is known as a link diagram. A link invariant is a function from the set of all link diagrams into another set whose value remains constant over all possible link diagrams of the same link. The celebrated Jones polynomial is an example of a link invariant whose output is a Laurent polynomial in t. While the Jones polynomial is an important link invariant in the history of knot theory, much of its significance lies in its links to other fields such as Chern-Simons theory and its generalisations to stronger link invariants such as Khovanov homology and the HOMFLY polynomial. Khovanov homology categorifies the Jones polynomial in that Khovanov homology assigns to every link diagram a bi-graded cochain complex whose graded Euler characteristic returns the unnormalised Jones polynomial of the link. Given a link's cochain complex, we can then find the Khovanov homology of the link by taking the cohomology of the complex. Khovanov homology is a much stronger link invariant than the Jones polynomial. For example, the Jones polynomial fails to distinguish an infinite class of links from the n-component unlink whereas Khovanov homology can always distinguish the n component unlink from a non-trivial link of n components. Furthermore, Khovanov homology reveals richer information about a link, such as torsion, and is a topological quantum field theory.

Confrences and Contributed Talks