I am a fifth year graduate student and teaching assistant at the George Washington University pursuing a PhD in Mathematics under the supervision of Józef H. Przytycki. My research interests are in knot theory, low – dimensional topology, and algebraic and quantum invariants of manifolds. I focus primarily on the study of skein modules and skein algebras of 3 – manifolds and the Topological Quantum Field Theoretic description of the Witten – Reshetikhin – Turaev 3 – manifold invariant. I also study self – distributive structures, and their generalisations, whose axioms are motivated by the Reidemeister moves in knot theory. Quandles and racks are prime examples of such structures.
My current research involves categorifying the Kauffman bracket skein module for lens spaces and S1 – bundles over surfaces, providing a nontrivial generalisation of Khovanov homology leading to homological invariants of links in these manifolds with arbitrary rings of coefficients.
I am the coauthor of a scholarly book on knot theory, in which I have written six chapters devoted to the study of 3-manifold topology and skein modules. The book will soon be published by Springer, Universitext. I am also a coauthor of the chapters on skein modules in the Encyclopedia of Knot Theory.