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Research Interests:

My research is in a field known as 'higher-dimensional algebra.' My dissertation, Lie 2-Algebras, was written under the guidance of John Baez. My work blends Lie theory with elements of category theory and has connections to braid theory and Lie algebra cohomology. I am also interested in the relationship between Lie algebras and algebraic structures known as quandles. Thus, my interests lie in quantum algebra and quantum and geometric topology.

In my dissertation, I defined and explored generalized Lie algebras, called 'Lie 2-algebras,' and classified them up to equivalence in terms of Lie algebra cohomology. A Lie 2-algebra is a category equipped with algebraic structure much like that of a Lie algebra, but where the laws involving the bracket only hold up to isomorphism. In my dissertation, I focused on semistrict Lie 2-algebras which are those where only the Jacobi identity fails to hold as an equation. I showed that just as any Lie algebra gives a solution of the Yang-Baxter equation, a semistrict Lie 2-algebra gives a solution of the Zamolodchikov tetrahedron equation, which is the higher dimensional analog of the Yang-Baxter equation. Furthermore, I explored the relationship between groups, Lie algebras, quandles, and braids, and described a novel means of passing from a Lie group to its Lie algebra.

To learn more about my work, please read (pdf versions of) my:

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Publications:

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(for a complete list, please see my Curriculum Vitae)

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