Mathematics

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, Lie 2-Algebras, 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 `categorified' Lie algebra, meaning that is consists of 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. I focused on semistrict Lie 2-algebras which are those where only the Jacobi identity fails to hold as an equation and 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.

My current interests, supported by the Collaboration Grants for Mathematicians program through the Simons Foundation (2015 - 2020), focus on self-distributive structures such as quandles and racks. A quandle is a set equipped with two binary operations satisfying axioms that capture the essential properties of group conjugation and algebraically encode the Reidemeister moves from classical knot theory. I am interested in the relationships between self-distributive structures and their (co)homology and crossed modules, group and Lie algebra (co)homology theories, and knot and knotted surface invariants.

Selected Publications

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

Selected Grants

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Selected Presentations

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

Fun Stuff