Doctor of Philosophy, The Ohio State University, 2013, Mathematics

In three dimensions, a topological quantum field theory, or TQFT, is a functor from the category of 3-dimensional framed cobordisms to the category of vector spaces. Two well-known TQFTs are the Hennings TQFT and the Dijkgraaf-Witten TQFT. The Hennings TQFT is built from a link invariant, by applying elements of a Hopf algebra in a systematic way to tangle diagrams. The Dijkgraaf-Witten TQFT is built by counting principal bundles on a 3-manifold which have been weighted by a 3-cocycle. We prove that the Hennings TQFT applied on the double of the group algebra is equivalent to the Dijkgraaf-Witten TQFT applied on a trivial cocycle. In order to extend this result to the more general case of a non-trivial cocycle, we discuss the notion of a quasi-Hopf algebra, which is an almost-cocommutative Hopf algebra. We then extend the definition of the Hennings TQFT so that instead of applying elements of a Hopf algebra to the tangle, we instead apply elements of a quasi-Hopf algebra. The specific quasi-Hopf algebra in which we are interested is the twisted double of the group algebra, where the twisting occurs via a 3-cocycle. Finally, we conjecture that the Hennings TQFT applied on the twisted double of the group algebra is equivalent to the Dijkgraaf-Witten TQFT applied on the same cocycle.

Bachelor of Science (BS), Ohio University, 2023, Mathematics

We construct a new braided monoidal category so that we can define an extended TQFT for subcobordisms.

Doctor of Philosophy (PhD), Ohio University, 2022, Mathematics (Arts and Sciences)

In Topological Field Theories (TFTs), there is a well documented correlation between 3-dimensional TFTs and braided monoidal categories. While braided monoidal 2-categories have been expected to have applications to 4-dimensional TFTs, there are very few known examples of braided monoidal 2-categories. The major goal of this dissertation is to present a categorification of a result of Pareigis, 1995. Namely, that modules over a commutative algebra in a braided monoidal category form a braided monoidal category. The categorified statement is that pseudomodules over a braided pseudomonoid in a braided monoidal 2-category form a braided monoidal 2-category. This result would be an example of constructing braided monoidal 2-categories from existing braided monoidal 2-categories. We approach this by constructing a new language which simplifies some of the complexities coming from relative tensor products of pseudomodules. The relationships between monoidal categories and multicategories has been well documented in Leinster, 2004. We define the notion of a multi-2-category, as well as the notion of a braided multi-2-category. We then construct braided pseudomonoids in braided multi-2-categories and examine pseudomodules over them. The main theorem of the dissertation is that local pseudomodules over a braided pseudomonoid in a braided multi-2-category forms a braided multi-2-category. A second result is a revision to the Joyal-Street definition of a balanced 2-category. We give a coherence for balanced 2-categories which was omitted in the original definition.

Doctor of Philosophy, The Ohio State University, 2018, Mathematics

In this thesis, we study properties of the TQFT associated to the modular category SO(p)-level 2 for an odd prime p. We compute the associated representation of the central extension of the mapping class group of the surface of genus one with one boundary component specialized at a simple object a of SO(p)-level 2. Let ζ be a primitive p-th root of unity. We show that for each a, there exists a full-rank free lattice over the ring of integers Z[ζ, i] inside the genus 1 specialized space at a that is preserved by the extended mapping class group action. We also show that for each a, the image of the extended mapping class group representation is finite. Finally, we relate the representations to the Weil representation over finite fields.