Projects

Ph.D. Thesis: I study Chern-Simons bundles over traceless character varieties of punctured surfaces. In particular, I generalize a construction of the Chern-Simons bundle, due to Ramadas-Singer-Weitsman, to the case of traceless character varieties of evenly punctured surfaces. Given a tangle in a 3-manifold whose boundary is the union of a punctured surface and a collection of embedded annuli, we obtain a cross-section of the Chern-Simons bundle which is parallel to a natural connection 1-form. I specialize to the case of 2-stranded tangles in the 3-ball, and obtain a simplified description of the Chern-Simons bundle and its natural connection 1-form in terms of local coordinates on the pillowcase. This facilitates computations of so-called Chern-Simons invariants, via cut-and-paste methods, of traceless representations of knots. 

I am currently writing up my results. 

• Poster from GSTGC 2022 at Georgia Tech.

• Slides from a talk at the AMS Central Sectional Meeting, Spring 2022.

NASA SCaN Internship: In Summer 2022, I was a virtual intern at NASA Goddard Spaceflight Center as a part of the Space Communications and Navigation (SCaN) program. Our team of seven interns, lead by Alan Hylton, investigate applications of pure mathematics to space networks. We explored modeling of space networks using graphs, hypergraphs, cellular sheaves, and more. I was particularly involved in modeling networks using hypergraphs, extending sheafy constructions to hypergraphs, and investigating applications of sheaf Laplacians to certain sheaves, especially path sheaves.  

Our group has two papers based on our work accepted to the 2023 IEEE Aerospace Conference. Preprints coming soon!

Here are some slides of presentations that I gave during the internship.

• Sheaves on graphs

• Sheaf Laplacians

You can find the intern journal Signal here, or watch a video about the internship here.

Older Projects

• Master's Thesis, completed while a master's student at the University of Northern Iowa

• Discrete Dynamics of Contractions on Graphs, a paper published in the undergrad journal Involve.