Jacob H. Swenberg

Do Link Polynomials Detect Causality in Globally Hyperbolic Spacetimes? Joint with Samantha Allen. J. Math. Phys. 62 (2021), no. 3, 032503. doi:10.1063/5.0040956. https://arxiv.org/abs/2011.09415

A Heegner--Shimura Approach to Class Number One. (Based on work in my undergraduate thesis). Joint with John Voight. Motivated by the class number 1 problem of Gauss, we provide a complete description of the Galois action on the set of principally polarized abelian surfaces with QM by a maximal quaternion order O and CM by an imaginary quadratic order S. The set A of such abelian surfaces is in bijection with a set of classes of optimal embeddings of S into O. We describe an action of the absolute Galois group on these optimal embeddings and show that the action agrees with the action on abelian surfaces. By completely describing the action on optimal embeddings, we describe the action on abelian surfaces. As a consequence, we determine various fields of moduli of abelian surfaces. Using recent progress on finding rational points on curves using the technique of quadratic Chabauty, we apply the results to give a new solution to the Gauss class number 1 problem.