Jacob H. Swenberg

Email: jaswenberg [AT] math [DOT] ucla [DOT] edu
Office: MS 2951

Teaching

• Fall 2021: Math 31A (Differential and Integral Calculus) Disc 2E/2F
• Winter 2022: Math 33A (Linear Algebra and Applications) Disc 1A/1B
• Spring 2022: Math 33A (Linear Algebra and Applications) Disc 2G/2H
• Summer 2022 (Session C): Math 32B (Multivariable Calculus II) Disc 1A/1B
• Fall 2022: Math 110AH (Honors Algebra) Disc 1A
• Winter 2023: Math 110BH (Honors Algebra) Disc 1A
• Spring 2023: Math 33A (Linear Algebra and Applications) Disc 3E/3F

Other Activities

• UCLA Number Theory Learning Seminar

Publications and Preprints

• 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

Ongoing Projects

• 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.
• A Lambda-adic Eichler--Shimura isomorphism for quaternionic modular forms. Ohta proved, using work of Hida, that the ordinary part of an inverse limit of etale cohomology groups for a tower of modular curves has a local quotient isomorphic to a space of Lambda-adic modular forms. We generalize this result to Shimura curves associated to indefinite quaternion algebras over arbitrary totally real fields with the goal of generalizing the Upsilon map of Sharifi's conjecture. Using Jacquet--Langlands, a study of the cohomology of Shimura curves allows us to obtain (canonical) rank 1 local quotients in the cohomology of Hilbert modular varieties.

Miscellany

• Old basic exam solutions (with Jas Singh and Sam Qunell). Use at your own risk!
• Visualizing images of curves under self-maps of the plane.
• General visualization of self-maps of the plane
• Nim (and other games?)
• A factoring game