Research Interests
I am interested in arithmetic geometry (= algebraic geometry + number theory). I enjoy approaching problems from a computational perspective and developing explicit methods to study them. Much of my research has consisted of devising algorithms to compute given mathematical quantities or objects, implementing these algorithms on a computer, and then computing collections of data using these implementations. Here are some of my research projects.
- Gluing curves along their torsion subgroups.
- Classifying algebras of low rank and degree over a commutative ring.
- Computing invariants and equations of Hilbert modular varieties.
- Computation of explicit equations for Belyi maps.
Two other frequent collaborators are Mike Musty and Jeroen Sijsling.
Publications
| 7. | [PDF] |
Creating a dynamic database of finite groups (with Lewis Combes, John Jones, Jennifer Paulhus, David Roe, and Manami Roy), in preparation. |
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| 6. | [PDF] | [2402.03160] |
Equations of genus 4 curves from their theta constants (with Jeroen Hanselman and Andreas Pieper) |
| 5. | [PDF] | [2301.10302] |
A database of basic numerical invariants of Hilbert modular surfaces (with Eran Assaf, Angelica Babei, Ben Breen, Edgar Costa, Juanita Duque-Rosero, Aleksander Horawa, Jean Kieffer, Avinash Kulkarni, Grant Molnar, and John Voight), accepted to Contemporary Mathematics. |
| 4. | [PDF] |
[2005.03587] [Link] |
Gluing curves of genus 1 and 2 along their 2-torsion (with Jeroen Hanselman and Jeroen Sijsling), Math. Comp. 90 (2021), no. 331, 2333-2379. |
| 3. | [PDF] | [Link] |
On Algebras of Low Rank and on Belyi Maps PhD thesis, Dartmouth College, 2019. |
| 2. | [PDF] |
[1805.07751] |
A database of Belyi maps (with Michael Musty, Jeroen Sijsling, and John Voight), Proceedings of the Thirteenth Algorithmic Number Theory Symposium. Open Book Ser., 2, 375–392, Math. Sci. Publ., Berkeley, CA, 2019. |
| 1. | [PDF] |
[1311.2081] [Errata] [Link] |
Numerical calculation of three-point branched covers of the projective line (with Michael Klug, Michael Musty, and John Voight), LMS J. Comput. Math. 17 (2014), no. 1, 379-430. |
Development
I contribute to several mathematical development and coding projects.- I am a contributor to The L-functions and Modular Forms Database (LMFDB). In addition to other smaller contributions, I am one of the maintainers of the LMFDB's database of Belyi maps. To see current issues and development, see the LMFDB repo on GitHub.
- The raw data for our Belyi map computations is also available here.
- Our code for gluing curves along their 2-torsion is available here.
- I am part of a collaboration working to add a database of finite groups to the LMFDB. Our prelimiary version is now on the beta site of the LMFDB and our code is available on GitHub.
- I have worked on computing invariants of Hilbert modular surfaces. We have created a preliminary version of a database of Hilbert modular surfaces, currently available here; we plan to add it to the LMFDB in the near future. Our is available here and the raw data is available here.