Software

Mathematical Databases

Project		Active	Contributions
	browse code paper	2008-2023 1,480 commits 105,323 ++ 103,649 --	The LMFDB has been a major focus of my work at MIT since 2018, and I now serve on its editorial board. My contributions fall into two categories: improving LMFDB's infrastructure and adding mathematical content to the site. On the infrastructure front, I converted the backend database from MongoDB to PostgreSQL, spinning off a Python-SQL interface library, Psycodict, in the process. I have also worked on centralizing functionality (such as code for laying out search pages) that had been duplicated in many sections, easing the process of adding new site-wide functionality and adding new sections. My mathematical contributions are described below.
Abelian varieties over finite fields	browse code paper	2016-2020 44 commits 11,913 ++ 8,168 --	In part 1 of the project to build a database of abelian varieties over finite fields within the LMFDB (joint with Dupuy, Kedlaya and Vincent), we used the Honda-Tate theorem to enumerate isogeny classes and compute many quantities that are invariant across isogeny classes.
Classical modular forms	browse code paper	2018-2021 51 commits 852 ++ 233 --	As part of a large group effort, we rebuilt the classical modular form section of the LMFDB from scratch, using code from Magma, Pari and Sage. My contributions were mostly on front-end code, so are reflected in the commits to the LMFDB respository more than the CMF repository.
Abelian varieties over finite fields	browse code	2019-2023 105 commits 2,554 ++ 1,205 --	In part 2 of the project to build a database of abelian varieties over finite fields within the LMFDB (joint with Costa, Dupuy, Marseglia, Vincent and West) we are using Deligne modules and work of Centeleghe-Stix to compute isomorphism classes and polarizations within many of the isogeny classes.
Finite groups	browse code paper	2019-2023 948 commits 51,143 ++ 22,368 --	As part of a collaboration with Combes, Jones, Paulhus, Roy, and Schiavone, we have taken existing databases of finite groups, computed various quantities about them (most notably subgroup lattices, character tables and abstract isomorphism classes for permutation groups) and created a searchable database of groups, subgroups, and characters.
Modular curves	browse code	2022-2023 396 commits 43,795 ++ 8,823 --	Building on two workshops, we are building a database of modular curves using the bijection with finite index subgroups of GL(2, Zhat). I have been helping coordinate work on finding models, maps between them, and rational points.

Open Source Computation

Project		Active	Contributions
	try code	2006-2023 500 commits 34,949 ++ 16,871 --	I have been involved in Sage since Sage Days 1. Much of my effort has gone toward creating the p-adics module, but I have also worked on finite fields, elliptic curves, number fields, the transition to git and github, the coercion system and many other parts of the software.

Infrastructure & Collaboration

Project		Active	Contributions
	browse code paper	2020-2022 374 commits 22,028 ++ 7,647 --	With the start of the COVID-19 pandemic, many research seminars moved online. Together with Costa, Poonen and Sutherland, I created an online index of these seminars which helps researchers worldwide find talks of interest that are newly accessible. It also provides useful tools for creating lightweight homepages for seminars and conferences.
	try code paper	2021-2022 17 commits 12,276 ++ 402 --	The pandemic made it harder to form collaborations between students in undergraduate classes to work on homework. This site allows instructors at MIT to opt in and help their students find study groups. The project is mostly driven by Sutherland (and the linked paper is written by him), but I helped with the initial setup and the matching algorithm.
	code	2019-2022 118 commits 18,542 ++ 11,399 --	The most popular interface to PostgreSQL from Python is psycopg, which allows for the execution of SQL commands from python. While robust and fast, it requires users to know SQL. For the LMFDB, I developed a layer on top of psycopg that constructs queries from dictionaries and returns the output in dictionaries as well. In support of researchseminars and psetpartners, this part of the LMFDB was extracted out into its own library.

Talks

Please e-mail me to learn more about talks where I did not use slides.

Conferences

I use MathMeetings.net to find conferences. Number Theory Web is also useful.

Past and Future Conferences Attended

A gold row indicates a conference that I organized, while a blue row indicates one where I spoke.

Curriculum Vitae   PDF

Education

Employment

Honors and Awards

Publications

Software

Talks

Teaching Experience

Conferences Organized

Other Activities

Interests   PDF

Mathematical Databases

The LMFDB is a website that hosts dozens of collections of mathematical objects, from the eponymous L-functions and classical modular forms to elliptic curves, number fields, finite groups and Artin representations. Mathematically, it is motivated by the Langlands program, but many of the objects arise much more broadly.

These databases make a new kind of experimental mathematics possible, with a spectacular example occurring last year. While studying elliptic curves ordered by conductor, a group of mathematicians including an undergraduate student observed that the average point counts modulo primes displayed an unexpected oscillation. This "murmurations" phenomenon was only apparent when averaging over tens of thousands of curves, and has led to further theoretical exploration including a workshop at ICERM in July 2023, a proof of an analogue in the modular form setting, and investigations in non-arithmetic settings like Maass forms. As the use of data more broadly in society continues to grow, I believe that mathematical databases have a key role to play.

I have worked on most sections of the LMFDB, including major contributions to

• Finite groups (with accompanying paper)
• Classical modular forms (with accompanying paper)
• Abelian varieties over finite fields (with accompanying paper)
• Modular curves (paper in progress)

p-adic Computation

p-adic numbers have played a central role in many advances in modern number theory, from Deligne's proof of the Weil conjectures to the deformation theory underlying Wiles' proof of Fermat's Last Theorem to Scholze's perfectoid spaces. They can be defined either analytically, via the completion process that constructs the real numbers from the rationals, or algebraically, as the inverse limit of finite rings ℤ/p^nℤ as n increases. In either perspective, they provide a link between divisibility and modular arithmetic on the one hand and topology and analysis on the other.

From a computational point of view, there are two main challenges in implementing p-adic arithmetic: tracking precision through a computation, and handling algebraic extensions, which are far more complicated than for real numbers. I have led the development of p-adic arithmetic in Sage, including a lattice method for tracking precision developed in a series of papers with Xavier Caruso and Tristan Vaccon ([1] [2] [3] [4] [5]).

I am interested in applications of p-adics to number theory and have worked on point counting algorithms for surfaces over finite fields and average polynomial time algorithms for computing L-functions of hypergeometric motives.

Application Materials - Principal Research Scientist at MIT

• Curriculum Vitae
• Research Statement
• Statement on teaching, service and mentorship
• Representative Publications:

  			Title	With	Status
  PDF			Creating a Dynamic Database of Finite Groups	Lewis Combes, John Jones, Jennifer Paulhus, Manami Roy, Sam Schiavone	In preparation
  PDF	arXiv		Hypergeometric L-functions in average polynomial time II	Edgar Costa, Kiran Kedlaya	Accepted to ANTS XVI
  PDF	arXiv	DOI	Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDB	Taylor Dupuy, Kiran Kedlaya, Christelle Vincent	Arithmetic Geometry, Number Theory, and Computation, Simons Symp, Springer, 2021, pp 375-448
  PDF	arXiv	DOI	Tracking p-adic precision	Xavier Caruso, Tristan Vaccon	LMS Journal of Computation and Mathematics 17 (Special issue A) (2014), 274-294.

Courses

I have taught the following courses, and still maintain homepages for most of them.

The society of p-adicts was founded in 2012 by Clifton Cunningham and David Roe. You can find the current run of T-shirts here. Note that orders are only fulfilled if enough T-shirts are ordered within a two week period (see the counter at the bottom of the page). If you place an order but it's not fulfilled, you won't be charged. Gather together other p-adicts to make sure you can get a shirt!

Contact Info

The best way to reach me is by e-mail.

E-mail

roed@mit.edu

Mailing Address

Department of Mathematics
Room 2-336
Massachusetts Institute of Technology
Cambridge, MA 02139-4307