Simons Collaboration: Arithmetic Geometry, Number Theory, and Computation

Sato-Tate data: Andrew Sutherland

Principal investigators
Jennifer S. Balakrishnan, Boston University
Noam D. Elkies, Harvard University
Brendan Hassett, Brown Univeristy
Bjorn Poonen, MIT
Andrew V. Sutherland, MIT
John Voight, Dartmouth College

Scientific objectives
Our common perspective is that advances in computational techniques will drive research in arithmetic geometry and number theory, both as a source of data and examples, and as an impetus for effective results; and that tackling central mathematical problems requires a combination of different computational platforms and techniques.

The dynamic interplay between experiment, theory, and computation has played a pivotal role in the development of number theory. Systematic computations have driven progress in recent decades, from the Birch-Swinnerton-Dyer conjecture in the 1960s based on computations with curves of genus one to recent progress on the Sato-Tate conjecture in genus two.

The frontier of research is now studying curves of higher genus, abelian surfaces, and K3 surfaces. Here, however, the development and implementation of practical algorithms has lagged behind the theory, and we seek to correct this imbalance. Available computational resources have reached a point where algorithms are now technically feasible. In contrast to the situation with curves of low genus, brute force computation typically yields very little; to obtain practical algorithms one must exploit the theoretical infrastructure of modern arithmetic geometry.

Positions
We are hiring a number of mathematicians and programmers to support the work of the collaboration.
MIT has four openings: Mathjobs listing and HR posting
Brown/ICERM has one opening: Mathjobs listing

News and links
L-functions and modular forms database
ICERM Semester Program on Computational aspects of the Langlands program
Record-setting number theory computation on Google

Events
co-organized by Dr. Balakrishnan:
  Sage Days 87: p-adics in Sage and the LMFDB (July 17-22, 2017)
  Women in Numbers 4 (August 14-18, 2017)
  Sage Days 88: Opening workshop for a year of coding sprints at the IMA (August 21-25, 2017)
co-organized by Dr. Voight:
  LMFDB workshop (June 12-16, 2017)
Simons collaboration conference Arithmetic Geometry, Number Theory, and Computation at MIT (August 20-24, 2018)

We are grateful for the support of the Simons Foundation. See the Foundation's press release. See Brown University's press release.