Alexander Betts
Postdoctoral Researcher in Mathematics

Contact details

Postal address:
Department of Mathematics
Harvard University
1 Oxford Street
Cambridge, MA 02138

Email: abetts[at]math[dot]harvard[dot]edu

About me

Research interests

I am a postdoc at Harvard University, specialising in arithmetic and anabelian geometry. The main kinds of questions I'm interested involve relating fundamental groups of varieties to their arithmetic properties: heights, reduction types, and most importantly their rational points.

Recent research highlights

  • With Jakob Stix, we proved a partial finiteness theorem for Grothendieck's section set. What is particularly exciting about this result is that it says something about the section set for every curve of genus at least 2, whereas the section set was previously only understood for certain very special curves. The preprint is here.
  • I developed an ``effective'' version of the Chabauty--Kim method, using fundamental groups to give explicit upper bounds on the numbers of rational points on curves. This extends previous work of Coleman and Balakrishnan--Dogra, who had done this for abelian and quadratic Chabauty. The conceptual breakthrough that enabled this generalisation was identifying and exploiting a new structure, the weight filtration, on Selmer schemes. The preprint is here, and some follow-on work on making these bounds explicit under standard conjectures can be found here.
  • With the project group I led at the 2020 Arizona Winter School, we carried out the first explicit calculations using the refined Chabauty--Kim method I developed with Netan Dogra. Specifically, we computed solutions to the S-unit equation for S a set of two prime numbers. Previously, the only cases where this had been accessible to Chabauty--Kim techniques were when S consisted of a single prime. The preprint is here.
  • With Netan Dogra, we proved a far-reaching generalisation of Oda's Criterion, asserting that the stable reduction type of a hyperbolic curve over a finite extension of Qp is determined by its pro-unipotent étale fundamental groupoid. As an application, we produced a refinement of the Chabauty--Kim method, allowing one to compute S-integral points on affine curves in cases where the usual Chabauty--Kim method fails. The preprint is here.
  • With Daniel Litt, we proved several foundational properties of the Galois action on pro-unipotent étale fundamental groupoids, which are analogues of well-known properties of the first étale cohomology of varieties. This allowed us to study the geometry of non-abelian Bloch--Kato Selmer sets outside the case of good reduction, and can be viewed as a first step towards a ``Chabauty--Kim at primes of bad reduction''. The preprint is here.

What I'm thinking about now

  • Developing an analogue of the Weil height machine for motivic rational homotopy types.
  • Using the Chabauty--Kim method to study the section set.
  • Computing local heights on curves for use in quadratic Chabauty computations.

Math 283Z: Foundations of non-abelian Chabauty

I am currently teaching a course in Harvard (Spring 2023) on the non-abelian Chabauty method. The course page can be found here.

Outreach with UKMT

When I was in the UK, I also did a lot of volunteering for the United Kingdom Mathematics Trust, a charitable organisation providing mathematics enrichment for bright under-18s. As well as helping set and mark the British Mathematical Olympiads, I regularly taught at various summer schools and sometimes took teams of students abroad to international competitions.

You can find out more about the UKMT on their website, or the Olympiad subtrust website.