Alexander Betts' webpage

About me

Research interests

I am interested in the study of Diophantine equations, i.e. the problem of determining integer or rational solutions to polynomial equations. The solutions to these equations turn out to be connected in profound and subtle ways to the shape of the space cut out by the equations, giving a very surprising connection between the worlds of the continuous (topology) and the discrete (integer solutions). My research is primarily concerned with better understanding this connection between topology and arithmetic, and using it to improve our tools for solving Diophantine equations in practice.

Research keywords: Chabauty–Kim method, Section Conjecture, fundamental groups in arithmetic topology

Recent research highlights

• With Martin Lüdtke and Theresa Kumpitsch, we showed that the Chabauty–Kim method can be used to prove instances of the "locally geometric" part of Grothendieck's Section Conjecture. We carried this out for Z[1/2]-points on the thrice-punctured line, in the process proving infinitely many examples of Kim's Conjecture. The preprint is here.
  
  
• 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, and the paper is to appear in the Annals of Mathematics.
  
  
• 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 paper 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.

What I'm thinking about now

• The Chabauty–Kim loci for a curve over the rational numbers always contain the rational points by construction, but in some cases they can also contain unexpected points defined over number fields. In a project with Jen Balakrishnan, I am currently trying to find good explanations for when these points appear by giving purely "algebraic" interpretations of certain Chabauty–Kim loci.
  
  
• The Ceresa cycle is an algebraic cycle on the Jacobian of a curve, whose associated cohomology classes control the extension structure in the depth 2 unipotent fundamental group. For a curve over Qp, the l-adic cohomology class always vanishes for weight reasons when l is different from p, but for l = p the story is expected to be quite different. With Wanlin Li, we are working on a project to prove ``generic'' non-triviality of these p-adic classes.
  
  
• In my thesis project, I gave an interpretation of local components of Néron–Tate heights on abelian varieties in terms of various kinds of fundamental groups. With Ishai Dan-Cohen I am currently working on making a "global" version of this theory, giving an interpretation of global heights in terms of motivic rational homotopy types. An eventual application should be a Manin–Dem'janenko-type theorem for motivic rational homotopy types of curves.

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.