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
I will be on the job market for tenure-track positions starting in 2024. If you might be interested in having me apply to a position, please do let me know! My CV can be found here.
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 topologyRecent 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.
- 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. I am currently trying to find good explanations for when these points appear by giving purely "algebraic" interpretations of certain Chabauty–Kim loci.
- In explicit computations of rational points using quadratic Chabauty, one wants to compute the values attained by certain local height functions at primes of bad reduction. Most calculations to date have been limited to cases where one can guarantee that all these local heights are zero; I am working with Sachi Hashimoto, Juanita Duque Rosero and Pim Spelier to develop algorithms to compute these local heights in general for hyperelliptic curves.
- 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.