Research

Complete lists of my preprints and publications can be found on my arXiv page and on my MathSciNet author page (subscription required).

Under construction: research descriptions in progress

Algebraic points on curves

The arithmetic of a curve over a number field $k$ encapsulates all of its $\overline{k}$ points together with the action of the absolute Galois group. Points on curves can also be viewed as divisors, and this dual identity gives powerful geometric tools to approach the problem. My research in this direction aims to leverage these geometric tools to organize and perhaps even understand $\overline{k}$ points on a curve.

The article Isolated and parameterized points on curves, written jointly with Isabel Vogt, gives an introduction to these ideas with several examples.

Here are some talks I’ve given on these concepts:

Colloquium at CRM in April 2024
Workshop at ICERM in June 2025 video

and here are my and my students’ research papers in this direction:

Number fields generated by points in linear systems on curves, joint with Irmak Balçik, Stephanie Chan, and Yuan Liu
Superelliptic degree sets over Henselian fields, by Alex Galarraga and Alex Wang
Degrees of points on varieties over Henselian fields, joint with Brendan Creutz
On the level of modular curves that give rise to isolated j-invariants, joint with Abbey Bourdon, Ozlem Ejder, Yuan Liu and Frances Odumodu

The Brauer-Manin obstruction: capturing subgroups and behavior under extensions

In 2022, I gave a lecture series at the Park City Mathematics Institute (PCMI) Graduate Summer School: Number Theory Informed by Computation. The lecture series was titled: Rational points on varieties and the Brauer-Manin obstruction, and focused on (my perspective on) how the feedback loop between computation and theory currently manifests in the study of rational points and the Brauer-Manin obstruction. My lecture notes give an introduction to the Brauer-Manin obstruction, so-called “capturing” subgroups of the Brauer-Manin obstruction, and questions concerning the Brauer-Manin obstruction over extensions.

Here are my and my students’ research papers in this direction:

Capturing subgroups of the Brauer-Manin obstruction

Brauer-Manin obstructions requiring arbitrarily many Brauer classes, joint with Jennifer Berg, Carlo Pagano, Bjorn Poonen, Michael Stoll, Nicholas Triantafillou, and Isabel Vogt
The $d$-primary Brauer-Manin obstruction for curves, joint with Brendan Creutz and Felipe Voloch
Degree and the Brauer-Manin obstruction, joint with Brendan Creutz, with an appendix by Alexei Skorobogatov

The Brauer-Manin obstruction under extensions

On the Hasse principle for conic bundles over even extensions by Sam Roven and Alex Wang
Quartic del Pezzo surfaces without quadratic points, joint with Brendan Creutz
Quadratic points on intersections of two quadrics, joint with Brendan Creuz
Persistence of the Brauer-Manin obstruction on cubic surfaces, joint with Carlos Rivera

Rational points on surfaces

In 2015, I gave a lecture series at the Arizona Winter School about explicit computation of the Brauer group and the Brauer-Manin obstruction on surfaces. Lecture notes are available on the Arizona Winter School website.

Here are my research papers in this direction:

Insufficiency of the Brauer-Manin obstruction for Enriques surfaces, joint with Francesca Balestrieri, Jennifer Berg, Michelle Manes, and Jennifer Park
Unramified Brauer classes on cyclic covers of the projective plane, joint with Colin Ingalls, Andrew Obus, and Ekin Ozman
On Brauer groups of double covers of ruled surfaces, joint with Brendan Creutz
Two torsion in the Brauer group of a hyperelliptic curve, joint with Brendan Creutz
Vertical Brauer groups and del Pezzo surfaces of degree $$, joint with Anthony Várilly-Alvarado
Higher dimensional analogs of Châtelet surfaces, joint with Anthony Várilly-Alvarado
Failure of the Hasse principle for Enriques surfaces, joint with Anthony Várilly-Alvarado
Failure of the Hasse principle for Châtelet surfaces in characteristic $2$