Brian Lawrence


Department of Mathematics
UCLA

I am currently at UCLA. Previously I was at UChicago, and before that Columbia, and before that I was a graduate student at Stanford. My advisor was Akshay Venkatesh (now here).

I'm interested in arithmetic geometry.

Papers

Sparsity of integral points on moduli spaces of varieties (with Jordan Ellenberg and Akshay Venkatesh, 19 pages) Integral points are sparse on interesting moduli spaces. More precisely, if X is any variety over a number field admitting a variation of Hodge structure whose associated period map is finite-to-one, then the number of S-integral points of X of height at most H grows slower than any positive power of H. (A general Shafarevich-type conjecture would predict that there are only finitely many S-integral points.) The main input to the proof is a point-counting result of Broberg, combined with a passage to large etale covers to increase the degree of X.

The Shafarevich conjecture for hypersurfaces in abelian varieties (with Will Sawin, submitted; 111 pages). This paper builds on the methods of my paper with Akshay Venkatesh to prove a Shafarevich-type result for hypersurfaces in abelian varieties. The key new input is perverse sheaf techniques, based on work of Kramer and Weissauer. Specifically, we use a sheaf convolution Tannakian category to prove a uniform big monodromy result.

Diophantine problems and p-adic period mappings (with Akshay Venkatesh; appeared in Inventiones; 76 pages). A new proof of Mordell's conjecture (finiteness of rational points on a curve of genus at least 2). The idea is to use p-adic Hodge theory to study how Galois representations vary p-adically in a family. Mordell's conjecture is proved by considering a family of abelian varieties, over the given curve as base. This work differs from previous proofs in that our methods don't use specific features of abelian varieties; our methods could potentially apply to more general families of higher-dimensional varieties. The paper concludes with a result on families of hypersurfaces in projective space.

Representations of surface groups with universally finite mapping class group orbit (with Daniel Litt, to appear in Math. Research Letters; 13 pages). Gives a condition for a representation of a surface group to have finite image, in terms of the action of the mapping class group on representations. The result is purely topological, though it is motivated by the Grothendieck-Katz p-curvature conjecture in arithmetic geometry.

A counterexample to an optimistic guess about étale local systems (with Shizhang Li, to appear in Comptes Rendus; 1 page). A counterexample in relative p-adic Hodge theory. It is known that, for p-adic Galois representations, de Rham implies potentially semistable; one expects that the same is true of étale local systems, for a suitable definition of "potentially semistable". We show that this is not true for a particularly restrictive definition of "potentially semistable", answering a question of Liu and Zhu.

Two recent p-adic approaches towards the (effective) Mordell conjecture (with Jennifer Balakrishnan, Alex Best, Francesca Bianchi, Steffen Müller, Nicholas Triantafillou, and Jan Vonk; 34 pages). A survey of recent work on the Mordell conjecture, including Minhyong Kim's nonabelian Chabauty program and my work with Akshay Venkatesh.

On the p-adic distribution of torsion values for a section of an abelian scheme (with Umberto Zannier; appeared in Rendiconti Lincei; 8 pages). A discreteness result for the torsion values of a section of an abelian scheme, in the p-adic topology.

A density result for real hyperelliptic curves (appeared in Comptes rendus). I have recently learned that Andrey Bogatyrev proved this result some years ago. See Theorem 5 and Lemma 1 of this paper (Effective computation of Chebyshev polynomials for several intervals, 1999, Sbornik: Mathematics, vol. 190, no. 11).

Teaching and Student Work

UChicago Math 203: Analysis in R^n, regular track. An introduction to metric spaces. Handouts, quizzes, and exams from Winter 2019.

Yuchen Chen (a UChicago undergrad) wrote a proof of the Hodge decomposition theorem on a Riemannian manifold. (Note that no complex structure is needed to define the Hodge decomposition: this form of the Hodge decomposition theorem only requires a Riemannian metric on a real manifold.) (2020)

Anuj Sakarda (a high-school student) has written a proof of the theorem on symmetric polynomials. (2020)

Spencer Dembner (a UChicago undergrad) has written some notes on the relationship between Galois groups and branched covers of Riemann surfaces. (2019)

Seminars, Past and Present

UChicago Number Theory Seminar (organized with Lue Pan, 2019-2020)

Crystalline Cohomology (organized with Shizhang Li, 2018)

Lecture Notes, etc.

Some expository notes I wrote as a student.

Here are some notes from an introductory talk on schemes.

Here are some notes on polynomials on lattices.

Fibonacci Numbers mod p (Stanford SUMO talk, Apr. 10, 2014)

Introduction to Heights (Arithmetic Dynamics talk, Apr. 29, 2014)

In 2013-2014 I ran Stanford's Student Algebraic Geometry Seminar.