Selected Papers

36. Canonical representations of surface groups
with Daniel Litt
Annals of Mathematics, arxiv version, notes

35. Geometric local systems on very general curves and isomonodromy
with Daniel Litt
Journal of the American Mathematical Society, arxiv version, unipotent monodromy version, notes

Preprints

34. The stable homology of Hurwitz modules and applications
with Ishan Levy
arxiv version, course notes

33. Homological stability for Hurwitz spaces and applications
with Ishan Levy
arxiv version, notes, course notes

32. The Cohen--Lenstra moments over function fields via the stable homology of non-splitting Hurwitz spaces
with Ishan Levy
arxiv version, notes, course notes

31. An alternate computation of the stable homology of dihedral group Hurwitz spaces
with Ishan Levy
arxiv version

30. Homological stability for generalized Hurwitz spaces and Selmer groups in quadratic twist families over function fields
with Jordan Ellenberg
arxiv version, notes

29. Finite braid group orbits on SL₂-character varieties
with Josh Lam and Daniel Litt
arxiv version

28. The locus of plane curves in the moduli stack of curves
arxiv version

Accepted Papers

27. Obstructions to applying the Baker--Bilu method for determining integral points on curves
with Bjorn Poonen
Accepted to American Journal of Mathematics, arxiv version, notes

26. Big monodromy for higher Prym representations
with Daniel Litt and Will Sawin
accepted to Geometry & Topology, Geometry & Topology, arxiv version

25. Low degree Hurwitz stacks in the Grothendieck ring
with Ravi Vakil and Melanie Matchett Wood,
Compositio Mathematica, arxiv version, notes

24. Invariance of the tame fundamental group under base change between algebraically closed fields,
Essential Number Theory, arxiv version.

23. Prill's problem
with Daniel Litt
Compositio Algebraic Geometry, arxiv version

22. Applications of the algebraic geometry of the Putman-Wieland conjecture
with Daniel Litt
Proceedings of the London Mathematical Society, arxiv version

21. An introduction to the algebraic geometry of the Putman-Wieland conjecture
with Daniel Litt
European Journal of Mathematics, arxiv version, notes

20. Stacky heights on elliptic curves in characteristic 3
Annales l'Institut Fourier, arxiv version, notes

19. A geometric approach to the Cohen-Lenstra heuristics
Journal de Théorie des Nombres de Bordeaux, arxiv version, longer version, notes for an algebro-geometric audience, summary for Oberwolfach algebraic geometry conference, summary for Oberwolfach number theory conference

18. A Fulton-Hansen theorem for almost homogeneous spaces
with János Kollár
Bollettino dell'Unione Matematica Italiana, arxiv version, prior version

17. The geometric distribution of Selmer groups of elliptic curves over function fields
with Tony Feng and Eric Rains
Mathematische Annalen, arxiv version, notes

16. The geometric average size of Selmer groups over function fields
Algebra & Number Theory, arxiv version, notes

15. The Torelli map restricted to the hyperelliptic locus
Transactions of the American Mathematical Society, arxiv version

14. Improved lower bounds for the number of fields with alternating Galois group
with Robert J. Lemke Oliver and Frank Thorne
Bulletin of the London Mathematical Society, arxiv version

13. An explicit Abelian surface with maximal Galois action
with Quinn Greicius
Experimental Mathematics, arxiv version

12. Surjectivity of Galois Representations in Rational Families of Abelian Varieties
with Ashvin Swaminathan, James Tao, and Yujie Xu
Algebra & Number Theory, arxiv version

11. Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians
with Ashvin Swaminathan, James Tao, and Yujie Xu
Indiana University Mathematics Journal, arxiv version

10. Lifting Subgroups of Symplectic Groups over ℤ/ℓℤ
with Ashvin Swaminathan, James Tao, and Yujie Xu
Research in Number Theory, arxiv version

9. The twisted algebraic Sato-Tate group of the curve y² = x⁸ - 14x⁴ + 1
with Sonny Arora, Victoria Cantoral-Farfan, Davide Lombardo, and Jackson S. Morrow
Mathematische Zeitschrift, arxiv version

8. Interpolation of Varieties of Minimal Degree
International Mathematics Research Notices, arxiv version

7. Interpolation Problems: Del Pezzo Surfaces
with Anand Patel
Annali della Scuola Normale Superiore di Pisa, arxiv version

6. Spin Canonical Rings of Log Stacky Curves
with Peter Ruhm and Robin Zhang
Annales de l'Insitut Fourier, arxiv version

5. Generalized Canonical rings of ℚ-divisors on Minimal Rational Surfaces
with Peter Ruhm and Robin Zhang
Mathematical Research Letters, arxiv version

4. A New Proof of Serre's Homological Characterization of Regular Local Rings
with Ravi Jagadeesan
Research in Number Theory

3. Peckness of Edge Posets
with David Hemminger and Zijian Yao
Order, arxiv version, expanded REU version

2. Artin Group Presentations Arising from Cluster Algebras
with Jacob Haley, David Hemminger and Hailee Peck
Algebras and Representation Theory, arxiv version

1. Proof of Stasinski and Voll's Hyperoctahedral Group Conjecture
Australasian Journal of Combinatorics, arxiv version

Videos

A video explaining the classification of finite fields in the little-known language of twelve halves tongue, where one can only use six letter words.

A talk from Nick Katz's 80th birthday conference on "Ranks of abelian varieties in quadratic twist families and monodromy."

An online talk on Malle's conjecture and Bhargava's conjecture for the Arithmetic & Homotopic Galois Theory seminar

A talk from the Princeton/IAS Number theory seminar on "The Cohen-Lenstra moments over function fields."

A talk on Malle's conjecture and the Picard rank conjecture from the conference "Monodromy in Chicago" at University of Chicago.

A colloquium I gave at UCLA on canonical representations.

A talk from the CMSA algebraic geometry in string theory seminar entitled "Geometric local systems on very general curves"

A talk on "The Grothendieck ring of varieties, and stabilization in the algebro-geometric setting" at PIMS. This is the second part of a two-part lecture series; the first lecture was given by Ravi Vakil.

A talk on "The average size of 2-Selmer groups over function fields" at ICTP.

A talk on "The Torelli map restricted to the hyperelliptic locus" at the Stanford algebraic geometry seminar.

Other Research

Notes for a topics course I taught at Harvard in Fall 2025 on arithmetic statistics over function fields. They focus on explaining progress toward the Cohen-Lenstra heuristics, Malle's conjecture, and the Poonen-Rains heuristics. They explain the main ideas of my work with Ishan Levy including work coming before ours, homological stability inputs, and consequences toward the above mentioned conjectures.

An explanation of why an infinite product of zeta values appears both in Bhargava's conjecture on the number of S_d degree d number fields and in the stable homology of the moduli space of curves.

An investigation of the interplay between the Poonen-Rains heuristics for Selmer groups and the Darda-Yasuda heuristics for heights on stacks.

My PhD dissertation, "A thesis of minimal degree: two" focusing on geometric parameterizations of moduli spaces relating to the Cohen-Lenstra heuristics.

A verification that Poincaré duality holds in the particular example of the projection map from a punctured affine plane to an affine line.

Properness of the Torelli map on compact type curves, a construction of the Torelli map on compact type curves and verification that it is proper.

"The Infinitesimal Torelli Problem," an identification of the Torelli map on tangent spaces as dual to the multiplication map on differential forms, available on arxiv.

A proof of the Shioda-Tate formula over finite fields, a brief proof of the Shioda-Tate formula for non-constant elliptic surfaces over finite fields, using the theory of Neron models.

A coincidence between genus 1 curves and canonical curves of low degree, an investigation of a curious coincidence between models of genus 1 curves and canonical curves in low degree.

My senior thesis, "Interpolation in Algebraic Geometry," available on arxiv. This is an expanded version of the two papers "Interpolation of Varieties of Minimal Degree" and "Interpolation Problems: Del Pezzo Surfaces." The last chapter is cowritten with Anand Patel. Here is a picture with my advisers.