Writing

Research

• A converse to geometric Manin's conjecture for general low degree hypersurfaces: in preparation!
• Terminal singularities of the moduli space of curves on low degree hypersurfaces and the circle method (with Jakob Glas):

  Shows the moduli space of (arbitrary genus) curves on a smooth low-degree hypersurface has at worst terminal singularities by using a geometric interpretation of the circle method to count rational points on iterated jet schemes.

• Non-smoothness of moduli spaces of higher genus curves on low degree hypersurfaces (with Amal Mattoo):

  Proves that moduli spaces of smooth higher genus curves on smooth low degree hypersurfaces are essentially always singular (unless the hypersurface is a linear subspace, in which case we show the moduli spaces are always smooth).

• A geometric approach to functional equations for general multiple Dirichlet series over function fields:

  Proves that Sawin's general multiple Dirichlet series over function fields are analytic and establishes a few functional equations. The former uses the decomposition theorem and general bounds for the cohomology groups of lisse sheaves on compactifications of configuration spaces, and the latter extends one of Sawin's examples employing a density trick with simple perverse sheaves.

• A higher genus circle method and an application to geometric Manin's conjecture:

  Shows the moduli space of (arbitrary genus) curves on a smooth low-degree hypersurface is irreducible of the expected dimension by geometrically re-interpreting the Browning-Vishe circle method strategy. Applies this to obtain a converse of geometric Manin's conjecture.

• Sum-product phenomena for planar hypercomplex numbers:

  Proves a sum-product bound for dual numbers and double numbers by extending Elekes's original strategy for complex numbers in a bootstrapping manner.

Expository

• Notes on geometric aspects of multiple Dirichlet series:

  Talk notes for the Philadelphia area NT seminar on analyticity and functional equations for Sawin's perverse sheaf-theoretic interpretation of multiple Dirichlet series.

• Notes on a geometric interpretation of the circle method applied to moduli spaces:

  Talk notes for the Harvard-MIT AG seminar on an application of the circle method to moduli spaces of curves on hypersurfaces.

• Notes on Artin-Verdier duality:

  Elementary notes on Artin-Verdier duality that should be pretty readable; I'd recommend reading the original sources though.