Research

My PhD dissertation was titled Automorphic Forms in String Theory: From Moonshine to Wall Crossing.

My current research is in physical mathematics and lies on the interface of computation, number theory, algebraic geometry and applications to natural and applied sciences, mostly physics (quantum field theories).

The topics I work with include the theory of motives, Hodge theory, zeta/L-functions, automorphic forms, Galois theory, arithmetic statistics, differential equations, special and transcendental functions. I adopt the following research philosophy:

  1. Use physical insight to formally prove results in arithmetic geometry. This also means formalizing results in physics.
  2. Use tools in arithmetic geometry correctly and rigorously to study mathematical structures in physical theories.

Mathematics

  • Langlands reciprocity beyond \(GL_2\) and computational aspects of the Langlands program. This includes:
    • Computing automorphic and motivic L-functions
    • Arithmetic statistics and distributions of traces of Frobenius
    • Computational methods for Calabi-Yau varieties and motives, and their arithmetic
    • Theory of automorphic forms (Mock, Real Analytic, Siegel, Hilbert, Exceptional, Bianchi)
  • Theory of complex multiplication of abelian varieties, its generalizations to algebraic varieties

Physics

  • Arithmetic and motivic structures of Feynman integrals and string amplitudes
  • Mathematical structure of BPS attractor varieties
  • Construction and characterization of rational vertex operator algebras with moduli spaces
  • Enumerative geometry and phenomena like wall crossing, conifold transisitons, flops, jumping

Formalization and Benchmarks

One approach to understanding CM for motives is to formalize it. I am attempting to formalize the notion of complex multiplication for my own research purposes. I have learned a great deal from the 2025 Clay School on Formalizing Class Field Theory and the Mathlib repo mainitained by Kevin Buzzard.1

Footnotes

  1. Any progress on my part will be represented by a PR to this repo↩︎