Research

My research lies broadly in physical mathematics, on the interface of computation, number theory, algebraic geometry and drawing insight from physics (quantum field theories and string theory) as well as mathematical physics (applications of number theory and algebraic geometry to quantum field theories and string theory).

I am mostly interested in arithmetic and computational aspects of the Langlands program beyond \(GL_2\). 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. (An interactive, formalized textbook is underway)

with inspiration drawn from/applications to the following topics in 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 transitions, flops, jumping

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