Research

I would categorize my research as “physical and computational mathematics”, broadly on the interface of number theory, geometry, physics and computation, mostly from an analytic and arithmetic perspective. My PhD dissertation was on Automorphic Forms in String Theory.

During my PhD and a few years that followed, I worked on problems in combinatorial enumerative geometry of CY manifolds and K3 surfaces, wall crossing, automorphic forms (Siegel, Jacobi, Mock, Hilbert), mirror symmetry with applications to black holes in string theory and vertex operator algebras.

In pursuing this line of research, one encounters strong hints of the interplay between arithmetic geometry (motives, Hodge theory, zeta functions, Shimura varieties), harmonic analysis (automorphic forms), number theory (L-functions), Galois theory (equi-distribution, complex multiplication), special and transcendental functions, and physics, thereby placing placing several problems in physics within the scope of the Langlands program. But computing these relations is not as straightforward.

To study these problems, one needs a blend of rigorous mathematics, technical computational tools and physical insight/motivation. In particular this requires developing tools/ideas in:

  • Computational aspects of the Langlands program. This includes:
    • Computing of automorphic and analytic L functions and zeta functions
    • Arithmetic statistics and distributions of traces of Frobenius (Sato-Tate, Lang-Trotter)
    • 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

which in turn can be applied to study physics problems such as:

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