Research

I would categorize my research as “physical mathematics”, broadly on the interface of number theory, geometry, physics and computation, mostly from an analytic and arithmetic perspective. I am interested in how the notion of physics changes if you change underlying mathematical (arithmetic, geometric, topological) structures. Conversely, I am interested in using physical intuition to understand how mathematical objects behave when their underlying over structures are changed.

I have worked on problems in enumerative geometry of CY manifolds and K3 surfaces, Moonshine, vertex operator algebras, BPS state counts and wall crossing and Hodge theory of rational superconformal field theories.

In particular, I am intersted in the theory of complex multiplication of abelian varieties, its generalizations to algebraic varieties and motives, zeta functions, computational methods, and how these toptics arise in physics (rationality of vertex operator algebras, attractor mechanism for black holes and Feynman integrals). I am also interested in how these structures are tied to random matrix theories and arithmetic distributions of their zeta functions. So I look at interdisciplinary mathematical problems in the following areas:

• Computational aspects of the Langlands program
• Computing of automorphic and analytic L functions
• Theory of automorphic forms (Mock, Real Analytic, Siegel, Hilbert, Exceptional, Bianchi)
• Arithmetic statistics of algebraic varieties
• Computational methods of Calabi-Yau geometry and arithmetic

and how can be applied to/intuitively studied from the following areas of physics and computing:

• Mathematical structure of attractor varieties
• Construction and characterization of rational conformal field theories with moduli spaces
• Enumerative BPS invariants in string theory and associated phenomena like wall crossing, conifold transisitons, flops, jumping
• Arithmetic and motivic structures underlying Feynman integrals
• Cryptography