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Number theory, automorphic forms, and mathematical physics
My research is in number theory and related areas. One can think of automorphic forms as quantum states and I find it fascinating that number theory is at all related to quantum mechanics.
A major theme in my work has been to show instances of randomness in number theory. A guiding principle is that when an answer to a given problem is difficult (e.g. finding prime numbers), it exhibits randomness according to some probabilistic law (e.g. prime numbers seem to distribute according to the Poisson process).
Families of automorphic representations are central to the resolution of certain algebraic and asymptotic questions. I am interested in the arithmetic statistics of families, the Katz-Sarnak heuristics, arithmetic quantum chaos and p-adic families of automorphic forms.
I have been also working on the asymptotic properties of special functions motivated by periods and central values. Symplectic geometry and integrable systems provide the relevant tools to investigate the Bessel, Airy and hypergeometric functions and their generalizations notably arising from branching laws of Lie groups representations. In this direction I recently proved the mirror symmetry conjecture for minuscule flag varieties.
- Families of L-functions and their symmetry (with P. Sarnak and S.W. Shin), proceedings of Simons symposium on the trace formula, Springer-Verlag (2016).
- Sato-Tate theorem for families and low-lying zeros of automorphic L-functions (with S.W. Shin), Invent. Math., 203 no.1 (2016), 1-177.
- Hybrid sup-norm bounds for Hecke-Maass cusp forms, J. Eur. Math. Soc. 17 no.8 (2015), 2069--2082.
- Large values of modular forms, Cambridge Journal of Mathematics, 2 no.1 (2014), 91–116.
- A non-split sum of coefficients of modular forms, Duke Math. J., 157 no.1 (2011), 109–165.