Arithmetic quantum chaos and random wave conjecture
A fundamental problem in the area of quantum chaos is to understand the distribution of large frequency eigenfunctions of the Laplacian on certain negatively curved Riemannian manifolds. Arithmetic quantum chaos refers to quantum systems that have arithmetic structure and so, are of interest to both number theorists and mathematical physicists. Such examples arise as hyperbolic surfaces obtained as quotients of the upper half-plane by a discrete arithmetic subgroup of SL(2,R). The random wave conjecture says that an eigenform of large Laplacian eigenvalue (which is also a joint eigenform of all Hecke operators) should behave like a random wave, that is, its distribution should be Gaussian. In this talk in particular we focus on this conjecture in the case of Eisenstein series. This is based on the joint work with Rizwanur Khan.