Beijing Institute of Mathematical Sciences and Applications

A programmatic conjecture of Bourgain, Gamburd, and Sarnak states that, under suitable conditions, the orbit of a group action should contain infinitely many points with prime coordinates. Specializing to quadratic forms, this conjecture suggests that equations of the form $f(x_1,…,x_s )=t$ should admit infinitely many prime solutions, where $f$ is an integral indefinite quadratic form in $s$ variables and $t$ is an integer. In this talk, I will present results and key ideas toward this conjecture, including joint work with Sarnak and with Jiamin Li.

Studies of quantum chaos in the physics community during the 1980's have resulted in the belief that the "local statistics" of energy levels of quantum systems with chaotic classical limit are described by Random Matrix Theory. There are no rigorous results in this direction, the closest being our understanding of the zeros of the Riemann zeta function, and indeed there are known counterexamples. Nonetheless, we conjecture that the above is true in some generic sense.<br>After surveying the general picture, I will describe recent progress towards a particular case of this conjecture for random hyperbolic surfaces of large genus, when the particular statistic studied is the variance of the number of eigenvalues in a spectral window, the subject of a detailed conjecture of Sir Michael Berry. We do this firstly for generic hyperbolic surfaces chosen randomly with respect to the Weil-Petersson measure, addressing number variance, a Central Limit Theorem and ergodicity, (works with Igor Wigman, and by Marklof and Monk), and secondly we will also describe analogous results due to Frederic Naud (Paris), to Yotam Maoz (Tel Aviv), and to Julien Moy (Paris) for the uniform random cover model, where one fixes a compact negatively curved surface, and takes covers of degree $N$ chosen with uniform probability, as $N$ grows.

In this talk, we will discuss certain asymptotic behaviors of the regularized determinant $\log⁡\det(\Delta_X)$ of Laplacian on Weil-Petersson random hyperbolic surfaces. This is a joint work with Yuxin He.

In this talk, I will introduce the random wave conjecture for Maass forms on the modular surface, and present some results on the quantum unique ergodicity, the $L^4$ norm, and the cubic moment of Hecke-Maass cusp forms. I will also discuss the joint distribution of Maass forms. The proofs use the analytic theory of $L$-functions. (Based on joint work with Shenghao Hua and Liangxun Li.)

We prove an Erdos-Turan type inequality for compact Lie groups, which is a quantitative Weyl criterion for equidistribution of sequences of conjugacy classes on compact Lie groups. As a corollary, we get an effective version of Deligne’s equidistribution theorem. We explain this corollary by showing the equidistribution of angles for hyper Kloosterman sums. This is a joint work with Yuk-Kam Lau and Ping Xi.