Activities This Week

Let $\Gamma=G_1*G_2*\dots *G_r$ be a free product of a finite number of finite groups and a finite number of copies of the infinite cyclic group. We sample uniformly at random an action of $\Gamma$ on $N$ elements. In this talk, we will discuss a few tools we developed to help answer some natural questions involving the configuration described above, such as: For $\gamma\in \Gamma$, what is the expected number of fixed points of $\gamma$ in the action we sampled? What is the the typical behavior of the cycle structure of the permutation corresponding to $\gamma$ etc.

This is a joint with Doron Puder.

The Gross-Zagier formula equates (up to an explicit non-zero constant) the central value of the first derivative of the Rankin-Selberg L-function of a weight 2 eigenform and the theta series of a class group character of an imaginary quadratic field (satisfying the Heegner hypothesis) with the height of a Heegner point on the corresponding modular curve. This equality is a key ingredient in the proof of the Birch and Swinnerton-Dyer conjecture for elliptic curves over the rationals in analytic rank 0 and 1. Two important generalizations present themselves: to allow eigenforms of higher weight, and to allow Hecke characters of infinite order. The former one is due to Shou-Wu Zhang. The latter one is the subject of a joint work in progress with Ari Shnidman and requires the calculation of the Beilinson-Bloch heights of generalized Heegner cycles. In this talk, I will report on the calculation of the archimedean local heights of these cycles.

I will discuss some of the interactions between number theory and the spectral theory of the Laplacian. Some have very classical background, such as the connection with lattice point problems. Others are newer, including connections with Random Matrix Theory, the zeros of the Riemann zeta function, and the work of Maryam Mirzakhani on the moduli space of hyperbolic surfaces.