Activities This Week

The group of diffeomorphisms acts on the space of symplectic structures on a given manifold. Taking a quotient by isotopies, we obtain the mapping class group action on the Teichmuller space of symplectic structures; the latter is a finite-dimensional manifold. The mapping class group action on the Teichmuller space is quite often ergodic, which leads to important consequences for symplectic invariants, such as symplectic packing problems. I would describe some of the problems which were solved using this approach. This is a joint work with Michael Entov.

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.