Ribbon graphs capture the topology of open Riemann surfaces in an elementary combinatorial form. One can hope this is the first step toward a general theory for open symplectic manifolds such as Stein manifolds. We will discuss progress toward such a higher dimensional theory (joint work with Alvarez-Gavela, Eliashberg, and Starkston), and in particular, what kind of topological spaces might generalize graphs. We will also discuss applications to the calculation of symplectic invariants.
is expected to behave like a random monochromatic wave .
We will discuss this in connection with the question of the nodal
domains of such forms on arithmetic hyperbolic surfaces with a reflection symmetry .
( Joint work with A.Ghosh and A.Reznikov we will also discuss a recent result of
J.Jang and J.Jung ) .
study periods of automorphic forms of unitary groups that show
up in Ichino-Ikeda conjecture. In this talk, I will report on the
present state of the Jacquet-Rallis trace formula. Then I will
discuss the problem of the spectral expansion. (joint work with Michal
simple algebras with an involution of second kind. We study some local questions arising from the relative trace formula approach.
we proved that a certain generating series for the classes of arithmetic divisors on a regular integral model M of a Shimura variety
for a unitary group of signature (n-1,1) for an imaginary quadratic field is a modular form of weight n valued in the
first arithmetic Chow group of M. I will discuss how this is proved, highlighting the main steps.
Key ingredients include information about the divisors of Borcherds forms on the integral model
Boolean function analysis traditionally studies Boolean functions on the Boolean cube, using Fourier analysis on the group Z_2^n. Other domains of interest include the biased Boolean cube, other abelian groups, and Gaussian space. In all cases, the focus is on results which are independent of the dimension.