On Real Zeros of Holomorphic Hecke Cusp Forms and Sieving Short Intervals

Kaisa Matomäki
University of Turku, Finland
October 13, 2011

A. Ghosh and P. Sarnak have recently initiated the study of so-called real zeros of holomorphic Hecke cusp forms, that is zeros on certain geodesic segments on which the cusp form (or a multiple of it) takes real values. In the talk I'll first introduce the problem and outline their argument that many such zeros exist if many short intervals contain numbers whose all prime factors belong to a certain subset of primes. Then I'll speak about new results on this sieving problem which lead to improved lower bounds for the number of real zeros.

The Universal Relation Between Exponents in First-Passage Percolation

Sourav Chatterjee
Courant Institute; NYU
October 18, 2011

It has been conjectured in numerous physics papers that in ordinary first-passage percolation on integer lattices, the fluctuation exponent \chi and the wandering exponent \xi are related through the universal relation \chi=2\xi -1, irrespective of the dimension. This is sometimes called the KPZ relation between the two exponents. I will give a rigorous proof of this conjecture assuming that the exponents exist in a certain sense.


Riemannian Exponential Map on the Group of Volume-Preserving Diffeomorphisms

Gerard Misiolek
University of Notre Dame; Institute for Advanced Study
October 19, 2011

In 1966 V. Arnold showed how solutions of the Euler equations of hydrodynamics can be viewed as geodesics in the group of volume-preserving diffeomorphisms. This provided a motivation to study the geometry of this group equipped with the \(L^2\) metric. I will describe some recent work on the structure of singularities of the associated exponential map and related results