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

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.

## On the Number of Hamilton Cycles in Psdueo-Random Graphs

A pseudo-random graph is a graph G resembling a typical random graph of the same edge density. Pseudo-random graphs are expected naturally to share many properties of their random counterparts. In particular, many of their enumerative properties should be similar to those of random graphs.

## How to Construct Topological Invariants via Decompositions and the Symplectic Category

## Rigidity of 3-Colorings of the d-Dimensional Discrete Torus

## The Universal Relation Between Exponents in First-Passage Percolation

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.

## On the Long-Time Behavior of 2-D Flows

## Riemannian Exponential Map on the Group of Volume-Preserving Diffeomorphisms

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

## On the Instability for 2D Fluids

For 2D Euler equation, we prove a double exponential lower bound on the vorticity gradient. We will also discus some further results on the singularity formation for other models.