Special day on nodal sets of random functions

We study two types of random Taylor series with independent complex Gaussian coefficients. These Taylor series are distinguished by the invariance of their zero sets with respect to isometries of the complex plane ('flat' model) or the unit disk ('hyperbolic'). In the flat case, we consider the very rare event where the number of zeros in a large disk is far from the expected value. We are interested in the probability of this event, and also in the conditional distribution of the zeros, given this event. For the hyperbolic model, we focus on the 'hole' event - when there are no zeros in a disk centered at the origin. We find asymptotic bounds for the logarithm of the probability of this event, as the radius of the disk goes to 1. The talk is based on joint works with J. Buckley, R. Peled and M. Sodin, and S. Ghosh.

A rational lemniscate is the level set of the modulus of a rational function. While sampling from an ensemble of random lemniscates that is invariant under rotations of the Riemann sphere, we study basic geometric and topological properties. For instance, what is the average (spherical) length of a random lemniscate? How many connected components are there and how are they arrangement in the plane? We will make these questions precise and address each of them. This is joint work with Antonio Lerario.

There are several questions about the zero set of Laplace eigenfunctions that have proved to be extremely hard to deal with and remain unsolved. Among these are the study of the size of the zero set, the study of the number of connected components, and the study of the topology of such components. A natural approach is to randomize the problem and ask the same questions for the zero sets of random linear combinations of eigenfunctions (known as monochromatic random waves). In this talk I will present some recent results in this direction related to the study of the topology and the nesting of the components of the zero sets of these monochromatic random waves. The results I'll present are based on joint works with Boris Hanin and Peter Sarnak.

We obtain a large deviations principle (in the space of probability measures on $\C$) for the empirical measure of zeroes of random polynomials with i.i.d. exponential coefficients. One of the key challenges here is the fact that the coefficients are a.s. all positive, which enforces a growing number of highly non-linear constraints on the locations of the zeroes. En route, we will discuss a recent characterization theorem of Bergweiler and Eremenko, and its application in the proof of our main theorem. Based on joint work with Ofer Zeitouni.