Welcome to the Special day on Nodal sets of random functions
This workshop is part of the 2015/2016 emphasis year in probability theory. No registration is needed.
Invited Speakers
 Yaiza Canzani (Harvard)
 Subhro Ghosh (Princeton)
 Erik Lundberg (Florida Atlantic Univ.)
 Alon Nishry (Univ. of Michigan)
Schedule
All talks in Lunt Hall, Room 105
Saturday, April 2nd:
9.30am  10.00am

Coffee

10.00am  11.00am

Alon Nishry (Univ. of Michigan)
Large fluctuations in the number of zeros for Gaussian Taylor series
 Abstract
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.

11.30am  12.30pm

Eric Lundberg (Florida Atlantic University)
On the geometry of random lemniscates.  Abstract
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.

2.30pm  3.30pm

Yaiza Canzani (Harvard University)
Structure of the zero set of monochromatic random waves  Abstract
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.

4.00pm  5.00pm

Subhro Ghosh (Princeton University)
Large deviations and random polynomials  Abstract
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 nonlinear
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.

Organizers
The scientific organizers are Antonio Aufffinger, Elton Hsu and Steve Zelditch.
Acknowledgements
This meeting is partially supported by a grant from the National Science Foundation to the probability group at Northwestern University
and by the Northwestern Mathematics Department as part of the 2015/2016 emphasis year in probability theory.