Zeros of random polynomials
August 12 to August 16, 2019
at the
American Institute of Mathematics,
San Jose, California
organized by
Norman Levenberg,
Doron Lubinsky,
Igor Pritsker,
and Maxim Yattselev
Original Announcement
This workshop will be devoted to the zero
distribution of random polynomials spanned by various deterministic bases. The prototypical classical example is the Kac polynomials, where the
coefficients are i.i.d. real (or complex) Gaussian random variables,
and the basis is given by standard monomials. Recent trends include studies of more general ensembles of random polynomials with nonGaussian coefficients that are spanned by various
polynomial bases, e.g., trigonometric and orthogonal polynomials.
The main topics for the workshop are
 Global asymptotic distribution of zeros: We plan to study the limiting measures
for the zero counting (empirical) measures, and provide the necessary and
sufficient conditions on the coefficients and the basis functions to guarantee
almost sure convergence. We also consider quantitative approaches to convergence
of the zero counting measures either through the large deviation methods or the
discrepancy results.

Intensity functions for real and complex zeros: It is of interest to develop
results on sharp asymptotics and explicit intensity functions for the zeros of
random polynomials spanned by various bases with general random coefficients. It
is known that different choices of bases and coefficients produce dramatically
different phenomena in terms of intensity functions and the asymptotic behavior
for the number of real and complex zeros.

Local asymptotic results on zeros: We aim at the local universality (scaling)
limits for the correlation functions of zeros of random polynomials here.
Several recent results (such as the replacement principle of Tao and Vu) for the
classical ensembles of random algebraic and trigonometric polynomials suggest
methods for addressing local universality questions for general random
orthogonal polynomials.
Material from the workshop
A list of participants.
The workshop schedule.
A report on the workshop activities.
A list of open problems.
Workshop Videos
Papers arising from the workshop: