#
Painleve equations and their applications

February 6 to February 10, 2017
at the

American Institute of Mathematics,
San Jose, California

organized by

Estelle Basor,
Yang Chen,
and Michael Rubinstein

## Original Announcement

This workshop will bring together experts in random matrix theory and number theory to study
the role of Painleve equations in number theory, random matrix theory, and the
many interesting connections between the two theories.
Random matrix theory has provided a rich set of statistical
results and tools for analyzing the eigenvalues and value distribution of
characteristic polynomials of Hermitian matrices and related statistics of
matrices in the classical compact groups and the connections to Painleve
equations in many settings is now well understood.

Several problems in number theory also have behaviors that are governed
by Painleve equations. Besides the spacing distribution of zeros of
$L$-functions, the
Painleve arise in the asymptotics of the moments of the derivative of the
Riemann zeta function, and in the moments of the average of the $k$-th divisor
function in short intervals. One goal at the workshop would be to bring
techniques from the world of Painleve to better understand the behavior of
these asymptotic formulas, for example to obtain, in the latter problem,
uniform asymptotics for the constant factor (as a function say of $k$ and the
size of the interval) in the moments. Another area worth exploring is the
appearance of the Chazy equation, a third order non-linear differential
equation, in relation to the Eisenstein series $E_2$. Finally, Bourgain,
Gamburd, and
Sarnak have found connections between the non-linear affine sieve and the
classification of
Painleve VI equation that have finite monodromy.

Other questions of interest involve describing statistical quantities that exist
more directly in random matrix theory at the level of finite, fixed size
matrices. For example, finding expressions for determinants of non-classical
Hankel matrices that depend on parameters, (or with perturbed weights) are
related to Painleve equations. This problem is in turn related to the study of
gap probability for finite ensembles of matrices. In this setting, a double
scaling limit is the quantity of interest. Recent results have used a 'ladder
technique' to find such expressions. We hope to discover how to make this
technique as systematic and transparent as possible.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.

A list of open problems.