for this workshop
Painlevé equations and their applications
American Institute of Mathematics, San Jose, California
Estelle Basor, Yang Chen, and Michael Rubinstein
This workshop, sponsored by AIM and the NSF, will bring together experts in random matrix theory and number theory to study the role of Painlevé 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 Painlevé equations in many settings is now well understood.
Several problems in number theory also have behaviors that are governed by Painlevé equations. Besides the spacing distribution of zeros of $L$-functions, the Painlevé 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 Painlevé 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 Painlevé VI equations 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 Painlevé 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.
The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
The deadline to apply for support to participate in this workshop has passed.
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