数学研究所

中国科学院华罗庚数学重点实验室

学术报告

多复变与复几何研讨班

SCV&CG Seminar

Speaker: Aron Wennman (KU Leuven, Belgium)

Title: Asymptotics and zeros of orthogonal polynomials in the plane

Time&Venue: 2025.1.18（星期六）15:00-16:00 & Zoom会议号: 844 8633 9906 密码: 097357

Abstract: In this talk I plan to describe some old and new results on orthogonal polynomials with respect to (weighted) area measure on Jordan domains and on the plane. The study of their large-degree asymptotics is a classical topic pioneered by Carleman in the 1920’s, and these questions have recently received increased attention due to connections to 2D Coulomb gas and random matrix models. After outlining the Coulomb gas connection, I will focus on two recent approaches. The first is PDE-based and shares some resemblance with the Riemann-Hilbert approach to orthogonal polynomials on the real line, and is suited to the setting of varying exponential weights. The second approach is adapted to the unweighted setting on a Jordan domain, and is loosely speaking based on extracting information about the orthogonal polynomials from the associated Bergman kernel, which is known explicitly in terms of a conformal map. The main result is a proof of a curious dichotomy exhibited by the zeros of orthogonal polynomials for Jordan domains with corners, going back to observations by Eiermann and Stahl. The talk is based on joint works with Erwin Miña-Díaz (University of Mississippi), and with Haakan Hedenmalm (KTH).