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WKB analysis and hypergeometric differential equation
Exact WKB analysis of the Pearcey system
Exact WKB analysis of the Pearcey system
Organizers
Speaker
Time
Saturday, November 29, 2025 2:00 PM - 4:00 PM
Venue
A3-3-301
Online
Zoom 442 374 5045
(BIMSA)
Abstract
In this talk we use the Pearcey integral as a model example to illustrate exact WKB analysis in a genuinely multivariable setting. we first derive the holonomic system of partial differential equations (the Pearcey system) that it satisfies, and explain how this system defines a rank–three D-module. Treating \eta as a large parameter, we then introduce logarithmic derivatives S and T, obtain a Hamilton–Jacobi–type nonlinear system, and construct WKB-type formal solutions. The leading terms of these solutions are governed by algebraic branches of a cubic equation, whose discriminant describes the turning point set, while the differences of the corresponding phase integrals define the Stokes set in the (x_1,x_2)-plane. Passing to the Borel transform in the WKB parameter, we show that the Borel transforms of the Pearcey WKB solutions satisfy a polynomial equation in the Borel variable, and hence are algebraic functions with finitely many branch points. This yields Borel summability and a clear resurgent structure, and makes it possible to interpret Stokes phenomena in terms of discontinuities across branch cuts in the Borel plane. Overall, the Pearcey system provides a concrete prototype for the synthesis of oscillatory integrals, holonomic PDEs, WKB theory, and Borel–Laplace analysis in several variables.
Speaker Intro
Xinxing Tang, received a bachelor's degree in basic mathematics from the School of Mathematics, Sichuan University in 2013, and received a doctorate from Beijing International Center for Mathematical Research, Peking University in 2018. From 2018 to 2021, she worked as a postdoctoral fellow at the Yau Mathematical Sciences Center, Tsinghua University, and joined Beijing Institute of Mathematical Sciences and Applications in 2021 as assistant professor. Research interests include: integrable systems, especially infinite-dimensional integrable systems that appear in GW theory and LG theory, and are interested in understanding the algebraic structure of infinite symmetries and related calculations. Other interests include: mixed Hodge structures, isomonodromic deformation theory, KZ equations.