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WKB analysis and hypergeometric differential equation
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.