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Quadratic differentials, WKB analysis, and cluster coordinates

Introductory Workshop: Holomorphic Differentials in Mathematics and Physics August 19, 2019 - August 23, 2019

August 22, 2019 (09:30 AM PDT - 10:30 AM PDT)
Speaker(s): Dylan Allegretti (University of Sheffield)
Location: MSRI: Simons Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC



The WKB method was originally introduced by Wentzel, Kramers, and Brillouin in 1926 as a way of finding approximate solutions of the Schrodinger equation in the semiclassical limit in quantum mechanics. The modern theory of WKB analysis is a refinement of this method which is deeply related to the theory of quadratic differentials and the associated spectral networks on Riemann surfaces. In this talk, I will review the notion of a Voros symbol from WKB analysis. Voros symbols are non-convergent formal series whose Borel sums define analytic functions under certain conditions. Recently, Iwaki and Nakanishi observed that the wall-crossing behavior of Voros symbols is governed by cluster transformations. I will present an extension of their result, which says that in fact the Borel sums of Voros symbols arise naturally as cluster coordinates on certain moduli spaces.

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H.264 Video 895_27252_7876_11-Allegretti.mp4
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