# Quadratic differentials, WKB analysis, and cluster coordinates

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

**Speaker(s):**Dylan Allegretti (University of Sheffield)

**Location:**SLMath: Eisenbud Auditorium

**Primary Mathematics Subject Classification**No Primary AMS MSC

**Secondary Mathematics Subject Classification**No Secondary AMS MSC

#### 11-Allegretti

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.

#### 11-Allegretti

H.264 Video | 895_27252_7876_11-Allegretti.mp4 |

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