Buildings, spectral networks, and the Riemann-Hilbert correspondence at infinity
Dynamics on Moduli Spaces April 13, 2015 - April 17, 2015
Location: SLMath: Eisenbud Auditorium
folations - leaves
universal covers of Riemann surface
Kontsevich-Soibelman deformation theory
Derived Algebraic Geometry
wall-crossing formula
stability conditions
Fukaya category
WKB theory
34C20 - Transformation and reduction of ordinary differential equations and systems, normal forms
34C60 - Qualitative investigation and simulation of ordinary differential equation models
14C21 - Pencils, nets, webs in algebraic geometry [See also 53A60]
34C45 - Invariant manifolds for ordinary differential equations
34C37 - Homoclinic and heteroclinic solutions to ordinary differential equations
34C29 - Averaging method for ordinary differential equations
14220
I will describe joint work with Katzarkov, Noll, and Simpson, which introduces the notion of a versal harmonic map to a building associated with a given spectral cover of a Riemann surface, generalizing to higher rank the leaf space of the foliation defined by a quadratic differential. A motivating goal is to develop a geometric framework for studying spectral networks that affords a new perspective on their role in the theory of Bridgeland stability structures and the WKB theory of differential equations depending on a small parameter. This talk will focus on the WKB aspect: I will discuss the sense in which the asymptotic behavior of the Riemann-Hilbert correspondence is governed by versal harmonic maps to buildings.
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