Buildings, spectral networks, and the Riemann-Hilbert correspondence at infinity
Dynamics on Moduli Spaces April 13, 2015 - April 17, 2015
Location: MSRI: Simons Auditorium
folations - leaves
universal covers of Riemann surface
Kontsevich-Soibelman deformation theory
Derived Algebraic Geometry
34E20 - Singular perturbations, turning point theory, WKB methods
34Mxx - Differential equations in the complex domain [See also 30Dxx, 32G34]
34M60 - Singular perturbation problems in the complex domain (complex WKB, turning points, steepest descent) [See also 34E20]
14B07 - Deformations of singularities [See also 14D15, 32S30]
14D15 - Formal methods; deformations [See also 13D10, 14B07, 32Gxx]
34M50 - Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.)
34M35 - Singularities, monodromy, local behavior of solutions, normal forms
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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