Mathematics Colloquia and Seminars

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Let C be a compact Riemann surface. A holomorphic quadratic differential on C determines a spectral curve S over C. Given a holomorphic quadratic differential, we can associate two types of differential operators. One is an oper on C, also known as a holomorphic Schrödinger operator. The other is a family of first-order differential equations related to Teichmüller theory, which appear in the study of Hitchin's integrable system. Recently, physicist Gaiotto conjectured a precise relationship between these two. I will describe the proof of this conjecture, in recent joint work with Olivia Dumitrescu, Georgios Kydonakis, Rafe Mazzeo, Motohico Mulase and Andrew Neitzke.

A spectral curve S is also the input data for topological recursion. It has been conjectured that the invariants produced by topological recursion provide a quantization of the spectral curve. Dumitrescu-Mulase package the invariants together into a formal power series which they call a "quantum curve.'' It is expected that the WKB analysis of our oper should be given by the topological recursion formulated by Dumitrescu-Mulase.