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The analytic backbone of the GFF/SLE coupling on Riemann surfaces

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If you have a question about this talk, please contact Renato Velozo.

This will be an introductory talk to Liouville Conformal Field Theory (LCFT) with an emphasis on its underlying geometric features. I will start by reviewing some facts about Riemann surfaces (uniformisation, Teichmüller theory, etc) and conformal welding, which can be viewed as a deformation of the pants decomposition of hyperbolic surfaces. Then I will introduce the Gaussian Free Field (GFF) and Gaussian Multiplicative Chaos (GMC), the main probabilistic tools needed to construct the partition function of LCFT . The latter is a mapping class group invariant function defined on Teichmüller space and I will explain how it is possible to solve a conformal welding problem involving GMC in order to construct this function by induction on the Euler characteristic of the surface. Doing so, a natural probability measure arises on homotopy classes of simple closed curves, which is interpreted as a Schramm-Loewner Evolution (SLE).

This talk is part of the Cambridge Analysts' Knowledge Exchange series.

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