Online: https://bbb.mpim-bonn.mpg.de/b/gae-a7y-hhd

The moduli space of complex curves has several descriptions, giving the same topological space but different geometric structures. The description in terms of metric ribbon graphs gives it a polytopal complex structure, and Kontsevich gave it an (almost everywhere) symplectic structure used in his proof of Witten's conjecture. I will revisit the associated geometry of this space (or rather of its universal cover, ie Teichmuller space) making it parallel to the Weil-Petersson geometry coming from hyperbolic metrics on surfaces: we will see how to define Fenchel-Nielsen coordinates that are Darboux for Kontsevich symplectic structure. There is in fact a flow, originally studied by Bowditch-Epstein, Mondello and Do, taking hyperbolic geometry to combinatorial geometry, and I will present stronger results about the convergence of this flow.

Based on joint work with Jorgen Andersen, Severin Charbonnier, Alessandro Giacchetto, Danilo Lewanski, Campbell Wheeler.

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