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We will introduce Teichmüller space and its different interpretations. We will discuss its topology and its geometry, and talk about how to construct a boundary. We will concentrate on closed surfaces and give hints at added difficulties when the surface is not closed.

57K20 ; 30F10 ; 30F60

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We will introduce Teichmüller space and its different interpretations. We will discuss its topology and its geometry, and talk about how to construct a boundary. We will concentrate on closed surfaces and give hints at added difficulties when the surface is not closed.

57K20 ; 30F10 ; 30F60

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Going back at least to the works of Witten and Kontsevich, it is known that (symplectic or Weil-Petersson) volumes of moduli spaces of Riemann surfaces share many features with the enumeration of maps. It is therefore natural to expect that the theory of random hyperbolic metrics sampled according to the Weil-Petersson measure on, say, punctured spheres is closely related to the theory of random planar maps. I will highlight some similarities and show that tree bijections, which are ubiquitous in the study of random planar maps, have analogues for hyperbolic surfaces. As an application, jointly with Nicolas Curien, we show that these random hyperbolic surfaces with properly rescaled metric admit a scaling limit towards the Brownian sphere when the number of punctures increases.[-]

Going back at least to the works of Witten and Kontsevich, it is known that (symplectic or Weil-Petersson) volumes of moduli spaces of Riemann surfaces share many features with the enumeration of maps. It is therefore natural to expect that the theory of random hyperbolic metrics sampled according to the Weil-Petersson measure on, say, punctured spheres is closely related to the theory of random planar maps. I will highlight some similarities ...[+]

05C80 ; 82B41 ; 30F60

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We will present a geometric criterion for the ergodicity of the billiard flow in a polygon with non-rational angles and discuss its application to the Diophantine case.

37D40 ; 37D50 ; 30F10 ; 30F60 ; 32G15

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In this somewhat speculative talk I will briefly describe recent results with Xuwen Zhu on the boundary behaviour of the Weil-Petersson metric (on the moduli space of Riemann surfaces) and ongoing work with Jesse Gell-Redman on the associated Laplacian. I will then describe what I think happens for the wave equation in this context and what needs to be done to prove it.

30F60 ; 32G15 ; 35L05

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We consider "higher dimensional Teichmüller discs", by which we mean complex submanifolds of Teichmüller space that contain the Teichmüller disc joining any two of its points. We prove results in the higher dimensional setting that are opposite to the one dimensional behavior: every "higher dimensional Teichmüller disc" covers a "higher dimensional Teichmüller curve" and there are only finitely many "higher dimensional Teichmüller curves" in each moduli space. The proofs use recent results in Teichmüller dynamics, especially joint work with Eskin and Filip on the Kontsevich-Zorich cocycle. Joint work with McMullen and Mukamel as well as Eskin, McMullen and Mukamel shows that exotic examples of "higher dimensional Teichmüller discs" do exist.[-]

We consider "higher dimensional Teichmüller discs", by which we mean complex submanifolds of Teichmüller space that contain the Teichmüller disc joining any two of its points. We prove results in the higher dimensional setting that are opposite to the one dimensional behavior: every "higher dimensional Teichmüller disc" covers a "higher dimensional Teichmüller curve" and there are only finitely many "higher dimensional Teichmüller curves" in ...[+]

30F60 ; 32G15

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We discuss singularities of Teichmüller harmonic map flow, which is a geometric flow that changes maps from surfaces into branched minimal immersions, and explain in particular how winding singularities of the map component can lead to singular behaviour of the metric component.

53C44 ; 30F60

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Mirzakhani wrote two papers studying the asymptotic behaviour of the number of curves of a given type (simple or not) and with length at most $L$. In this talk I will explain a new independent proof of Mirzakhani's results.
This is joint work with Viveka Erlandsson.

57N05 ; 30F45 ; 30F60 ; 32G15 ; 57M50

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Satake duality is an isomorphism between the algebra of characters of a simple Lie group G and the Hecke algebra of the affine Lie group corresponding to the Langlands dual to G. We will suggest a (conjectural) generalisation of this isomorphism replacing the characters by functions on local systems on surfaces and Hecke algebra by the algebra on cells on local systems with values in an affine Lie group.

17B20 ; 30F60

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I will describe a program to describe Hitchin components as the moduli space of some new geometric structure on the surface. This geometric structure generalizes the complex structure. Its construction uses the punctual Hilbert scheme of the plane. It should give a unified description of Hitchin components without fixed complex structure on the surface. I also present a generalization to character varieties for non split real groups in the spirit of G-Higgs bundles.[-]

I will describe a program to describe Hitchin components as the moduli space of some new geometric structure on the surface. This geometric structure generalizes the complex structure. Its construction uses the punctual Hilbert scheme of the plane. It should give a unified description of Hitchin components without fixed complex structure on the surface. I also present a generalization to character varieties for non split real groups in the ...[+]

30F60 ; 14D21 ; 53C15 ; 14C05

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Introduction to Teichmüller theory Lecture 1 - Fanoni, Federica (Auteur de la Conférence) | CIRM H Nouveau

Introduction to Teichmüller theory Lecture 2 - Fanoni, Federica (Auteur de la Conférence) | CIRM H Nouveau

Random hyperbolic surfaces - Budd, Timothy (Auteur de la Conférence) | CIRM H Nouveau

On the ergodicity of billiards in non-rational polygons - Forni, Giovanni (Auteur de la Conférence) | CIRM H Nouveau

The wave equation for Weil-Petersson metrics - Melrose, Richard (Auteur de la Conférence) | CIRM H Nouveau

Totally geodesic submanifolds of Teichmüller space and moduli space - Wright, Alexander (Auteur de la Conférence) | CIRM H Nouveau

Singularities of Teichmüller harmonic map flow - Rupflin, Melanie (Auteur de la Conférence) | CIRM H Nouveau

Counting curves of given type, revisited - Souto, Juan (Auteur de la Conférence) | CIRM H Nouveau

Local systems and Satake duality - Fock, Vladimir (Auteur de la Conférence) | CIRM H Nouveau

Geometric approach to Hitchin components via punctual Hilbert schemes - Thomas, Alexander (Auteur de la Conférence) | CIRM H Nouveau