Geometry and Topology of Surfaces

Summer School Lectures: Harmonic maps between surfaces, measured foliations and geodesic laminations

Abstract: I will explain how certain techniques from Geometric Analysis can be used to describe topological objects like foliations and laminations which appear in topology and in particular in Thurston theory.

Conference Talk: Harmonic maps in singular geometry and rigidity

Abstract: In this talk, I will present an overview of several results in geometric superrigidity. More precisely, I will describe the application of harmonic maps to the rigidity of buildings and Teichmüller space.

Summer School Lectures:

Let us see how simple ideas and observations coming from doing trigonometry with hyperbolic polygons have blossomed into important developments in modern mathematics.

Conference Talk: Moduli Spaces of crowned hyperbolic surfaces have polylogarithmic Mirzakhani volumes

Abstract: Mirzakhani showed that the Weil-Petersson volume of the moduli space of bordered hyperbolic surfaces with fixed boundary lengths is a polynomial, and used this to prove Witten's conjecture. We derive analogous volumes for moduli spaces of crowned hyperbolic surfaces (i.e.: Riemann surfaces with points on the boundary). For the purposes of this talk, we focus predominantly on the case of punctured disks.

Title: Function theory, Dynamics, and Ergodic theory in Teichmuller theory

Abstract: The first part of this talk concerns function theory on Teichmuller space. We will prove the existence of radial limits of bounded pluriharmonic functions at the Thurston boundary, and derive a Poisson integral formula expressed in terms of these radial limits. In the second part, we will discuss dynamical properties of the Torelli group on the space of projective measured laminations. In particular, we will show that the action of the Torelli group is not ergodic, and that its conical limit set has measure zero.

Summer School Lectures:

structures hyperboliques de surfaces, les laminations géodésiques, les applications d’étirement et de tremblements de terre.

Conference Talk: Unit tangent spheres of Teichmüller space with Thurston’s metric

Abstract: In this talk, I will give a formula to determine the dimensions of faces of unit tangent spheres of Teichmüller space with Thurston’s metric. As its application, I will show that the combinatorial structure of the unit tangent sphere does not depend on the base point. This is a part of joint work with Athanase Papadopoulos.

Surface homeomorphisms and mapping classes. Measured foliations. The space of measured foliations. The piecewise-linear structure of the space of measured foliations. Thurston's classification of mapping classes.

Title: Playing with moduli spaces

Abstract: Roughly speaking, a moduli space is a space of geometric objects up to some equivalence. For example, the space of triangles up to rigid motion is a simple moduli space. More complicated examples include moduli spaces of hyperbolic surfaces, which play an important role in geometric topology. These ideas can be very sophisticated, but some of the underlying principles can be approached without being an expert.

We’ll begin by playing around with examples coming from puzzles and drawings. It will be an excuse to explore how combinatorics often dovetails with geometry in surprising ways.

N.B. Participants are invited to “prepare" for class by playing Quadratis:

http://quadratis.app

A Finsler metric is equivalent to distributing a convex set to each tangent space of a manifold, and therefore a natural extension of the classical convex geometry. We will look into the relevance of convex geometry and concave geometry in this context, and discuss Finsler geometry as well as timelike finsler geometry.

Title: On the Finsler structures of some weak metric

Abstract: A Finsler structure on a manifold generalizes a Riemannian structure by assigning, in each tangent space, a Minkowski norm rather than an inner product. As a consequence it induces a length structure and a weak metric on the manifold. On the other hand, the Funk weak metric on a convex set in R^n was introduced as supporting examples to solve the Hilbert fourth problem: Find all the metrics for which line segments are geodesics. It is well-known that the Finsler structure associated with the Funk weak metric on the unit ball in the Euclidean space is of constant flag curvature −1/4 and constant S-curvature 3/2. In this talk the geometric properties of the Finsler structure are associated with Funk and Apollonian weak metric on the open unit disc will be discussed..