Geometry and Topology of Surfaces

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

To be announced.

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

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.

To be announced.