Low Dimensional Geometry and Topology

Analytic aspects of low dimensional geometry

Francis Bonahon (University of Southern California) The quantum Teichmüller space and its representations

Abstract: The quantum Teichmüller space of a surface S is a deformation of the algebra of functions on the Teichmüller space, namely on the space of complex structures on the surface. There are two competing versions of this quantization, one based on C*-algebras and one using non-commutative algebraic geometry. We will focus on the algebro-geometric version which, for the mathematician, has the advantage of having an interesting finite-dimensional representation theory. We will show how this purely algebraic representation theory is related to 3-dimensional hyperbolic geometry.

Maryam Mirzakhani (Clay/Princeton, USA) Simple curves and the volume of moduli space.

In these lectures, we will discuss the geometry of the moduli space of Riemann surfaces by using properties of simple geodesics on hyperbolic surfaces. We will also discuss some related topics including the Weil-Petersson volumes of moduli spaces, the action on the mapping class group on the space of measured laminations, ergodic properties of the earthquake flow, and its relationship with the Teichmüller horocycle flow.

Anton Zorich (Rennes) Flat surfaces and Teichmüller dynamics

Topics to be convered:

Billiards in rational polygons and geodesic flow on flat surfaces. Calabi Theorem. Families of flat surfaces versus moduli spaces of Abelian differentials. Volume element in the moduli spaces. Ergodicity of SL(2,R)-action. "Ratner theory" programm.

Generic geodesics on a translation surface. Asymptotic cycle. Multiplicative ergodic theorem. Lyapunov exponents. Asymptotic flag of an orientable measured foliation. Lyapunov exponents of the Teichmüller geodesic flow.

Interval exchange transformations. Renormalization as a time-acceleration machine. Zippered rectangles. Analogy with continued fractions. Rauzy-Veech induction and Teichmuller geodesic flow.

Closed geodesics and closed saddle connections on a translation surface. Counting problems. Eskin-Masur quadratic asymptotics. Principal boundary of the moduli space of Abelian and quadratic differentials. Volumes of the moduli spaces and volumes of the "cusps". Evaluation of Siegel-Veech constants.

Recent results and open problems.

Fax:
+44 (0)24 7652 3548

Phone:
+44 (0)24 7652 2681

E-Mail:
mrc@maths.warwick.ac.uk

How to get to Warwick

Main Symposium Page