Low Dimensional Geometry and Topology
Workshop on
Analytic aspects of low dimensional geometry
Monday September 4  Saturday September 9, 2006
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 noncommutative algebraic geometry. We will focus on the algebrogeometric version which, for the mathematician, has the advantage of having an interesting finitedimensional representation theory. We will show how this purely algebraic representation theory is related to 3dimensional 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 WeilPetersson 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 timeacceleration machine. Zippered rectangles. Analogy with continued fractions. RauzyVeech induction and Teichmuller geodesic flow.
Closed geodesics and closed saddle connections on a translation surface. Counting problems. EskinMasur 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 SiegelVeech constants.
Recent results and open problems.
For participation and further information please contact the
Warwick Mathematics Research Centre, mrc@maths.warwick.ac.uk
Travel information

For participation and further information please contact the:
Mathematics Research Centre
University of Warwick
Coventry CV4 7AL  UK
Fax:
+44 (0)24 7652 3548
Phone: +44 (0)24 7652 2681
EMail:
mrc@maths.warwick.ac.uk
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