Translation surfaces
Translation surfaces are obtained by gluing together finitely many polygons in the Euclidean plane by identifying their sides via translations. This simple definition gives rise to a rich theory, which is an active topic of research. This course is an introduction to this subject.
More precisely: we will talk about different equivalent definitions of translation surfaces and introduce their moduli spaces. We will then discuss some results about geodesics on translation surfaces.
Prerequisites: basics in topology and in complex analysis, basic knowledge of manifolds.
Lectures:
- Wednesday 14-16, SR 8
- Thursday 14-16, Room 0.200
Registration is possible on Müsli.
Some references: the first four are surveys about translation surfaces (and various other related topics). Miranda's book is a reference for the Riemannian surfaces part. Hubbard's book is a quite advanced reference for Teichmüller theory. The last is the PhD thesis of Anja Randecker on infinite translation surfaces
- Alex Wright, Translation surfaces and their orbit closures: an introduction for a broad audience, EMS Surveys in Mathematical Sciences
- Anton Zorich, Flat surfaces, Frontiers in number theory, physics, and geometry, Volume I. Available on the arXiv here
- Giovanni Forni and Carlos Matheus, Introduction to Teichmüller dynamics and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards, Journal of Modern Dynamics. Available on the arXiv here
- Jean-Christophe Yoccoz, Interval exchange maps and translation surfaces, Clay Mathematics Proceedings. Available here
- Rick Miranda, Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics
- John Hubbard, Teichmüller Theory And Applications to Geometry, Topology and Dynamics, Volume 1: Teichmüller theory, Matrix editions
- Anja Randecker, Geometry and topology of wild translation surfaces, PhD thesis
Handwritten notes
Important disclaimer: I do not guarantee that the notes are correct and that there is no discrepancy between what is in the notes and what is covered during the lectures.
- Introduction
- Some basics of surface topology
- An introduction to Riemann surfaces and Abelian differentials
- Definitions of translation surfaces
- Parameter spaces of Riemann and translation surfaces
- The GL+(2,R)-action on the strata
- The straight line flow: periodicity, minimality and ergodicity properties