Translation surfaces

Translation surfaces

 

Dates

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Lecture

Tuesday, 10:15-11:45, SR10, Gebäude E 2.5
First lecture on Tuesday, April 7.

Organisational remark: There is a possibility that part of the lecture will be held as a block session during the lecture-free period due to scheduling conflicts during some weeks. We will discuss this in the first lecture. If you are interested in the lecture but unable to attend the first lecture, please send me an email in advance.

 

Requirements

Linear Algebra I, II + Analyis, I,II + Algebra (recommended), Complex Analysis (recommended) 

 

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Content

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Translation surfaces form a fascinating bridge between geometry, topology, and dynamics. Emerging in the 1990s as a vibrant new mathematical field, their study intertwines ideas from geometric group theory, algebraic geometry, and dynamical systems. A finite translation surface arises from a simple geometric construction: by gluing together finitely many Euclidean polygons along parallel edges, one obtains a closed surface endowed with a flat metric and conical singularities. Despite this elementary origin, translation surfaces lead to deep and far-reaching mathematics. They naturally define algebraic curves in the moduli space M_g​ of Riemann surfaces of genus g. They also play a central role in the study of dynamical systems such as billiard flows in polygonal tables and the Teichmüller flow on Teichmüller space. In this lecture, we will introduce the foundational concepts of translation surface theory and explore some of its most significant and recent developments.

 

Literature

• Daniel Massart:  A short introduction to translation surfaces, Veech surfaces, and Teichmueller dynamics, arxiv.org/abs/2107.11581
• Anton Zorich: Flat Surfaces, https://arxiv.org/abs/math/0609392
• Anja Randecker: Skript zur Vortragsreihe 'Unendliche Translationsflächen', https://modellansatz.de/pages/media/060-ebfc21e7-unendliche_translationsflaechen.pdf
• Vincent Delecroix, Pascal Hubert, Ferrán Valdez: Infinite Translation Surfaces in the Wild, https://arxiv.org/abs/2403.05424
• Frank Herrlich, Gabriela Schmithuesen: On the boundary of Teichmueller disks in Teichmueller and in Schottky space, https://arxiv.org/abs/math/0702496

Additional Topics

• zu (X,G)-Mannigfaltigkeiten: John G. Ratcliffe: Foundations of Hyperbolic Manifolds (Kap. 8)