Advanced Topics in Geometry (V5D3), Wintersemester 2019/20

Dozent: Dr. Matteo Costantini

Sprechstunde: Nach Vereinbarung per email.

Lectures:
Dienstag 16:00 s.t. - 17:30 Uhr, MATH / N 0.008 - Neubau
Mittwoch 14:30 s.t. - 16:00 Uhr, MATH / N 0.003 - Neubau

The main topic of this course is flat surfaces and their moduli spaces. A flat surface is obtained by gluing together finitely many polygons in the Euclidean plane by identifying their sides via translations. A flat surface is naturally equipped with a flat metric with conical singularities. The datum of a flat surface is equivalent to the one of a complex structure on a topological surface together with a holomorphic one form. During the course we will show this identification and we will use it in order to study different aspects of flat surfaces and their parameter space.

This is an active topic of research in which many areas of mathematics come together, for example complex algebraic geometry and the study of moduli spaces of Riemann surfaces, Teichmüller theory, dynamical systems and the study of billiard trajectories and homogenous dynamics. The course is an introduction to the topic. Depending on the backgrouond and the interests of the students, some areas will be more deeply presented than other.

Literatur: The following references are surveys about flat surfaces and Teichmüller curves and some more specific references. Handwritten notes of the course are available here.

• Alex Wright, From rational billiards to dynamics on moduli spaces, Bull. Amer. Math. Soc. 53 (2016), 41-46
• Alex Wright, Translation surfaces and their orbit closures: An introduction for a broad audience, EMS Surv. Math. Sci. 2015
• Martin Möller, Teichmueller curves from an algebraic viewpoint, Park City Lecture notes
• Anton Zorich, Flat surfaces, Frontiers in number theory, physics, and geometry, Volume I. Available on the arXiv here
• Robert A. Kucharczyk, Real Multiplication on Jacobian Varieties, Available on the arXiv here
• Pascal Hubert and Thomas Schmidt, An introduction to Veech surfaces
• Howard Masur and Serge Tabachnikov, Rational billiards and flat structures, Handbook of dynamical systems, Vol. 1A, 1015–1089, 2002.
• Benson Farb and Dan Margalit, A primer on mapping class groups, 49. Princeton University Press, Princeton, NJ, 2012
• Claire Voisin, Hodge theory and complex algebraic geometry I, 76. Cambridge University Press, Cambridge, 2007.