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.