Special topics in Algebra Summer Term 18

Dates: Thursday, 10:00 (c.t.) - 12:00, Zeichensaal

Lecturer: Prof. Dr. Gabriela Weitze-Schmithüsen

Topics:

In this lecture we will study Veech groups of translation surfaces. These are discrete subgroups of the matrix group SL(2,R). They are dened by a very down to earth construction which is easy to understand. Although they were intensively studied in the last thirty year, there are still a lot of open questions. In particular it is not at all known which discrete subgroups of SL(2,R) occur as Veech groups.

One reason why Veech groups are so popular is that they play a crucial role in the solution of very dierent problems: They help to understand the long term behaviour of a billiard ball on a polygonal shaped billiard table. They are used to approximate how many closed geodesics of a given length do exist on a translation surface. Furthermore they code information about geodesics in Teichmüller space and so-called Teichmüller curves which are special complex algebraic curves in moduli space of closed Riemann surfaces of genus g. These relations lead in an appealing way to links between topics in geometry, algebra and number theory.

In the course we will learn in detail the dierent methods used to study Veech groups as subgroups of SL(2,R) and discuss some of the links to the theory of translation surfaces, Teichmüller spaces and moduli spaces mentioned above in more detail.

Prerequisites:

Good knowledge in Algebra.