Lectures are a sequel to ''Advanced topics of Analysis E1'' in the previous quarter. A Riemann surface is a two-real-dimensional manifold with holomorphic coordinate transformations. The moduli space of Riemann surfaces is a geometric space whose points represent classes of conformally equivalent Riemann surfaces. The Teichmüller space is a universal covering of the moduli space.
Each point in it is an isomorphism class of 'marked' Riemann surfaces. Ahlfors was the first to derive the complex structure of Teichmüller space. We will study his method. Topics include Teichmüller spaces, quasiconformal maps, Teichmüller’s theorem,
and the complex structure of Teichmüller space.
At the end of this course, students are expected to:
-- be familiar with quasiconformal maps
-- understand Teichmüller’s theorem
-- understand Ahlfors' approach for the complex structure of Teichmüller space
Riemann surfaces, moduli spaces of Riemann surfaces, Teichmüller spaces
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
Standard lecture course
|Course schedule||Required learning|
|Class 1||Teichmüller spaces||Details will be provided during each class session.|
|Class 2||Quasiconformal maps|
|Class 3||The Teichmüller distance|
|Class 4||Teichmüller modular groups|
|Class 5||Quadratic differentials|
|Class 6||Teichmüller’s theorem|
|Class 7||Ahlfors' approach I|
|Class 8||Ahlfors' approach II|
H. M. Farkas and I. Kra, Riemann surfaces, GTM 71, Springer-Verlag
Y. Imayoshi and M. Taniguchi, An Introduction to Teichmüller Spaces, Springer-Verlag
L. V. Ahlfors, The complex analytic structure of the space of closed Riemann surfaces. In Rolf Nevanlinna et. al., editor, Analytic Functions, pages 45-66. Princeton University Press, 1960.