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. Teichmüller space is a universal covering of the moduli space. Each point in it is an isomorphism class of 'marked' Riemann surfaces.
This course will be completed with ''Advanced courses in Analysis D'' in the next quarter. Our goal is to understand Ahlfors’ method
which is the first to derive the complex structure of Teichmüller space. To prepare for it, several basic tools and theorems about Riemann surfaces will be introduced in this course.
At the end of this course, students are expected to:
-- understand the moduli space and the Teichmüller space of the torus
-- be familiar with differential forms on Riemann surfaces
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||Riemann surfaces||Details will be provided during each class session.|
|Class 2||The moduli space of the torus||Details will be provided during each class session.|
|Class 3||The Teichmüller space of the torus||Details will be provided during each class session.|
|Class 4||Topology of Riemann surfaces||Details will be provided during each class session.|
|Class 5||Differential forms||Details will be provided during each class session.|
|Class 6||Harmonic differetials, holomorphic differetials||Details will be provided during each class session.|
|Class 7||Bilinear relations,Period matrices||Details will be provided during each class session.|
To enhance effective learning, students are encouraged to spend approximately 100 minutes reviewing class content afterwards for each class.
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.