2023 Students Enrolled in or before 2015 School of Science Mathematics
Advanced courses in Analysis C
- Academic unit or major
- Mathematics
- Instructor(s)
- Masaharu Tanabe
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Fri (M-102(H115))
- Class
- -
- Course Code
- ZUA.C333
- Number of credits
- 100
- Course offered
- 2023
- Offered quarter
- 3Q
- Syllabus updated
- Jul 8, 2025
- Language
- English
Syllabus
Course overview and goals
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.
Course description and aims
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
Keywords
Riemann surfaces, moduli spaces of Riemann surfaces, Teichmüller spaces
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| 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. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes reviewing class content afterwards for each class.
Textbook(s)
None
Reference books, course materials, etc.
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.
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.C301 ： Complex Analysis I
- MTH.C302 ： Complex Analysis II
Prerequisites
Not required