2023 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Analysis D1
- Academic unit or major
- Graduate major in 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
- MTH.C408
- Number of credits
- 100
- Course offered
- 2023
- Offered quarter
- 4Q
- Syllabus updated
- Jul 8, 2025
- Language
- English
Syllabus
Course overview and goals
Lectures are a sequel to ''Advanced topics in Analysis C1'' 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.
Course description and aims
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
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 | Teichmüller spaces | Details will be provided during each class session |
| Class 2 | Quasiconformal maps | Details will be provided during each class session |
| Class 3 | The Teichmüller distance | Details will be provided during each class session |
| Class 4 | Teichmüller modular groups | Details will be provided during each class session |
| Class 5 | Quadratic differentials | Details will be provided during each class session |
| Class 6 | Teichmüller’s theorem | Details will be provided during each class session |
| Class 7 | Ahlfors' approach | 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.C407 ： Advanced topics in Analysis C1
Prerequisites
Understanding of advanced topics in analysis C1 is required.