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2023　Advanced topics in Analysis D1

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Tanabe Masaharu 
Class Format
Lecture    (Face-to-face)
Media-enhanced courses
Day/Period(Room No.)
Fri3-4(M-102(H115))  
Group
-
Course number
MTH.C408
Credits
1
Academic year
2023
Offered quarter
4Q
Syllabus updated
2023/3/20
Lecture notes updated
-
Language used
English
Access Index

Course description and aims

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.

Student learning outcomes

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 that will be developed

✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Standard lecture course

Course schedule/Required learning

  Course schedule Required learning
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

Out-of-Class Study Time (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.

Assessment criteria and methods

Assignments (100%).

Related courses

  • MTH.C407 ： Advanced topics in Analysis C1

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Understanding of advanced topics in analysis C1 is required.

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