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2023　Advanced topics in Analysis C1

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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.C407
Credits
1
Academic year
2023
Offered quarter
3Q
Syllabus updated
2023/3/20
Lecture notes updated
-
Language used
English
Access Index

Course description and aims

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 topics in Analysis D1'' 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.

Student learning outcomes

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 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 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

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.C301 ： Complex Analysis I
  • MTH.C302 ： Complex Analysis II

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

Not required

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