2022 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Analysis A
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Masaharu Tanabe
- Class Format
- Lecture (Livestream)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Mon
- Class
- -
- Course Code
- MTH.C401
- Number of credits
- 100
- Course offered
- 2022
- Offered quarter
- 1Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
A Riemann surface is a two-real-dimensional manifold with holomorphic coordinate transformations.
The theory of Riemann surfaces has been a source of inspiration and examples for many fields of mathematics.
We will study the most important theorems concerning closed Riemann surfaces: the Riemann-Roch theorem,
Abel’s theorem, and the Jacobi inversion theorem.
This course will be completed with ''Advanced topics in Analysis B'' in the next quarter.
In this course, our goal is to understand and hence to prove the Riemann-Roch theorem.
Course description and aims
At the end of this course, students are expected to understand the Riemann-Roch theorem.
Keywords
Riemann surfaces, the Riemann-Roch theorem
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 | Topology of Riemann surfaces | |
| Class 3 | Differential forms | |
| Class 4 | Harmonic differetials, holomorphic differetials | |
| Class 5 | Bilinear relations | |
| Class 6 | Divisors | |
| Class 7 | The Riemann-Roch theorem |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None in particular
Reference books, course materials, etc.
H. M. Farkas and I. Kra, Riemann surfaces, GTM 71, Springer-Verlag
Evaluation methods and criteria
Assignments. Details will be announced during the session.
Related courses
- ZUA.C301 ： Complex Analysis I
- MTH.C301 ： Complex Analysis I
- MTH.C302 ： Complex Analysis II
- MTH.C402 ： Advanced topics in Analysis B
Prerequisites
None
Other
None in particular