2022 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Special lectures on current topics in Mathematics E
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Toshiyuki Sugawa / Ege Fujikawa
- Class Format
- Lecture (Livestream)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive
- Class
- -
- Course Code
- MTH.E635
- Number of credits
- 200
- Course offered
- 2022
- Offered quarter
- 2Q
- Syllabus updated
- Jul 10, 2025
- Language
- Japanese
Syllabus
Course overview and goals
We introduce to plane quasiconformal mappings. The notion of quasiconformal mappings generalizes that of conformal mappings and it plays an important role in complex dynamics, Kleinian groups and Teichmueller theory as well as the classical function theory. The quasiconformal mappings look artificial in its definition. However, it appears naturally in complex analysis. In this series of lectures, after a brief introduction to the definition and basic properties of quasiconformal mappings, we will prove the measurable Riemann mapping theorem, which is the most fundamental existence result in the theory. If time permits, we will mention the lambda lemma and Teichmueller spaces.
In order to understand quasiconformal mappings and their existence result, we need knowledge about advanced analysis such as theories of measures, extremal lengths and singular integral operators. Therefore, trying to understand the theory of quasiconformal mappings will, in turn, provide opportunities to observe how such advanced theories are applied to concrete examples.
Course description and aims
1. Understand the definitions of quasiconformal mappings and be able to determine whether a given map is quasiconformal.
2. Understand the proof of the measurable Riemann mapping theorem.
3. Know about applications of quasiconformal mappings.
Keywords
quasiconformal mappings, extremal length, Beltrami equation, measurable Riemann mapping theorem
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is an intensive lecture course. There will be some assignments.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | We will give lectures on the following items: 1. Conformal mappings and (smooth) quasiconformal mappings 2. Moduli of quadrilaterals and geometric definition of quasiconformal mappings 3. Extremal length 4. Analytic definition of quasiconformal mappings and complex dilatations 5. Solution to the Beltrami equation on the plane (Proof of the measurable Riemann mapping theorem). 6. Lambda lemmas 7. The universal Teichmueller space | Problems for excercises will be assigned during course lectures. |
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 required
Reference books, course materials, etc.
L.V. Ahlfors, Lectures on Quasiconformal Mappings, van Nostrand
Evaluation methods and criteria
Assignments (100%).
Related courses
- none
Prerequisites
None Required