2024 Students Enrolled in or before 2015 School of Science Mathematics
Advanced courses in Analysis A
- Academic unit or major
- Mathematics
- Instructor(s)
- Yoshihiro Tonegawa
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Mon
- Class
- -
- Course Code
- ZUA.C331
- Number of credits
- 100
- Course offered
- 2024
- Offered quarter
- 1Q
- Syllabus updated
- Mar 14, 2025
- Language
- English
Syllabus
Course overview and goals
This lecture focuses on the mean curvature flow within the framework of geometric measure theory, an object called Brakke flow, and discusses its definitions and recent research results.
A time-parameterized family of surfaces is a mean curvature flow if the velocity of the surface is given by the mean curvature of the surface itself at each point and time. The mean curvature flow may be considered as a gradient flow of surface measure, and its static (or time-independent) object is precisely a minimal surface. In general, the mean curvature flow develops singularities, and it is natural to consider the flow within the framework which allows singularities. The convenient notion for that purpose is a varifold. In this lecture, starting from the definition of Brakke flow, the up-to-date research results will be explained.
Course description and aims
Understanding on the notions of mean curvature flow and Brakke flow within the framework of geometric measure theory is the goal.
Keywords
mean curvature flow, geometric measure theory
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course. Problems for reports are given occasionally.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Definitions of mean curvature flow and Brakke flow | Details will be provided during each class session. |
| Class 2 | Some basic notions from geometric measure theory | |
| Class 3 | Huisken's monotonicity formula | |
| Class 4 | Compactness theorem of Brakke fllow | |
| Class 5 | Tangent flow of Brakke flow | |
| Class 6 | Overview on existence and regularity theorems of Brakke flow | |
| Class 7 | Outline of proof of Kim-Tonegawa's existence theorem | |
| Class 8 | Other topics |
Study advice (preparation and review)
To enhance effective learning, by referring to textbooks and other course materials, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class contents afterwards (including assignments) for each class.
Textbook(s)
Brakke's mean curvature flow: an introduction, Springerbrief, Yoshihiro Tonegawa
Reference books, course materials, etc.
None in particular
Evaluation methods and criteria
Evaluation is based on attendance and assignments.
Related courses
- ZUA.C332 ： Advanced courses in Analysis B
- ZUA.C305 ： Real Analysis I
- MTH.C351 ： Functional Analysis
Prerequisites
Students are required to take Advanced course in Analysis B.