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- Common courses
- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Tonegawa Yoshihiro
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- Day/Period(Room No.)
- Mon3-4(M-112(H117))
- Group
- -
- Course number
- MTH.C401
- Credits
- 1
- Academic year
- 2024
- Offered quarter
- 1Q
- Syllabus updated
- 2024/3/14
- Lecture notes updated
- -
- Language used
- English
- Access Index
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Course description and aims
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.
Student learning outcomes
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 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 | 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 |
Out-of-Class Study Time (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)
Brakke's mean curvature flow: an introduction, Springerbrief, Yoshihiro Tonegawa
Reference books, course materials, etc.
None in particular
Assessment criteria and methods
Assignments. Details will be announced during the session.
Related courses
- MTH.C305 ： Real Analysis I
- MTH.C306 ： Real Analysis II
- MTH.C402 ： Advanced topics in Analysis B
- MTH.C351 ： Functional Analysis
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Familiarity with general measure theory is desirable but not necessary.
Other
None in particular
