▶Japanese
Post to SNS
- Home
- > School of Science
- > Mathematics
- > Advanced courses in Analysis B
- School of Science
-
School of Engineering
- Group 2
- Group 3
- Group 4
- Group 5
- Group 6
- Metallurgical Engineering
- Organic and Polymeric Materials
- Inorganic Materials
- Chemical Engineering Course
- Chemical Engineering Course(Chemical Engineering)
- Applied Chemistry Course(Chemical Engineering)
- Polymer Chemistry
- Mechanical Engineering and Science
- Mechanical and Intelligent Systems Engineering
- Mechano-Aerospace Engineering
- International Development Engineering
- Control and Systems Engineering
- Industrial and Systems Engineering
- Electrical and Electronic Engineering
- Computer Science
- Civil Engineering
- Architecture and Building Engineering
- Social Engineering
- Common Cource of Engineering
- School of Bioscience and Biotechnology
- Common Course of Undergraduate School
-
Graduate School of Science and Engineering
- Mathematics
- Physics(Particle- Nuclear- and Astro-Physics)
- Physics(Condensed Matter Physics)
- Chemistry
- Earth and Planetary Sciences
- Chemistry and Materials Science
- Metallurgy and Ceramics Science
- Organic and Polymeric Materials
- Applied Chemistry
- Chemical Engineering
- Common Course of Mechanical Engineering
- Mechanical Sciences and Engineering
- Mechanical and Control Engineering
- Mechanical and Aerospace Engineering
- Common Course of Electronic Engineering
- Electrical and Electronic Engineering
- Physical Electronics
- Communications and Integrated Systems
- Communications and Computer Engineering
- Civil Engineering
- Architecture and Building Engineering
- International Development Engineering
- Nuclear Engineering
- Graduate School of Bioscience and Biotechnology
-
Interdisciplinary Graduate School of Science and Engineering
- Built Environment
- Energy Sciences
- Computational Intelligence and Systems Science
- Electronic Chemistry
- Materials Science and Engineering
- Innovative and Engineered Materials
- Environmental Chemistry and Engineering
- Environmental Science and Technology
- Mechano-Micro Engineering
- Information Processing
- Electronics and Applied Physics
- Graduate School of Information Science and Engineering
- Graduate School of Decision Science and Technology
- Graduate School of Innovation Management
- Common Course of Graduate School
- Program for Leading Graduate Schools
- Academic unit or major
- Mathematics
- Instructor(s)
- Tonegawa Yoshihiro
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- Day/Period(Room No.)
- Mon3-4(M-102(H115))
- Group
- -
- Course number
- ZUA.C332
- Credits
- 1
- Academic year
- 2024
- Offered quarter
- 2Q
- Syllabus updated
- 2024/3/14
- Lecture notes updated
- -
- Language used
- English
- Access Index
-
Course description and aims
Lectures are a sequel to ''Advanced course in Analysis A'' in the previous quarter. 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 | Approximate mean curvature vector | Details will be provided during each class session. |
| Class 2 | Estimates satisfied by approximate mean curvature vector (1) | |
| Class 3 | Estimates satisfied by approximate mean curvature vector (2) | |
| Class 4 | construction of time-discrete approximate mean curvature flow | |
| Class 5 | Rectifiability theorem of limit measure | |
| Class 6 | Integrality theorem of limit measure | |
| Class 7 | Regularity theorem of Brakke flow | |
| Class 8 | Other topics |
Out-of-Class Study Time (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
Assessment criteria and methods
Assignments. Details will be announced during the session.
Related courses
- ZUA.C331 ： Advanced courses in Analysis A
- ZUA.C305 ： Real Analysis I
- MTH.C351 ： Functional Analysis
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students are required to take Advanced course in Analysis A.
