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2024 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)
Yoshihiro Tonegawa
Class Format
Lecture (Face-to-face)
Media-enhanced courses
-
Day of week/Period
(Classrooms)
3-4 Mon
Class
-
Course Code
MTH.C401
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

Standard lecture course.

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

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)

Brakke's mean curvature flow: an introduction, Springerbrief, Yoshihiro Tonegawa

Reference books, course materials, etc.

None in particular

Evaluation methods and criteria

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

Familiarity with general measure theory is desirable but not necessary.

Other

None in particular