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- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Akutagawa Kazuo Takakuwa Shoichiro
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Intensive (H201)
- Group
- -
- Course number
- MTH.E433
- Credits
- 2
- Academic year
- 2016
- Offered quarter
- 2Q
- Syllabus updated
- 2016/12/14
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
This course covers two typical topics in geometric analysis. The first is harmonic mapping, and the second is Ricci flow. Harmonic mapping is a generalization of harmonic functions, geodesic lines, and minimal surfaces, and was the first topic developed in geometric analysis. Ricci flow is a non-linear partial differential equation for Riemannian metrics, introduced in 1982 by Richard Hamilton to find standard Riemannian metrics for manifolds. The instructor will explain basic results for both topics.
Students will learn from these two topics how analytic and geometric methods are used in tandem to solve problems in geometric analysis.
Student learning outcomes
・Be familiar with the basic techniques in partial differential equations.
・Be familiar with many basic examples of harmonic maps and solutions of Ricci flow.
・Understand basic theory of harmonic maps and associated heat flow.
・Understand basic theory of Ricci flow.
Keywords
Riemannian manifold, curvature tensor, geodesic, energy functional, harmonic map, heat equation, Einstein-Hilbert functional,
Ricci flow, Ricci soliton
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
This is a standard lecture course. There will be some assignments.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Basic concepts of Riemannian geometry | Details will be provided during each class session |
| Class 2 | Length of curves and geodesics | Details will be provided during each class session |
| Class 3 | Existence theorem of geodesics | Details will be provided during each class session |
| Class 4 | The 1st variation formula of energy of curves | Details will be provided during each class session |
| Class 5 | The 2nd variation formula of energy of curves | Details will be provided during each class session |
| Class 6 | Applications to geometry of geodesics | Details will be provided during each class session |
| Class 7 | Definitions and examples of harmonic maps | Details will be provided during each class session |
| Class 8 | Existence theorem of harmonic maps | Details will be provided during each class session |
| Class 9 | Existence theorem of harmonic maps | Details will be provided during each class session |
| Class 10 | Overview of Hamilton's Ricci flow ( definition, background and history ) | Details will be provided during each class session |
| Class 11 | Various solutions of Ricci flow | Details will be provided during each class session |
| Class 12 | Existence and uniqueness of short time solutions of Ricci flow | Details will be provided during each class session |
| Class 13 | Evolution of curvature | Details will be provided during each class session |
| Class 14 | Convergence of Ricci lfow | Details will be provided during each class session |
| Class 15 | Applications to geometry of Ricci flow | Details will be provided during each class session |
Textbook(s)
none required
Reference books, course materials, etc.
・B．Andrews-C．Hopper, The Ricci flow in Riemannian Geometry, Lect. Notes in Math. Vol. 2011, Springer, 2011
Assessment criteria and methods
reports 100%
Related courses
- ZUA.B301 ： Geometry I
- MTH.B302 ： Geometry II
- MTH.B331 ： Geometry III
Prerequisites (i.e., required knowledge, skills, courses, etc.)
manifold theory, calculus of several variables, functional analysis
