2022 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Special lectures on advanced topics in Mathematics C
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Shinichiro Matsuo / Kiyonori Gomi
- Class Format
- Lecture (Livestream)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive
- Class
- -
- Course Code
- MTH.E433
- Number of credits
- 200
- Course offered
- 2022
- Offered quarter
- 2Q
- Syllabus updated
- Jul 10, 2025
- Language
- Japanese
Syllabus
Course overview and goals
This is an introductory course on geometric analysis. We will fous on the bubbling analysis of Uhlenbeck and explain her compactness theorem.
Through the detailed explanation of the Uhlenbeck compactness, we will become familiar with many concepts of geometric analysis.
Course description and aims
Be familiar with connections, curvature, and gauge transformations.
Be familiar with the anti-self-dual equations and instantons.
Be familiar with Sobolev spaces.
Be familiar with conformal invariance and bubbling analysis.
Keywords
anti-self-dual equations, instantons, moduli spaces, Uhlenbeck compactness, bubbling analysis
Competencies
- 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/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | We will cover the following topics: 1. Connections, curvatures, and gauge transformations. 2. The anti-self-dual equations and instantons 3. Bubbling analyisis 4. Global slices and the Coulomb gauge 5. Curvature is proper. 6. Mean-value theorem, Chern-Simons invariants, and the anti-self-dual equations 7. The proof of the Uhlenbeck compactness theorem | Details will be provided during each class session. |
Study advice (preparation and review)
Textbook(s)
None required.
Reference books, course materials, etc.
Details will be provided during each class session.
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.B341 ： Topology
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
- MTH.B331 ： Geometry III
- MTH.C305 ： Real Analysis I
- MTH.C306 ： Real Analysis II
- MTH.C351 ： Functional Analysis
Prerequisites
Basic knowledge about smooth manifolds