2026 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Geometry F
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Kiyonori Gomi
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Tue (M-102(H115))
- Class
- -
- Course Code
- MTH.B502
- Number of credits
- 100
- Course offered
- 2026
- Offered quarter
- 2Q
- Syllabus updated
- Mar 5, 2026
- Language
- English
Syllabus
Course overview and goals
The most basic characteristic classes of vector bundles are introduced. Their basic properties and applications are also explained.
Course description and aims
- to understand a definition and properties of the most basic characteristic classes of vector bundles.
- to learn applications of these characterisitic classes.
Keywords
vector bundle, Euler class, Stiefel-Whiteny class, Chern class, Pontryagin class, index theorem, exotic sphere
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Thom class and Euler class |
Details will be provided during each class |
| Class 2 | Applications of Euler class |
Details will be provided during each class |
| Class 3 | Stiefel-Whiteny class |
Details will be provided during each class |
| Class 4 | Chern class |
Details will be provided during each class |
| Class 5 | Pontryagin class |
Details will be provided during each class |
| Class 6 | Index theorem |
Details will be provided during each class |
| Class 7 | Exiotic sphere |
Details will be provided during each class |
Study advice (preparation and review)
Official Message: 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)
None
Reference books, course materials, etc.
John Milnor, James D. Stasheff, Characteristic Classes. Volume 76 (Annals of Mathematics Studies), Princeton University Press.
Ichiro Tamura, Differential Topology, Iwanami.
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
- MTH.B331 ： Geometry III
- MTH.B501 ： Advanced topics in Geometry E
Prerequisites
Knowledge on topology (MTH.B341) and maniofolds (MTH.B301, MTH.B302) are required. Also, students are supposed to have attended Advanced topics in Geometry E(MTH.B501).