2026 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Geometry E
- 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-B107(H104))
- Class
- -
- Course Code
- MTH.B501
- Number of credits
- 100
- Course offered
- 2026
- Offered quarter
- 1Q
- Syllabus updated
- Mar 5, 2026
- Language
- English
Syllabus
Course overview and goals
Characteristic classes of vector bundles are invariants which have been applied universally in geometry. Basic properties of cohomology required for the introduction of the characteristic classes, vector bundles and their related notions will be explained.
Course description and aims
- to get deeper understanding of cohomology of topological spaces.
- to understand vector bundles and related notions.
Keywords
vector bundle
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
A standard lecture course. Homeworks will be assined for each lesson.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | review of homology |
Details will be provided during each class |
| Class 2 | review of cohomology |
Details will be provided during each class |
| Class 3 | definition of vector bundles |
Details will be provided during each class |
| Class 4 | Remannian metric |
Details will be provided during each class |
| Class 5 | maps of vector bundles and subbundles |
Details will be provided during each class |
| Class 6 | orientation of vector bundle |
Details will be provided during each class |
| Class 7 | theorem of Leray-Hirsch |
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.B341 ： Topology
Prerequisites
Knowledge on topology (MTH.B341) and maniofolds (MTH.B301, MTH.B302) are required.