2025 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Special lectures on current topics in Mathematics D
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Hokuto Konno / Hisaaki Endo
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive
- Class
- -
- Course Code
- MTH.E634
- Number of credits
- 200
- Course offered
- 2025
- Offered quarter
- 2Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
This lecture provides an overview of recent developments in studying the structure of the diffeomorphism groups of 4-manifolds using gauge-theoretic methods. The diffeomorphism group is a fundamentally important object, yet its structure is generally difficult to analyze. However, advancements in "family gauge theory," which extends gauge theory to families of 4-manifolds, have revealed that certain phenomena unique to dimension 4 also manifest in diffeomorphism groups. This lecture will introduce these results and outline their proofs.
By examining the proofs of several key results on the diffeomorphism groups of 4-manifolds, we will gain an understanding of the fundamental ideas behind family gauge theory and typical approaches used in its arguments.
The aim of this course is to provide an understanding of the basics.
Course description and aims
・To acquire a fundamental understanding of family gauge theory.
・To develop basic knowledge of the diffeomorphism groups of 4-manifolds.
Keywords
4-manifold, diffeomorphism group, family gauge theory
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 | The following topics will be covered in this lecture: ・Exotic diffeomorphisms of 4-manifolds ・Family Seiberg-Witten invariants ・Gauge-theoretic characteristic classes ・Homological instability of the moduli space of 4-manifolds ・Infinite generation of the mapping class group of 4-manifolds | to be specified in each lecture |
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)
None required.
Reference books, course materials, etc.
None.
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
- MTH.B331 ： Geometry III
- MTH.B341 ： Topology
Prerequisites
To have basic knowledge in the theory of differentiable manifolds