2024 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Geometry G
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Hisaaki Endo
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Fri
- Class
- -
- Course Code
- MTH.B503
- Number of credits
- 100
- Course offered
- 2024
- Offered quarter
- 3Q
- Syllabus updated
- Mar 14, 2025
- Language
- English
Syllabus
Course overview and goals
The aim of this lecture is to familiarize the students with the basic language of and some fundamental theorems for Lefschetz fibrations on 4-manifolds. This course will be succeeded by [MTH.B504 : Advanced topics in Geometry H].
Course description and aims
Students are expected to
・understand the definitions of Lefschetz fibrations, monodromy representations and Hurwitz systems.
Keywords
4-manifolds, Lefschetz fibrations, monodromy representations, Hurwitz systems, mapping class groups, signatures
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | 4-manifolds and their intersection forms | Details will be provided in class. |
| Class 2 | Definition of Lefschetz fibrations | Details will be provided in class. |
| Class 3 | Singular fibers and their neighborhoods | Details will be provided in class. |
| Class 4 | Monodromy representations and classification theorems | Details will be provided in class. |
| Class 5 | Hurwitz systems and elementary transformations | Details will be provided in class. |
| Class 6 | Meyer's signature cocycle and local signatures | Details will be provided in class. |
| Class 7 | Relators in mapping class groups and their signatures | Details will be provided in class. |
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 contents 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.
R. I. Gompf and A. I. Stipsicz, 4-Manifolds and Kirby Calculus, American Mathematical Society, 1999.
H. Endo and K. Hayano, 4-manifolds and fibrations, in Japanese, Kyoritsu Shuppan, 2024.
Evaluation methods and criteria
Homework assignments (100％)
Related courses
- MTH.B301 ： Geometry I
- MTH.B202 ： Introduction to Topology II
- MTH.B302 ： Geometry II
- MTH.B341 ： Topology
- MTH.B504 ： Advanced topics in Geometry H
Prerequisites
Basic algebraic topology (homology, cohomology, and the fundamental group) and smooth manifolds.
Other
To be announced.