2024 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Special lectures on advanced topics in Mathematics D
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Yuta Nozaki / Takefumi Nosaka
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive
- Class
- -
- Course Code
- MTH.E434
- Number of credits
- 200
- Course offered
- 2024
- Offered quarter
- 3Q
- Syllabus updated
- Mar 14, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The main theme of this lecture is to understand the structure of groups consisting of homology cylinders. A homology cylinder is a certain 3-manifold with boundary, closely related to the mapping class groups of surfaces and their subgroups known as the Torelli group. They also have connections with the homology cobordism groups defined via 4-manifolds. To investigate these topics, we will introduce the LMO functor defined by Cheptea, Habiro, and Massuyeau, and learn about its properties and applications.
As mentioned above, homology cylinders have various connections to the study of low-dimensional topology. In this lecture, we will explore important topics and fundamental research methods in low-dimensional topology among these relationships. For example, Dehn surgery on links and its special case known as clasper surgery are fundamental with many applications. Regarding the LMO functor, we will ensure a solid understanding through simple computational examples.
Course description and aims
・Be familiar with Dehn surgery on links.
・Understand a relation between the Torelli group and homology cylinders.
・Be able to compute the Kontsevich invariant and the LMO functor in lower degree.
・Understand a relation between the LMO functor and clasper surgery.
・Understand groups consisting of homology cylinders with lower degree.
Keywords
quantum topology, mapping class group, knot, homology cobordism group, Kontsevich invariant, LMO functor, clasper surgery
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 order: ・Homology cylinders and mapping class groups ・The lower central series of the Torelli group and its graded quotients ・A category of cobordisms and that of tangles ・Jacobi diagrams and the Kontsevich invariant ・The LMO functor and clasper surgery ・Homomorphisms induced by the LMO functor ・The structure of groups related to homology cylinders | Details will be provided during each class session. |
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.
Textbook(s)
None required
Reference books, course materials, etc.
T. Ohtsuki, Quantum invariants: a study of knots, 3-manifolds, and their sets, Series on Knots and Everything 29, World Sci., River Edge, NJ (2002)
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
- MTH.B331 ： Geometry III
Prerequisites
Good understanding on the materials in the "related courses" is expected
Other
Not in particular