2021 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Special lectures on advanced topics in Mathematics J
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Yusuke Kuno / Takefumi Nosaka
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive
- Class
- -
- Course Code
- MTH.E534
- Number of credits
- 200
- Course offered
- 2021
- Offered quarter
- 3Q
- Syllabus updated
- Jul 10, 2025
- Language
- Japanese
Syllabus
Course overview and goals
This lecture is about a certain (infinite-dimensional) Lie algebra, called the Goldman Lie algebra, which is associated with an oriented surface. This Lie algebra is defined in terms of intersections of curves in the surface, and thus one can think of it as an object in low-dimensional topology. Actually, behind its definition there is a geometrical context related to the moduli space of flat bundles over the surface. In this lecture, I will first explain the definition of the Goldman Lie algebra, and then discuss the following two topics: (1) the description of Dehn twists in terms of the Goldman Lie algebra, and (2) the formality of the Goldman bracket (and the Turaev cobracket).
The aim of this lecture is to explain that the intersections of curves on an oriented surface give rise to interesting algebraic structures, which are useful to the study of self-diffeomorphisms and the mapping class group of the surface.
Course description and aims
・Understand the definition of the Goldman Lie algebra.
・Understand the logarithm of Dehn twists.
・Understand the definition of symplectic expansions.
・Understand the definition of the Turaev cobracket.
・Understand the formality of the Goldman bracket and the Turaev cobracket.
Keywords
Goldman bracket, Turaev cobracket, Dehn twists, mapping class group
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. ・The fundamental group and the homology group of sufaces ・The lower central series of a free group of finite rank ・Goldman bracket ・The logarithm of Dehn twists ・Generalized Dehn twits ・Symplectic expansions ・Turaev cobracket ・Formality of the Goldman bracket and the Turaev cobracket | 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.
They should do so by referring to textbooks and other course material.
Textbook(s)
None required.
Reference books, course materials, etc.
None required
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