2023 Faculty Courses School of Computing Department of Mathematical and Computing Science Graduate major in Mathematical and Computing Science
Discrete, Algebraic and Geometric Structures
- Academic unit or major
- Graduate major in Mathematical and Computing Science
- Instructor(s)
- Sakie Suzuki / Shinya Nishibata / Masaaki Umehara / Hideyuki Miura / Toshiaki Murofushi
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Mon (W9-322(W932)) / 5-6 Thu (W9-322(W932))
- Class
- -
- Course Code
- MCS.T408
- Number of credits
- 200
- Course offered
- 2023
- Offered quarter
- 3Q
- Syllabus updated
- Jul 8, 2025
- Language
- Japanese
Syllabus
Course overview and goals
Discrete, algebraic and geometric structures appear in many stages of the study in mathematical and computing science. The objective of this course is to describe some advanced topics, and for students to know mathematical structures behind them.
Course description and aims
The students are expected to learn advanced mathematical methods to analyze discrete, algebraic and geometric structures appeared in mathematical and computing science, and to be able to apply them to some practical problems.
Keywords
discrete structure, algebraic structure, geometric structure
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
The course provides advanced topics of discrete, algebraic and geometric structures.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | PL manifolds | Understand the contents covered by the lecture. |
| Class 2 | Knots and links | Understand the contents covered by the lecture. |
| Class 3 | Knot groups | Understand the contents covered by the lecture. |
| Class 4 | Kauffman bracket and Jones polynomial | Understand the contents covered by the lecture. |
| Class 5 | Jones polynomial as functor | Understand the contents covered by the lecture. |
| Class 6 | Hopf algebras and quantum groups | Understand the contents covered by the lecture. |
| Class 7 | Colored Jones polynomial | Understand the contents covered by the lecture. |
| Class 8 | Universal quantum invariant | Understand the contents covered by the lecture. |
| Class 9 | Equivariance of universal quantum invariant under Hopf algebra morphism action | Understand the contents covered by the lecture. |
| Class 10 | Dehn surgery | Understand the contents covered by the lecture. |
| Class 11 | Witten-Reshetkhin-Turaev invariant | Understand the contents covered by the lecture. |
| Class 12 | Triangulations | Understand the contents covered by the lecture. |
| Class 13 | Dijkgraaf-Witten invariant | Understand the contents covered by the lecture. |
| Class 14 | Turaev-Viro invariant | Understand the contents covered by the 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)
Not specified in particular.
Reference books, course materials, etc.
References and some handouts will be provided in the lectures.
Evaluation methods and criteria
Students to write a report on some aspects of the course.
Related courses
- MCS.T231 ： Algebra
- MCS.T201 ： Set and Topology I
Prerequisites
None.