2024 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 / Zin Arai / Toshiaki Murofushi
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Mon / 5-6 Thu
- Class
- -
- Course Code
- MCS.T408
- Number of credits
- 200
- Course offered
- 2024
- Offered quarter
- 3Q
- Syllabus updated
- Mar 14, 2025
- Language
- English
Syllabus
Course overview and goals
In various studies of mathematical and computational sciences, discrete, algebraic, and geometric structures appear ubiquitously. This course will introduce advanced topics related to discrete, algebraic, and geometric structures, focusing on knot theory and quantum topology. The aim is to give participants an opportunity to explore some of the mathematical structures underlying research in mathematical and computational sciences.
Course description and aims
By taking this course, students will aim to acquire advanced knowledge related to discrete, algebraic, and geometric structures in knot theory and quantum topology, and further be able to apply this knowledge to several concrete problems.
Keywords
knot theory, quantum topology, discrete structure, algebraic structure, geometric structure
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
In this lecture, advanced topics in knot theory and quantum topology will be explained in a lecture format.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Categories and functors | Understand the contents covered by the lecture. |
| Class 2 | Monoidal categories | Understand the contents covered by the lecture. |
| Class 3 | Category of braids and braided categories. | Understand the contents covered by the lecture. |
| Class 4 | Pivotal categories and ribbon categories | Understand the contents covered by the lecture. |
| Class 5 | Category of tangles | Understand the contents covered by the lecture. |
| Class 6 | Reshetikhin-Turaev invariants and Witten-Reshetikhin-Turaev TQFT | Understand the contents covered by the lecture. |
| Class 7 | String links and bottom tangles | Understand the contents covered by the lecture. |
| Class 8 | "Hopf algebra actions" by external Hopf algebras in braided categories | Understand the contents covered by the lecture. |
| Class 9 | Equivariance property of Reshetikhin-Turaev invariants under Hopf algebra actions | Understand the contents covered by the lecture. |
| Class 10 | Invariant of 3-manifold using Hopf algebra and ideal triangulations 1 | Understand the contents covered by the lecture. |
| Class 11 | Invariant of 3-manifold using Hopf algebra and ideal triangulations 2 | Understand the contents covered by the lecture. |
| Class 12 | Burau representation, Alexander polynomial and cluster algebras 1 | Understand the contents covered by the lecture. |
| Class 13 | Burau representation, Alexander polynomial and cluster algebras 2 | Understand the contents covered by the lecture. |
| Class 14 | Exercises | Understand the contents covered by the lecture. |
Study advice (preparation and review)
In order to enhance learning outcomes, students should refer to the relevant sections of textbooks or distributed materials, and spend approximately 100 minutes each on preparation and review (including assignments) related to the content of each class.
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.