2026 (Current Year) 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
Class Format: Lecture
Media-enhanced courses: -
Day of week/Period (Classrooms): unknown
Class: -
Course Code: MCS.T408
Number of credits: 200
Course offered: 2026
Offered quarter: 3Q
Syllabus updated: Aug 28, 2026
Language: English

Syllabus

Course overview and goals

This course introduces algebraic and geometric structures arising in knot theory and quantum topology from the viewpoint of category theory. Topics include monoidal categories, braid and tangle categories, dualities, pivotal categories, and ribbon categories. We also study Hopf algebras and their representation categories to understand how these categorical structures arise from algebra. Through concrete examples and graphical calculus, students will explore the common mathematical structures underlying discrete, algebraic, and geometric objects.

Course description and aims

By taking this course, students will understand the basic concepts of categories, functors, natural transformations, monoidal categories, braided categories, dualities, pivotal categories, and ribbon categories, and will be able to work with these structures in concrete examples. Students will also learn to use graphical calculus and understand monoidal structures, dualities, and braidings arising in representation categories of Hopf algebras.

Keywords

category theory, monoidal category, braids and tangles, pivotal and ribbon categories, Hopf algebra, quantum topology

Competencies

Specialist skills
Intercultural skills
Communication skills
Critical thinking skills
Practical and/or problem-solving skills

Class flow

In addition to lectures, the course includes pair-work exercises. Students will discuss basic definitions and concrete examples with their partners and present solutions to selected exercises on the board.

Course schedule/Objectives

	Course schedule	Objectives
Class 1	Categories and functors	Understand the contents covered by the lecture.
Class 2	Natural transformations	Understand the contents covered by the lecture.
Class 3	Monoidal categories	Understand the contents covered by the lecture.
Class 4	Monoidal functors	Understand the contents covered by the lecture.
Class 5	Category of braids	Understand the contents covered by the lecture.
Class 6	Braided categories	Understand the contents covered by the lecture.
Class 7	Category of tangles and its dualities	Understand the contents covered by the lecture.
Class 8	Category of framed tangles and its twists	Understand the contents covered by the lecture.
Class 9	Dualities, rigid monoidal categories, and dual functors	Understand the contents covered by the lecture.
Class 10	Pivotal categories and graphical calculus	Understand the contents covered by the lecture.
Class 11	Ribbon categories	Understand the contents covered by the lecture.
Class 12	Algebras, bialgebras, and their modules	Understand the contents covered by the lecture.
Class 13	Ribbon Hopf algebras and their modules	Understand the contents covered by the lecture.
Class 14	Reshetikhin-Turaev functor	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

Grades will be based on in-class pair-work exercises, board presentations, and the final examination. Pair work will be evaluated based on understanding of the course material and participation in discussion, board presentations on understanding of the problems and clarity of explanation, and the final examination on overall understanding of the course material.

Related courses

MCS.T231 ： Algebra
MCS.T201 ： Set and Topology I

Prerequisites

None.