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.
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.
discrete structure, algebraic structure, geometric structure
Intercultural skills | Communication skills | Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
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✔ | - | ✔ | ✔ | ✔ |
The course provides advanced topics of discrete, algebraic and geometric structures.
Course schedule | Required learning | |
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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 a functor | Understand the contents covered by the lecture. |
Class 6 | Hopf algebras | Understand the contents covered by the lecture. |
Class 7 | Quantum groups | Understand the contents covered by the lecture. |
Class 8 | Colored Jones polynomial | Understand the contents covered by the lecture. |
Class 9 | Universal quantum invariant | Understand the contents covered by the lecture. |
Class 10 | Commutativity of the Hopf algebra structure of quantum groups and that of bottom tangles | Understand the contents covered by the lecture. |
Class 11 | Dehn surgery | Understand the contents covered by the lecture. |
Class 12 | unified Witten-Reshetkhin-Turaev invariant | Understand the contents covered by the lecture. |
Class 13 | triangulation | Understand the contents covered by the lecture. |
Class 14 | Dijkgraaf-Witten invariant | Understand the contents covered by the lecture. |
Class 15 | Hennings invariant | Understand the contents covered by the lecture. |
Not specified in particular.
References and some handouts will be provided in the lectures.
Students to write a report on some aspects of the course.
None.