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 audiences are expected to gain advanced mathematical methods to analyze discrete, algebraic and geometric structures.
Poisson algebras and quantizations, quantum groups, quantum invariants of knots and 3-dimensional manifolds
✔ Specialist skills | Intercultural skills | Communication skills | ✔ Critical thinking skills | ✔ Practical and/or problem-solving skills |
Lectures and exercises
Course schedule | Required learning | |
---|---|---|
Class 1 | Lie algebras and symplectic manifolds | Understand the contents covered by the lecture. |
Class 2 | Modules over rings of power series and completions | Understand the contents covered by the lecture. |
Class 3 | Poisson algebras | Understand the contents covered by the lecture. |
Class 4 | Quantizations of Poisson algebras | Understand the contents covered by the lecture. |
Class 5 | Poisson manifolds | Understand the contents covered by the lecture. |
Class 6 | Quantization of Poisson manifolds | Understand the contents covered by the lecture. |
Class 7 | Physical meaning of quantization | Understand the contents covered by the lecture. |
Class 8 | Knots and links | Understand the contents covered by the lecture. |
Class 9 | Kauffman bracket and Jones polynomial | Understand the contents covered by the lecture. |
Class 10 | Hopf algebras | Understand the contents covered by the lecture. |
Class 11 | Quantum groups | Understand the contents covered by the lecture. |
Class 12 | Colored Jones polynomial | Understand the contents covered by the lecture. |
Class 13 | Universal quantum invariant | Understand the contents covered by the lecture. |
Class 14 | Quantum invariants of 3-dimensional manifolds | Understand the contents covered by the lecture. |
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.
Not specified.
References are provided in the lectures.
Evaluate by homework.
Not specified.