2025 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra C1
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Hironori Oya
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- MTH.A407
- Number of credits
- 100
- Course offered
- 2025
- Offered quarter
- 3Q
- Syllabus updated
- Mar 19, 2025
- Language
- English
Syllabus
Course overview and goals
This is an introductory course on cluster algebras. Cluster algebras, introduced by Fomin and Zelevinsky in the early 2000s, are algebras defined through certain combinatorial procedure. They have been found to be related to various branches of mathematics and have been studied from multiple perspectives.
In this course, we will begin with the definition of cluster algebras and explore their general theory. Additionally, we will study concrete examples of mathematical objects that possess a cluster algebra structure. Through this course, together with the subsequent "Advanced Topics in Algebra D1", students are expected to learn the definition of cluster algebras and their ubiquity.
Course description and aims
- To be able to explain the definition of cluster algebras.
- To be able to explain examples of cluster algebras.
- To be able to explain the statement of fundamental theorems in the theory of cluster algebras.
Keywords
Mutation of quivers, Cluster algebras, Laurent phenomenon, c/g-vectors, F-polynomials
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course. Assignments will be given during the classes.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Introduction to cluster algebras by examples | Details will be provided during each class session |
| Class 2 | Definition of cluster algebras | Details will be provided during each class session |
| Class 3 | Laurent phenomenon: part 1 | Details will be provided during each class session |
| Class 4 | Laurent phenomenon: part 2 | Details will be provided during each class session |
| Class 5 | Cluster algebras of finite type | Details will be provided during each class session |
| Class 6 | c/g-vectors, F-polynomials: part 1 | Details will be provided during each class session |
| Class 7 | c/g-vectors, F-polynomials: part 2 | Details will be provided during each class session |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to explore references provided in the lectures and other materials.
Textbook(s)
None in particular.
Reference books, course materials, etc.
・T. Nakanishi, Cluster Algebras and Scattering Diagrams, MSJ Mem., 41, Mathematical Society of Japan, Tokyo, 2023. xiv+279 pp.
・S. Fomin and A. Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no.2, 497-529.
・S. Fomin and A. Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no.1, 63-121.
・A. Berenstein, S. Fomin, and A. Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), no.1, 1-52.
・S. Fomin and A. Zelevinsky, Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), no.1, 112-164.
Evaluation methods and criteria
Assignments (100%)
Related courses
- Algebra III
- MTH.A408 ： Advanced topics in Algebra D1
- ZUA.A334 ： Advanced courses in Algebra D
Prerequisites
It is desirable to have basic knowledge on algebra.
Other
None in particular.