2025 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra D1
- 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.A408
- Number of credits
- 100
- Course offered
- 2025
- Offered quarter
- 4Q
- Syllabus updated
- Mar 19, 2025
- Language
- English
Syllabus
Course overview and goals
This course follows "Advanced Topics in Algebra C1" and provides an introductory exposition 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 focus on concrete examples of mathematical objects that possess cluster algebra structures. The goal of this course is to understand how cluster algebra structures arise in mathematics.
Course description and aims
- To be able to explain a relation between cluster algebras and upper cluster algebras.
- To be able to explain a cluster algebra structure on the coordinate ring of a double Bruhat cell.
- To be able to understand mathematical statements that appear in advanced topics on cluster algebras.
Keywords
Cluster algebras, Upper cluster algebras, Locally acyclic cluster algebras, Double Bruhat cells, Categorification
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 | Locally acyclic cluster algebras | Details will be provided during each class session |
| Class 2 | Double Bruhat cells | Details will be provided during each class session |
| Class 3 | Cluster algebra structure on the coordinate ring of double Bruhat cells: part 1 | Details will be provided during each class session |
| Class 4 | Cluster algebra structure on the coordinate ring of double Bruhat cells: part 2 | Details will be provided during each class session |
| Class 5 | Categorification of cluster algebras | Details will be provided during each class session |
| Class 6 | Advanced topics: part 1 | Details will be provided during each class session |
| Class 7 | Advanced topics: 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
- MTH.A407 ： Advanced topics in Algebra C1
- ZUA.A334 ： Advanced courses in Algebra C1
- ZUA.A334 ： Advanced courses in Algebra D1
Prerequisites
It is desirable to have basic knowledge on algebra.
Other
None in particular.