MATH4123 代数学特論II ・ 代数学 特論C
Topics in Algebra: Introduction to Cluster Algebra

Time and Venue
月3 、6号館-302
(Monday 13:00 - 14:30. Room 302, Building No. 6, North Campus, Kyoto University)

Course Description
The purpose of the course is to give a gentle introduction to the theory of Cluster Algebra. Cluster Algebra was introduced by S. Fomin and A. Zelevinsky around 2000 as a tool for studying total positivity and dual canonical bases in Lie theory. The rapid development of the theory of Cluster Algebra in recent years revealed relations to diverse area of mathematics, including commutative and non-commutative algebraic geometry, quiver representations, discrete dynamical systems and Teichmuller theory. I hope students who successfully complete this course will acquire a working knowledge of Cluster Algebras which is central to many research area in Mathematics and Physics.

References
Lectures


Part 1: Definition and Examples
2017 - 04 - 10 Lecture 1 Introduction & Examples
2017 - 04 - 17Lecture 2Total Positivity
2017 - 04 - 24Lecture 3Cluster Algebra - Definition and Examples
2017 - 05 - 01No Class
Part 2: Properties and Classification
2017 - 05 - 08Lecture 4Semifield and Coefficients
Laurent Phenomenon
2017 - 05 - 15Lecture 5Review of Root Systems
2017 - 05 - 22Lecture 6Finite Type Classification of Cluster Algebra
2017 - 05 - 29Lecture 7Generalized Associahedron - Construction
2017 - 06 - 05Lecture 8Generalized Associahedron - Exchangeable Roots
Part 3: Upper Cluster Algebra & Double Bruhat Cells
2017 - 06 - 12 Lecture 9Upper Cluster Algebra
(Supplementary: Proof that U(S)=U(S') if S~S' are both coprime)
2017 - 06 - 19Lecture 10Double Bruhat Cells - Description of Cluster Algebra
2017 - 06 - 26Lecture 11Double Bruhat Cells - Generalized Minors
Part 4: Cluster Algebra from Surface and Grassmannian
2017 - 07 - 03Lecture 12Cluster Algebra from Surface
2017 - 07 - 10Lecture 13Lambda Length
2017 - 07 - 31Lecture 14Cluster Algebra from Grassmannian

Homework

Due Date
2017 - 05 - 08 Homework 1Solution 1
2017 - 06 - 12Homework 2Solution 2
2017 - 07 - 03Homework 3Solution 3