Cluster Algebra
Project Description
"How do you describe a matrix where the determinant of every submatrix is positive?"
The answer to this simple question has a rich algebraic structure. The theory of Cluster Algebra was introduced in 2000 to study the above problem of total positivity. This new theory quickly becomes one of the most important research areas in mathematics, which finds applications in representation theory, combinatorics, hyperbolic geometry, algebraic geometry, dynamical systems, quantum theory, and mathematical physics.
The projects aim to introduce the basics of this important subject and some possible research direction to motivated undergraduate students. Students who have excellent performances will be invited to continue on the project for more original research.
The answer to this simple question has a rich algebraic structure. The theory of Cluster Algebra was introduced in 2000 to study the above problem of total positivity. This new theory quickly becomes one of the most important research areas in mathematics, which finds applications in representation theory, combinatorics, hyperbolic geometry, algebraic geometry, dynamical systems, quantum theory, and mathematical physics.
The projects aim to introduce the basics of this important subject and some possible research direction to motivated undergraduate students. Students who have excellent performances will be invited to continue on the project for more original research.
Supervisor
IP, Ivan Chi Ho
Quota
10
Course type
UROP1100
UROP2100
UROP3100
UROP3200
UROP4100
Applicant's Roles
The students will be introduced to cluster algebra and read related research papers under the guidance of the supervisor. Students should have a solid background in linear algebra and abstract algebra (group and ring theory). At the end of the project, students in UROP 1100 are expected to write an expository article on the subject. For students in UROP 2100 or higher, some original research works in combinatorics, representation theory or mathematical physics are expected.
Applicant's Learning Objectives
1) Understand the motivation of cluster algebra and get working experience in the subject.
2) Improve critical thinking skills through reading pure mathematics paper.
3) Learn to search for references during research
4) Prepare themselves for postgraduate studies in mathematics or physics.
2) Improve critical thinking skills through reading pure mathematics paper.
3) Learn to search for references during research
4) Prepare themselves for postgraduate studies in mathematics or physics.
Complexity of the project
Moderate