Cluster algebras

MATH 264Y Subject & Catalog Number

Description

This course will survey one of the most exciting recent developments in algebraic combinatorics, namely, Fomin and Zelevinsky's theory of cluster algebras. Cluster algebras are a class of combinatorially defined commutative rings that provide a unifying structure for phenomena in a variety of algebraic and geometric contexts. Introduced in 2001, cluster algebras have already been shown to be related to a host of other fields of math, such as quiver representations, Teichmuller theory, Poisson geometry, and total positivity. Cluster structures in Grassmannians have in particular been linked to integrable systems and physics. In the first part of the course I will cover the basics of cluster algebras and total positivity. In the second part of the class I will discuss recent developments and applications of the theory (topics could include the positive Grassmannian, the amplituhedron, KP solitons, etc). I will assume that people have some familiarity with combinatorics. Familiarity with root systems would also be helpful. I will not assume prior knowledge of total positivity or cluster algebras.