Cluster Algebras

Cluster algebras were first defined by Fomin and Zelevinsky to study problems in total positivity and dual canonical bases. Since their birth, many other areas of mathematics and physics have found them useful! My favorite of these areas are algebraic combinatorics, representation theory, and dimer models.

Cluster algebras are certain subrings of the field of rational functions in n variables! The generators of these algebras are built by starting with an initial seed of data and recursively building more generators via a process called mutation.

This process of mutation can be unwieldy and becomes messy with large rational expressions in our n variables. One astounding fact is that these generators happen to simplify in nice ways! Finding ways to understand the resultant generators through a combinatorial interpretation (think pictures that represent messy algebraic expressions!) is an active area of research.