Cluster Algebras and Their Applications

First introduced by Fomin and Zelevinsky, cluster algebras are commutative rings that have many combinatorial properties. They have had many applications to both mathematics and physics. In this talk, I will first introduce cluster algebras and explore some of their properties. I will then move on to their applications, starting with dilogarithm identities and then moving to integrable systems and the thermodynamic Bethe ansatz (TBA). Time permitting, I will connect some of these ideas to the ODE/IM correspondence.