On 13th June 2024 Dr. Theodoros Kouloukas gave a talk on the Liouville integrability of cluster maps and integrable deformations at the Algebraic and Geometric Aspects of Differential and Difference Equations workshop at the University of Portsmouth.

Abstract: We introduce a class of birational maps derived by sequences of cluster mutations and study their Liouville integrability using the properties of the underlying cluster algebra structure. We also consider deformations of the cluster maps, with compatible presymplectic forms defined by the corresponding quiver, which preserve integrability. In this framework, we present examples of integrable discrete Hirota reductions, periodic reductions of integrable lattice equations (dKdV, mKdV and sine-Gordon), transfer maps of Yang-Baxter maps and parametric deformations of finite type cluster
algebras of type A_N .