Cluster Recurrences and Generalisations

Cluster algebras and their Laurent phenomenon can be used to produce certain integer sequences and recurrences e.g. the Markov numbers and Somos sequences and to prove the periodicity conjecture of Zamolodchikov. In this talk I will give describe the notion of a cluster recurrence in terms of an action of the symmetry group of a cluster algebra on its cluster variables and show how these examples arise. I will describe some recent work on understanding invariants of this action and (if time) some non-commutative generalisations. Based on joint work with Zach Greenberg.