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Cluster algebras and discrete integrable systems

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Discrete Integrable Systems

We consider a large family of nonlinear rational recurrence relations which arise from mutations in cluster algebras defined by quivers. The advantage of the cluster algebra formalism is that it immediately provides an invariant symplectic (or presymplectic) structure. The problem of determining which of the recurrences are integrable in the sense of Liouville’s theorem is related to the notion of algebraic entropy, and via a series of conjectures related to tropical (max-plus) algebra, this leads to a very sharp criterion for the allowed degrees of the terms in the recurrence. As a result, four infinite families of discrete integrable systems are obtained. This is joint work with Allan Fordy.

This talk is part of the Isaac Newton Institute Seminar Series series.

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