Integrable Systems Seminar: Fordy -- Periodic Cluster Mutation and Related Integrable Maps

Prof. Allan Fordy from University of Leeds, UK, will give a talk on 
Thursday 29th of August at 4pm in Carslaw room 829 (AGR). All welcome.

TITLE: Periodic Cluster Mutation and Related Integrable Maps

ABSTRACT: We consider nonlinear recurrences generated from cluster mutations applied
to quivers that have the property of being cluster mutation-periodic with period 1.
Such quivers were completely classified [1], being characterised in terms of the
skew-symmetric matrix that defines the quiver.

The associated nonlinear recurrences are equivalent to birational maps, which
generically preserve a nondegenerate Poisson bracket. In special cases the maps
have an invariant (degenerate) presymplectic structure, but they can be reduced to
symplectic maps on a space of lower dimension. Whilst the cluster property guarantees
the "Laurent property", it does not imply complete integrability.

Algebraic entropy can be used to identify the integrable cases, which fall into a
number of interesting classes [3].

Each of these can be analysed in the context of Liouville integrability.  Two of these
classes are "super-integrable" and even linearisable and give rise to interesting
Poisson algebras [2,3].

My aim is to review this work, but 1 hour will only permit a brief description.

1.  A.P. Fordy and R.J. Marsh,
     Cluster mutation-periodic quivers and associated Laurent sequences ,
     J Algebr Comb, 34, 19-66 (2011) , arXiv:0904.0200v5 [math.CO].

2.  A.P. Fordy,
     Mutation-Periodic Quivers, Integrable Maps and Associated Poisson Algebras,
     Phil. Trans. R. Soc. A 369 1264-1279 (2011) , arXiv:1003.3952v2 [nlin.SI].

3.  Allan Fordy and Andrew Hone,
     Discrete integrable systems and Poisson algebras from cluster maps,
     Comm Math Phys (In Press).  arXiv:1207.6072 [nlin.SI].