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Title: Cluster structures on triangulated non-orientable surfaces
Author: Wilson, Jonathan Michael
ISNI:       0000 0004 6350 613X
Awarding Body: Durham University
Current Institution: Durham University
Date of Award: 2017
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In 2002, Fomin and Zelevinsky introduced a cluster algebra; a dynamical system that has already proved to be ubiquitous within mathematics. In particular, it was shown by Fomin, Shapiro and Thurston that some cluster algebras arise from orientable surfaces. Subsequently, Dupont and Palesi extended this construction to non-orientable surfaces, giving birth to quasi-cluster algebras. The finite type cluster algebras possess the remarkable property of their exchanges graphs being polytopal. We discover that the finite type quasi-cluster algebras enjoy a closely related property, namely, their exchange graphs are spherical. Revealing yet more connections we unify these two frameworks via Lam and Pylyavskyy's Laurent phenomenon algebras, showing that both orientable and non-orientable marked surfaces have an associated LP-algebra. The integration of these structures is attempted in two ways. Firstly we show that the quasi-cluster algebras of unpunctured surfaces have LP structures. Secondly, to obtain a connection for all marked surfaces, we consider laminations, forging the notion of the laminated quasi-cluster algebra. We show that each marked surface exhibits a lamination which supplies the laminated quasi-cluster algebra with an LP structure.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available