MUTATION INVARIANT FUNCTIONS ON CLUSTER ENSEMBLES ASSOCIATED WITH SURFACES

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dc.contributor.advisorZickert, Christian Ken_US
dc.contributor.authorKaufman, Dani Toveen_US
dc.contributor.departmentMathematicsen_US
dc.contributor.publisherDigital Repository at the University of Marylanden_US
dc.contributor.publisherUniversity of Maryland (College Park, Md.)en_US
dc.date.accessioned2021-07-14T05:33:10Z
dc.date.available2021-07-14T05:33:10Z
dc.date.issued2021en_US
dc.description.abstractWe define the notion of an invariant function on a cluster ensemble with respect to a group action of the cluster modular group on its associated function fields. We realize many examples of previously studied functions as elements of this type of invariant ring and give many new examples. We construct invariants for cluster algebras associated with surfaces using hyperbolic geometry, Teichm\'uller theory and skein algebras of surfaces. We complete a classification of them for surface ensembles for the action of Dehn twists, and generalize this classification to the non-surface mutation finite setting. We use this classification to answer some questions about the structure of affine cluster algebras, to construct a correspondence between $\A$ and $\X$ invariants, and to propose an explanation for why many different computations of canonical bases of cluster algebras agree.en_US
dc.identifierhttps://doi.org/10.13016/qodc-xuau
dc.identifier.urihttp://hdl.handle.net/1903/27440
dc.language.isoenen_US
dc.subject.pqcontrolledMathematicsen_US
dc.subject.pquncontrolledCluster Algebraen_US
dc.subject.pquncontrolledCluster Ensembleen_US
dc.subject.pquncontrolledHyperbolic Geometryen_US
dc.subject.pquncontrolledTeichmuller Spaceen_US
dc.titleMUTATION INVARIANT FUNCTIONS ON CLUSTER ENSEMBLES ASSOCIATED WITH SURFACESen_US
dc.typeDissertationen_US

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