Sangam: A Confluence of Knowledge Streams

MUTATION INVARIANT FUNCTIONS ON CLUSTER ENSEMBLES ASSOCIATED WITH SURFACES

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dc.contributor Zickert, Christian K
dc.contributor Digital Repository at the University of Maryland
dc.contributor University of Maryland (College Park, Md.)
dc.contributor Mathematics
dc.creator Kaufman, Dani Tove
dc.date 2021-07-14T05:33:10Z
dc.date 2021-07-14T05:33:10Z
dc.date 2021
dc.date.accessioned 2022-05-20T08:38:08Z
dc.date.available 2022-05-20T08:38:08Z
dc.identifier https://doi.org/10.13016/qodc-xuau
dc.identifier http://hdl.handle.net/1903/27440
dc.identifier.uri http://localhost:8080/xmlui/handle/CUHPOERS/117595
dc.description We 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.
dc.format application/pdf
dc.language en
dc.subject Mathematics
dc.subject Cluster Algebra
dc.subject Cluster Ensemble
dc.subject Hyperbolic Geometry
dc.subject Teichmuller Space
dc.title MUTATION INVARIANT FUNCTIONS ON CLUSTER ENSEMBLES ASSOCIATED WITH SURFACES
dc.type Dissertation


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