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Title:
Microlocal Sheaves and Cluster Algebras
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Abstract:
Many interesting spaces, such as wild character varieties, positroid strata of the Grassmannian, and integrable systems associated to lattice polygons, can be described as moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the cluster structures on these spaces are systematically recovered by the Floer theory of exact Lagrangian fillings of these Legendrians. The bipartite graphs of e.g. Postnikov's work on the totally positive Grassmannian appear as retracts of certain canonical Lagrangian fillings. More general cluster varieties appear as spaces of microlocal sheaves on skeleta of Weinstein 4-manifolds more general than cotangent bundles. This is joint work with Vivek Shende, David Treumann, and Eric Zaslow.
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