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Cluster Algebra Via Non-Archimedean Enumerative Geometry

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CARW03 - Interdisciplinary applications of cluster algebras

I will discuss the mirror symmetry of cluster varieties via the enumeration of non-archimedean analytic curves. We construct a canonical scattering diagram by counting infinitesimal non-archimedean cylinders bypassing the Kontsevich-Soibelman algorithm. We prove adic convergence, ring homomorphism, finite polyhedral approximation, theta function consistency and Kontsevich-Soibelman consistency. Furthermore, we prove a comparison theorem with the combinatorial constructions of Gross-Hacking-Keel-Kontsevich. This has two-fold implications: First it gives a concrete combinatorial way for computing the abstract non-archimedean curve counting in the case of cluster varieties; conversely, we obtain geometric interpretations of various combinatorial constructions and answer several conjectures of GHKK . Joint work with S. Keel.

This talk is part of the Isaac Newton Institute Seminar Series series.

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