Cluster Varieties, Toric Degenerations and Convexity in Algebraic Geometry

Learn about cluster varieties, their role in algebraic geometric approaches to higher Teichmüller theory, and their connections to toric geometry in this 57-minute lecture from Universidad Nacional Autonoma de Mexico. Explore how the Gross-Siebert mirror symmetry program's ideas, developed by Gross, Hacking, Keel, and Kontsevich in 2014, revealed fundamental properties of cluster varieties and cluster algebras. Discover the generalization of key toric geometry constructions and their application to studying minimal models of cluster varieties and toric degenerations. Examine a specialized notion of convexity that enables the production of minimal models for cluster varieties, building upon the foundational work introduced by Fock and Goncharov in the early 2000s.