Compactifications of cluster varieties and convexity

Abstract:

Cluster varieties are log Calabi-Yau varieties which are unions of algebraic tori glued by birational  "mutation" maps. They can be seen as a generalization of the toric varieties. In toric geometry, projective toric varieties can be described by polytopes. We will see how to generalize the polytope construction to cluster convexity which satisfies piecewise linear structure. As an application, we will see the non-integral vertex in the Newton Okounkov body of Grassmannian comes from broken line convexity. We will also see links to the symplectic geometry and application to mirror symmetry. The talk will be based on a series of joint works with Bossinger, Lin, Magee, Najera-Chavez, and Vianna.