Abstract: Cluster varieties are log Calabi-Yau varieties which are unions of algebraic tori glued by birational "mutation" maps. In particular, they are blowups of toric varieties. We will show how to generalize the polytope construction of toric varieties to cluster varieties. As an application, we will see that the non-integral vertex in the Newton-Okounkov body of the Grassmannian comes from the so-called broken line convexity. We will also discuss mirror dualities of cluster varieties from the symplectic perspective. This talk will be based on a series of joint works with Bardwell-Evans, Bossinger, Hong, Lin, Magee, and Najera-Chavez.
