Abstract

In this talk, I will give a gentle introduction to the cluster varieties. I will first define the cluster varieties in detail. We will study their basic properties. Then I will talk about the tropical version of the construction. In the end, I will state the Fock-Goncharov conjecture which says roughly that the tropical points in a cluster variety parameterizes a basis for the regular functions on its mirror. Some nice sufficient conditions were obtained by Gross-Hacking-Keel-Kontsevich. If time permit, I will either illustrate how to visualize the definition from the quivers or give some concrete examples from the representation theory (upon audience’s request).