Title : Canonical basis, cluster variety, and Riemann's moduli space of curves

Speaker : Hyun Kyu Kim (KIAS)

Abstract : This is an introductory talk on the canonical basis of the algebra of regular functions on a cluster variety. I will start from the basic example of a split algebraic torus, then review a rank 2 example of a cluster variety. Then I will review Riemann's moduli space of curves, the Teichmüller space which is its universal cover, and discuss the cluster variety structure that this space has. I will describe Fock and Goncharov's canonical basis for this case, based on the trace-of-monodromy functions along loops. If time allows, I will discuss quantization, higher rank generalization, and some open questions.