Room 201, Astronomy-Mathematics Building, NTU

(台灣大學天文數學館 201室)

An Introduction to Cluster Varieties

Jia-rui Fei (NCTS)

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).

Reference

Fock, Vladimir V.; Goncharov, Alexander B. Cluster ensembles, quantization and the dilogarithm. *Ann. Sci. Éc. Norm. Supér. (4)* 42 (2009), no. 6, 865–930.

Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. I. Foundations. *J. Amer. Math. Soc.* 15 (2002), no. 2, 497–529.

Gross, Mark; Hacking, Paul; Keel, Sean Birational geometry of cluster algebras. *Algebr. Geom.* 2 (2015), no. 2, 137–175.

Gross, Mark; Hacking, Paul; Keel, Sean; Kontsevich, Maxim, Canonical bases for cluster algebras, arXiv:1411.1394