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NCTS Seminar in Algebraic Geometry
14:00 - 15:30, April 28, 2017 (Friday)
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)

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).

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, SergeyZelevinsky, Andrei Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15 (2002), no. 2, 497–529.
Gross, MarkHacking, PaulKeel, Sean Birational geometry of cluster algebras. Algebr. Geom. (2015), no. 2, 137–175.
Gross, Mark; Hacking, Paul; Keel, Sean; Kontsevich, Maxim, Canonical bases for cluster algebras, arXiv:1411.1394


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