Research Seminar: Complex Geometry

05.06.2015 Travis Mandel (QGM Aarhus)

Tropical curve counting and canonical bases

Abstract: Gross, Hacking, Keel, and Kontsevich recently constructed certain canonical bases for cluster algebras. The construction is combinatoric, but the bases are conjecturally controlled by the Gromov-Witten theory of the mirror cluster variety. I will discuss a new construction of these bases in terms of certain tropical curve counts which one expects to correspond to the predicted holomorphic curve counts. I will also discuss a refinement of the tropical counts which produces quantized versions of the canonical bases.