Speaker: Sergey Galkin

Title: Mirror symmetries of P2 enumerated by Markov triplets.

Date: Jan 19, 2010, 13:15 - 14:45

Place: IPMU, Balcony B on the 5th floor of the new building (the first talk in the new IPMU building!)

Abstract:

Triplets of integer numbers (x,y,z) satisfying Markov's equation
__x^2 + y^2 + z^2 = 3 xyz__
are in charge of two numerologies for the projective plane P2:
these numbers are the ranks of exceptional bundles
and their squares are the weights of Prokhorov-Hacking's degenerations
of the plane to weighted projective plane P(x2,y2,z2).
Batyrev's ansatz states that given a (good) toric degeneration of variety X
one may construct a Landau-Ginzburg model mirror dual to X as a
Laurent polynomial with the Newton polytope being the fan polytope of
the degeneration.

I'll show this ansatz holds in the situation of Prokhorov-Hacking's degenerations, and relate the polynomials constructed from different degenerations by birational symplectomorphic mutations.