Speaker: Sergey Galkin

Title: On conjectures of Dubrovin and Ostrover-Tyomkin

Date: May 30, 2014, 14:30-15:30

Place: Gyeongju

Abstract:

I will briefly review three recent works:

Next four algebraic properties of quantum cohomology of a Fano manifold are all distinct:

In 1404.7388 using mirror symmetry and a simple argument from Ginzburg-Landau theory I show that property (4) should also hold for all Fano manifolds, and prove it for all toric Fano manifolds, thus confirming conjecture of Ostrover-Tyomkin. This is also related to Conjecture O (of 1404.6407) that roughly says that the structure sheaf of a Fano manifold is mirror dual to the Lagrangian thimble formed by the locus of real positive points. Ostrover and Tyomkin shown that for some toric Fano fourfolds (3) fails, but (2) is true. In 1405.3857 we show that for isotropic Grassmannian IG(2,6) (2) fails, but (1) is true; and also the derived category of coherent sheaves has a full exceptional collection, so it is the first case where one needs big quantum cohomology to formulate the first part of Dubrovin's conjecture. Finally, in 1404.3857 we formulate Gamma Conjecture I (related to Conjecture O) and also Gamma Conjecture II (which is the exact formulation of the third part of Dubrovin's conjecture). I will provide some confirming examples, and give some ways that could lead to proving these conjectures.