Jakob Hultgren Umeå University: SYZ for hypersurfaces of toric Fano manifolds & optimal transport II

Jakob Hultgren, Umeå University: SYZ for hypersurfaces of toric Fano manifolds and optimal transport II
The original SYZ-conjecture in mirror symmetry asks for special Lagrangian torus fibrations in Calabi-Yau manifolds. I will focus on families of Calabi-Yau hypersurfaces in toric Fano manifolds. Recent work by Yang Li reduces a weak version of the SYZ-conjecture in this setting to the solvability of a real Monge-Ampère equation on the boundary of a polytope. I will give a brief introduction to this and then explain how, curiously, this Monge-Ampère equation is solvable for some families and not solvable for other families. I will explain how solvability can be described in terms of optimal transport theory, and how another subtle aspect of the PDE: the location of the discriminant locus, becomes less mysterious when viewed through the lens of optimal transport. Finally, I will highlight some of the many open problems related to this. This is based on joint work with Rolf Andreasson, Mattias Jonsson, Enrica Mazzon and Nick McCleerey.