Mathematics Colloquium

We will discuss some recent (and not-so-recent) progress on a program to establish Kontsevich's homological mirror symmetry conjectures for hypersurfaces in (C*)^N and, by extension, in toric varieties. We will first review a geometric construction of mirrors of these hypersurfaces, which turn out to be toric Landau-Ginzburg models.

In this limited setting, we define a fiberwise wrapped Fukaya category of the Landau-Ginzburg model and show (by constructing a mirror to the structure sheaf, or of other line bundles) that the derived category of the hypersurface embeds into it. This is joint work in progress with Mohammed Abouzaid.