Institute for Geometry and Physics Miami-Cinvestav-Campinas

Abstract: The topology of hypersurfaces in toric varieties is largely governed by the structure of triangulations of the corresponding newton polytopes. The combinatorial gadget encoding these triangulation is a by now classical polytope called the secondary polytope. I'll discuss various enhancements of this polytope which afford more refined information about toric hypersurfaces. This is part of joint work with Katzarkov and G. Kerr.

Abstract: I will explain the consequences of toric phase change (physically predicted by Herbst, Hori, and Page, and mathematically described by Herbst and Walcher, and generalized by Ballard, Katzarkov, and myself) on homological projective duality, as introduced by Kuznetsov. I will remind the audience of the toric phase change picture, and discuss the case of the homological projective dual to the Veronese embedding for arbitrary degree.

Abstract: In joint work with Anton Mellit we show that four-dimensional variety with a full exceptional collection of 5 coherent sheaves is biregular isomorphic to the projective fourspace.

Abstract: The caustic is an enveloppling hypersurface for a family of curves (cf. caustics in optics where family of curves is formed by light rays). Caustics form walls in configuration spaces where the number ofreal curves jump in positive genus. These walls disappear in the tropical limit. This disappearance is responcible for invariance of planar Welschinger numbers also in positive genus. The talk will explain this phenomenon.

Abstract: We discuss some complements of my joint work with Mark Gross and Ludmil Katzarkov on mirror symmetry for varieties of general type. In particular, we explain how the employed duality of Landau-Ginzburg models is related to the discrete Legendre transform of Gross and Siebert. We will address the generalized homological mirror symmetry conjecture, Hochschild (co-)homology and Orlov's theorem on hypersurfaces in projective space.