UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

March 07, 2017

939 Evans Hall

3:45PM: The maximum likelihood degree of a toric variety

Serkan Hosten

The maximum likelihood degree (ML degree) of a projective variety is the number of complex critical points of the likelihood function with generic data. We consider the ML degree of projective toric varieties. For this, we look at "scaled" toric varieties given by a monomial parametrization involving arbitrary complex coefficients. We show that for generic choice of coefficients the ML degree of such a scaled toric variety is equal to the degree of the original toric variety. We also prove that the ML degree drops if and only if the scaling vector is on the hypersurface defined by the principal A-determinant. With this tool in our hand, we will compute the ML degree of various classical toric varieties, of toric hypersurfaces, and toric varieties arising in graphical models.

5:00PM: Genus-One Landau-Ginzburg/Calabi-Yau Correspondence

Dustin Ross

First suggested by physicists in the late 1980's, the Landau-Ginzburg/Calabi-Yau correspondence studies a relationship between spaces of maps from curves to the quintic 3-fold (the Calabi-Yau side) and spaces of curves with 5th roots of their canonical bundle (the Landau-Ginzburg side). The correspondence was put on a firm mathematical footing in 2008 when Chiodo and Ruan proved a precise statement for the case of genus-zero curves, along with an explicit conjecture for the higher-genus correspondence, which is determined from genus-zero data alone. In this talk, I will begin by describing the motivation and the mathematical formulation of the LG/CY correspondence, and I will report on recent work with Shuai Guo that verifies the higher-genus correspondence in the case of genus-one curves.

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