Speaker: Sergey Galkin
Title: Minuscule varieties and mirror symmetry
Date: Feb 1, 2016, 14:55 (pre-talk at 14:00)
Place: LC303B, University of South Carolina. Algebraic Geometry, Arithmetic Geometry, and Commutative Algebra Seminar.
I will give a uniform construction of a polynomial that is mirror dual to minuscule homogeneous varieties (e.g. Grassmannians), that we developed with Alexey Bondal. The construction involves a symmetry breaking mechanism of minuscule descent, that I will also explain. Homological mirror symmetry in this case predicts that derived categories of coherent sheaves on these varieties admit full exceptional collections, and it is likely that a uniform construction similar to our mirror construction could produce them, but it have not been found yet. In the pre-talk I will give some background on interplay between projective geometry of homogeneous varieties, representation theory and combinatorics. In particular I'll define (co)minuscule varieties and representations, recall geometric classification of Lie groups (after Landsberg and Manivel), describe Schubert cells, and define Bruhat and Hasse partially ordered sets, and explain Gonciulea-Lakshmibai's toric degenerations to moduli spaces of quiver representations.