Thursday, April 12, 2018,
1:10-2:25 p.m. Waterman 402
Melvin, Middleburry College
Crystals, Langlands duality and mirror symmetry
The Langlands philosophy has motivated significant advances in representation theory and geometry. For a reductive algebraic group G, having Langlands dual group G', the slogan is: the geometry associated to G controls the representation theory of G'. In algebraic and symplectic geometry, the philosophy of mirror symmetry states that the complex geometry of a (Fano, Kahler) complex variety X is controlled by the symplectic geometry of a mirror partner X'. Recently, Lam-Templier have described explicit realizations of the Langlands philosophy in the mirror symmetry of flag varieties. In this talk, I will define a version of mirror symmetry for flag varieties and describe recent work concerning the appearance of combinatorial representation-theoretic data, known as Kashiwara crystals, in this story. I will then describe a conjectural program relating the appearance of this combinatorial data to hierarchies of toric degenerations of flag varieties. The discussion will be kept down-to-earth and be guided by examples.