Bridge Seminar: Margaret Symington (Mercer University)

In 1996, the emerging understanding of mirror symmetry motivated Batyrev to introduce reflexive polytopes. Generalizations of reflexivity due to Batyrev & Borisov (1997) and to Nill & Kasprzyk (2011) also enjoy duality properties. In both cases, understanding dualities of complex algebraic varieties is a guiding objective. Changing the guiding objective to be understanding (singular) Lagrangian fibrations of symplectic manifolds also leads one to generalizations of reflexivity. In this talk, I will explain a natural way to define an integral affine structure on the complement of a set of points on the boundary of a polytope in R³, describe the generalization of reflexive polytopes that this leads to, and how this generalization relates to the works of Batyrev & Borisov and Nill & Kasprzyk in dimension 3.