UWashington-PIMS Mathematics Colloquium: Lauren K. Williams

The framework of mirror symmetry, originally discovered by string theorists, asserts that geometric objects come in "mirror pairs" (X,Y), where the enumerative geometry (e.g. quantum cohomology) of X controls the complex geometry of the mirror dual Y. Mirror symmetry for toric varieties and flag varieties, including Grassmannians, has been extensively studied: a prototypical theorem is that the quantum cohomology of X is isomorphic to the Jacobian ring of the mirror dual. In joint work with Konstanze Rietsch, we prove a "polytopal mirror theorem'' for Grassmannians, which says that the Newton Okounkov body of X coincides with the superpotential polytope of its mirror dual Y. We then use this polytopal mirror statement as an ansatz to construct some new mirrors, in particular, the mirror of each Schubert variety in the Grassmannian. Finally, we observe that the lattice points of our Newton-Okounkov bodies have a mysterious interpretation in terms of quantum cohomology. This talk will be accessible to a general audience.