18.218   M.I.T.   Spring 2018

## Combinatorics of Grassmannian

Class meets: MWF 2-3 pm   Room 4-145         First class: Wednesday, February 7

Instructor: Alexander Postnikov

Course webpage: http://math.mit.edu/~apost/courses/18.218/

Keywords: Grassmannian, flag manifolds, Plucker coordinates, Schubert cells, Young diagrams, Littlewood-Richardson rule, matroids, MacPhersonian, hypersimplices, permutohedra, positive Grassmannian, positroids, plabic graphs, cluster algebras, quantum cohomology, Gromov-Witten invariants, scattering amplitudes, amplituhedron, tropical geometry, and more ...

Synopsis:

Have you ever wondered what the number of k-dimensional subspaces in the l-dimensional complex space that intersect with m generic n-dimensional subspaces is?

Questions like this are the subject of Schubert calculus, which studies beautiful geometric objects called Grassmannians and flag manifolds and related algebra and combinatorics. Many familiar combinatorial objects and constructions, such as Young diagrams and tableaux, the Littlewood-Richardson rule come into play.

Points of the Grassmannian can be thought of as linear subspaces in a vector space, or dually, as hyperplane arrangements. Combinatorics of hyperplane arrangements is the origin of matroids.

Grassmannian and flag manifold are closely related to the study of combinatorics of certain convex polytopes such as hypersimplices and permutohedra.

Speaking about more recent advances, the positive Grassmannian and related combinatorial objects, such as positroids and plabic graphs have appeared in various areas of mathematics and physics ranging from cluster algebras and quantum algebra to the study of solitons and scattering amplitudes. Physics leads to the study of the amplituhedron, which is an extension (or rather projection) of the positive Grassmannian that has even more mysterious combinatorial structure.

We'll talk about all this stuff and its generalizations in various directions: flag manifolds for other Cartan-Killing types, quantum cohomology and Gromov-Witten invariants, K-theory, extension of physics "beyond the planar limit" ...

The emphasis of the course will be on combinatorial objects and constructions that appear in all these areas.

The course should be accessible to first year graduate students.