Schubert calculus and geometric shifting: combinatorics meets combinatorics, via geometry-- Allen Knutson

A classic theorem in algebraic combinatorics is the Littlewood-Richardson rule, which computes in a manifestly positive way the multiplicative structure constants of the Schur function basis of the ring of symmetric functions. A classic theorem of extremal combinatorics is the Erdo"s-Ko-Rado theorem, which describes the maximum solutions to the following problem: consider k-subsets of {1,...,n} such that any two intersect. Then (for n>2k) the only way to maximize a collection of such is for there to exist a special element that lies in every subset. To prove their theorem, EKR invented the "shifting" technique. I'll recall its definition, and define a geometric version, which I'll use to prove the Littlewood-Richardson rule, thought of in terms of intersection theory on Grassmannians. This work is joint with Ravi Vakil.