## Topology & Geometric Group Theory Seminar

## Fall 2007

### 1:30 - 2:30, Malott 253

Tuesday, November 6

**Allen
Knutson**, UC San Diego

*Schubert calculus and geometric shifting: combinatorics meets
combinatorics, via geometry*

A classic formula 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 Erdos-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 a new
Littlewood-Richardson rule, thought of in terms of intersection theory
on Grassmannians.

This work is joint with Ravi Vakil.

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