Alladi Ramakrishnan Hall

The Cohomology Ring of the Complex Grassmannian

Vijay Ravikumar

TIFR

We give an introduction to Schubert Calculus by studying the cohomology ring of a complex Grassmannian X. We describe the Schubert varieties of X and show how the corresponding Schubert classes form an additive basis for H*(X). By relating the intersections of Schubert varieties to the products of Schubert classes, we determine the multiplicative structure on H*(X) with respect to the Schubert basis. In particular, we describe Pieri's rule for multiplying arbitrary Schubert classes with certain special Schubert classes, and show how it can be used to solve classical problems in enumerative geometry.