Frank Sottile  (University of Toronto and MSRI)

          "Is the Schubert Calculus Effective?"

	     Wednesday, 25 September (Colloquium)

The Schubert calculus, a basic tool in enumerative geometry, gives
algorithms for solving problems of enumerating linear subspaces of
projective space, such as: How many lines in projective 4-space meet 6
given planes?  (Answer: 5) Its core component is the intersection
form, also known as the Littlewood-Richardson rule, for the
cohomology (or Chow) ring of a Grassmann manifold in terms of its
basis of Schubert classes.

An effective version of the Schubert calculus would explicitly find
the 5 lines meeting 6 given planes in P^4. It may also explicitly
describe how to deform an intersection of two Schubert varieties into
a sum of distinct Schubert varieties, that is, a geometric
manifestation of the Littlewood-Richardson rule. This introductory
talk describes some results along these lines, including an effective
version of Pieri's formula.