 Thursday: October 26th, 2000
 Mathematics Colloquium:
Alex Postnikov, University of California at Berkeley, will speak on
"Schubert Calculus and Quantum Cohomology", at 4:00PM in Math 501.
Refreshments at 3:30PM in Math 402.
Abstract: A long standing open problem in Schubert
calculus is to give a combinatorial interpretation to the intersection
numbers of Schubert varieties in flag manifolds. Equivalently, these
numbers are the structure constants of the cohomology ring of the flag
manifold. They generalize the famous LittlewoodRichardson
coefficients. I will discuss the generalization of this problem to the
quantum cohomology ring of the flag manifold, which is a
multiparameter deformation of the usual cohomology ring. Its structure
constants are called the GromovWitten invariants. Many results from
the Schubert calculus, such as Monk's and Pieri's rules, naturally
extend to the quantum cohomology. Remarkably, the GromovWitten
invariants seem to be more symmetric objects than the Schubert
intersection numbers. They possess new properties that could not be
easily detected on the "classical'' level of the Schubert intersection
numbers.
