Gromov-Witten invariants and toric tableaux

Mathematics Colloquium, Purdue University

Alexander Postnikov

April 29, 2003

     The quantum cohomology ring is  a deformation of the multiplicative
structure of the usual cohomology.   This ring encodes the Gromov-Witten
invariants that  count numbers of  certain rational curves.   We present
a  simple  combinatorial  model  for   the  quantum  cohomology  of  the
Grassmannian.  We  give a formula  for the  GW-invariants in terms  of a
generalization of Schur symmetric polynomials.  The construction implies
several nontrivial symmetries of  the GW-invariants, including a certain
"strange duality" that inverts the quantum parameter q.  The constuction
also gives a simple solution to the problem posed by Fulton and Woodward
about characterization  of the  powers of  q that  occur in  the quantum
product.   The  talk  will  be  as elementary  as  possible.   No  prior
knowledge of the subject will be assumed.