  This article is an elaboration of a talk given at an international conference
on Operator Theory, Quantum Probability, and Noncommutative Geometry held
during December~20--23, 2004, at the Indian Statistical Institute, Kolkata. The
lecture was meant for a general audience, and also prospective research
students, the idea of the quantum cohomology based on the Gromov--Witten
invariants. Of course there are many important aspects that are not discussed
here.
