A theorem of B. Kim identified the relations of the quantum
cohomology ring of the (generalized) flag manifolds G/B with the conserved
quantities for the Toda lattice. M. Guest and T. Otofuji, and L. Mare,
showed that if a similar quantum cohomology ring exists for affine flag
manifolds, then its relations will be determined by the periodic Toda
lattice. I will show how to construct a quantum product which deforms the
usual quantum cohomology product and which depends on an additional affine
quantum parameter. It turns out that the conserved quantities of the
periodic Toda lattice give the ideal of relations in the new ring. The
construction involves a generalization of the notion of ``curve
neighborhoods" of Schubert varieties, which were defined and studied earlier
by the speaker in several joint works with A. Buch, P.E. Chaput, and N.
Perrin. It also requires a generalization of the ``Givental formalism"
(quantum differential equations, a flat connection etc) to this case. The
current project is joint with Liviu Mare.