### Abstract or Additional Information

A theorem of B. Kim identified the relations of the quantum cohomology ring of the (generalized) flag manifolds with the conserved quantities for the Toda lattice. There were expectations that a similar statement exists, relating a previously undefined quantum cohomology ring for the affine flag manifolds to the periodic Toda lattice. I will show how to construct such a quantum cohomology ring, which deforms the usual quantum cohomology ring and it depends on an additional affine quantum parameter. The construction uses the technique 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 turns out that the conserved quantities of the periodic Toda lattice give the ideal of relations in the new ring, at least in Lie types A-D and E6. The current project is joint with Liviu Mare.