K-theoretic mirror formulae

I will define permutation-equivariant K-theoretic Gromov-Witten invariants (introduced by Givental). Roughly speaking "mirror formulae" means that certain q-hypergeometric series associated to a manifold X are generating series of these invariants. I will write these series for certain X (toric,zero sections of convex bundles) and , time permitting , discuss the proofs of such results.