String theory, in the search for the ultimate laws of nature, places mathematical objects known as algebraic curves at the center of the modern framework of fundamental physics, generating new mathematical questions and pointing to plausible answers with an amazing pace and persistence. Questions to be investigated in this project lie at the crossroads of two major pathways in mathematics of the past two centuries. One of them is the in-depth pursuit of the intricate properties of algebraic curves, in the form inherited from works of Gauss, Abel, Jacobi, Riemann, Klein, and Poincare. The other is the broad conceptual landscape of mathematical physics, dictated by the progress of classical, statistical, and quantum mechanics, and often associated with the names of Hamilton, Maxwell, Gibbs, Poincare, Hilbert, Einstein, and Weyl. Some of the questions under study are motivated by mathematical questions arising out of string theory; in turn, the research is expected to provide feedback to string theorists inspiring previously unanticipated directions of research.The specific aim of this project is to develop a new chapter of the Gromov-Witten theory, that is, the theory of topological invariants of phase spaces of Hamiltonian systems, where the characters of permutation groups, acting by renumbering of marked points on the Cech cohomology of coherent sheaves over moduli spaces of stable maps of holomorphic curves to a target Kahler phase space, are studied and computed. The ongoing and forthcoming research of such permutation-equivariant K-theoretic Gromov-Witten invariants is to include: (a) constructing these invariants and exploring their general properties; (b) developing the symplectic loop-space quantization formalism for representing the invariants by generating functions; (c) establishing the appropriate Quantum Riemann-Roch Theorems to provide the complete adelic characterization of permutation-equivariant K-theoretic Gromov-Witten invariants in terms of cohomological Gromov-Witten invariants; (d) developing the fixed-point-localization techniques for computing permutation-equivariant Gromov-Witten invariants; (e) applying the techniques in order to obtain K-theoretic analogues of the mirror formulas (i.e., to identify toric q-hypergeometric functions with certain genus-0 permutation equivariant Gromov-Witten invariants of toric manifolds); (f) introducing and studying the K-theoretic mirrors (i.e., complex oscillatory representations of such q-hypergeometric functions, and the corresponding D_q-modules); (g) elucidating the role of the groups of q-difference operators acting by hidden symmetries in the permutation-equivariant quantum K-theory, and exploiting these symmetries to reconstruct all the genus-0 invariants of toric manifolds from the respective q-hypergeometric functions; (h) expressing twisted permutation-equivariant K-theoretic Gromov-Witten invariants in terms of the untwisted ones by combining the boson-fermion correspondence with the adelic characterization in higher genus; and (i) exploring the relationships between quantum K-theory of the point target space and the q-analogues of the KdV-hierarchy of integrable systems, anticipated by analogy with the Witten-Kontsevich theorem for cohomological Gromov-Witten invariants of the point.