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Twisted Gromov-Witten invariants are intersection numbers in moduli spaces of stable maps to a manifold or orbifold which depend in addition on a vector bundle over and an invertible multiplicative characteristic class. Special cases are closely related to local Gromov-Witten invariants of the bundle and to genus-zero one-point invariants of complete intersections in . We develop tools for computing genus-zero twisted Gromov-Witten invariants of orbifolds and apply them to several examples. We prove a “quantum Lefschetz theorem” that expresses genus-zero one-point Gromov-Witten invariants of a complete intersection in terms of those of the ambient orbifold . We determine the genus-zero Gromov-Witten potential of the type surface singularity . We also compute some genus-zero invariants of , verifying predictions of Aganagic, Bouchard, and Klemm . In a self-contained appendix, we determine the relationship between the quantum cohomology of the surface singularity and that of its crepant resolution, thereby proving the Crepant Resolution Conjectures of Ruan and of Bryan and Graber  in this case
We classify finite-dimensional simple spherical representations of rational double affine Hecke algebras, and we study a remarkable family of finite-dimensional simple spherical representations of double affine Hecke algebras.
Let be a totally real field of narrow class number one, and let be a modular, semistable elliptic curve of conductor . Let be a non-CM quadratic extension with such that the sign in the functional equation of is . Suppose further that there is a prime that is inert in . We describe a -adic construction of points on which we conjecture to be rational over ring class fields of and satisfy a Shimura reciprocity law. These points are expected to behave like classical Heegner points and can be viewed as new instances of the Stark-Heegner point construction of . The key idea in our construction is a reinterpretation of Darmon's theory of modular symbols and mixed period integrals in terms of group cohomology
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