Twisted Gromov-Witten invariants are intersection numbers in moduli spaces of stable maps to a manifold or orbifold $X$ which depend in addition on a vector bundle over $X$ 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 $X$. 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 $X$. We determine the genus-zero Gromov-Witten potential of the type $A$ surface singularity $[{\mathbb{C}}^2/{\mathbb{Z}}_n]$. We also compute some genus-zero invariants of $[{\mathbb{C}}^3/{\mathbb{Z}}_3]$, verifying predictions of Aganagic, Bouchard, and Klemm [3]. In a self-contained appendix, we determine the relationship between the quantum cohomology of the $A_n$ surface singularity and that of its crepant resolution, thereby proving the Crepant Resolution Conjectures of Ruan and of Bryan and Graber [12] 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 $F$ be a totally real field of narrow class number one, and let $E/F$ be a modular, semistable elliptic curve of conductor $N\ne (1)$. Let $K/F$ be a non-CM quadratic extension with $(\mathrm{Disc}K,N)=1$ such that the sign in the functional equation of $L(E/K,s)$ is $-1$. Suppose further that there is a prime $\mathfrak{p}|N$ that is inert in $K$. We describe a $\mathfrak{p}$-adic construction of points on $E$ which we conjecture to be rational over ring class fields of $K/F$ 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 [5]. 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