Let R_n be the ring of coinvariants for the diagonal action of the symmetric group S_n. It is known that the character of R_n as a doubly graded S_n-module can be expressed using the Frobenius characteristic map as $\nabla e_{n}$, where e_n is the nth elementary symmetric function and $\nabla $ is an operator from the theory of Macdonald polynomials. We conjecture a combinatorial formula for $\nabla e_{n}$ and prove that it has many desirable properties that support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. These results make use of the theory of ribbon tableau generating functions of Lascoux, Leclerc, and Thibon. We also show that a variety of earlier conjectures and theorems on $\nabla e_{n}$ are special cases of our conjecture.

Finally, we extend our conjectures on $\nabla e_{n}$ and several of the results supporting them to higher powers $\nabla^{m}e_{n}$.

The minimal degeneration singularities in the affine Grassmannians of simple simply laced algebraic groups are determined to be either Kleinian singularities of type A or closures of minimal nilpotent orbits. The singularities for non-simply laced types are studied by intersection cohomology and equivariant Chow group methods.

We classify the simple integrable modules of double affine Hecke algebras (DAHA) via perverse sheaves. We also get some estimates for the Jordan-Hölder multiplicities of induced modules.

We prove the André-Oort conjecture on special points of Shimura varieties for arbitrary products of modular curves, assuming the generalized Riemann hypothesis (GRH). More explicitly, this means the following. Let n ≥ 0, and let Σ be a subset of ℂⁿ consisting of points all of whose coordinates are j-invariants of elliptic curves with complex multiplications. Then we prove (under GRH) that the irreducible components of the Zariski closure of Σ are special subvarieties, that is, are determined by isogeny conditions on coordinates and pairs of coordinates. A weaker variant (Th. 1.3) is proved unconditionally.

In this paper we study the propagation of singularity for Schrödinger-type equations with variable coefficients. We introduce a new notion of wave propagation set, the homogeneous wave front set, which propagates along straight lines with finite speed away from x≠0. Then we show that it is related to the wave front set in a natural way. These results may be considered as a refinement of the microlocal smoothing property of Craig, Kappeler, and Strauss under more general assumptions.

We continue our study of the local Gromov-Witten invariants of curves in Calabi-Yau threefolds.

We define relative invariants for local theory which give rise to a (1+1)-dimensional topological quantum field theory (TQFT) taking values in the ring $\mathbb{Q}[[t]]$. The associated Frobenius algebra over $\mathbb{Q}[[t]]$ is semisimple. Consequently, we obtain a structure result for the local invariants. As an easy consequence of our structure formula, we recover the closed formulas for the local invariants in the case where either the target genus or the degree equals 1.