We introduce the concepts of harmonic stability and harmonic index for a complete minimal hypersurface in $R^{n+1}(n\leq3)$ and prove that the hypersurface has only finitely many ends if its harmonic index is finite. Furthermore, the number of ends is bounded from above by 1 plus the harmonic index. Each end has a representation of nonnegative harmonic function, and these functions form a partition of unity. We also give an explicit estimate of the harmonic index for a class of special minimal hypersurfaces, namely, minimal hypersurfaces with finite total scalar curvature. It is shown that for such a submanifold the space of bounded harmonic functions is exactly generated by the representation functions of the ends.

We extend certain aspects of C. Simpson's correspondence between harmonic metrics and variations of Hodge structure to the category of complex variations of mixed Hodge structure, and we prove an analog of W. Schmid's nilpotent orbit theorem for admissible variations of graded-polarized mixed Hodge structure $\mathscr {V} \Delta\sp \ast$.

We give an explicit description in terms of logarithmic differential forms of the isomorphism of P. Schneider and U. Stuhler relating de Rham cohomology of p-adic symmetric spaces to boundary distributions. As an application we prove a Hodge-type decomposition for the de Rham cohomology of varieties over p-adic fields which admit a uniformization by a p-adic symmetric space.

In this paper we establish C. Siegel's mass formulas with quadratic characters which give a mass formula for spinor genera. For ternary quadratic forms, the splitting numbers are completely determined by local Jordan decompositions.

In this paper we use Lie group actions on noncompact Riemannian manifolds with calibrations to construct calibrated submanifolds. In particular, if we have an $(n-1)$-torus acting on a noncompact Calabi-Yau $n$-fold with a trivial first cohomology, then we have a special Lagrangian fibration on that $n$-fold. We produce several families of examples for this construction and give some applications to special Lagrangian geometry on compact almost Calabi-Yau manifolds. We also use group actions on noncompact $G_2$-manifolds to construct coassociative submanifolds, and we exhibit some new examples of coassociative submanifolds via this setup.

Let $\mathscr {C}(S_n)$ be the $\mathbb {Z}$-module of integer-valued class functions on the symmetric group $S_n)$. We introduce a graded version of the convolution product on $\mathscr {C}(S_n)$, and we show that there is a degree-preserving ring isomorphism $\mathscr {C}(S_n)\longrightarrow H^\ast({\rm Hilb}^n(\mathbb {A_C}^2);\mathbb {Z})$ to the cohomology of the Hilbert scheme of points in the complex affine plane.

We prove the arithmetic Hodge index and hard Lefschetz conjectures for the Grassmannian G(2,N) parametrizing lines in projective space, for the natural arithmetic Lefschetz operator defined via the Plücker embedding of G in projective space. The analysis of the Hodge index inequality involves estimates on values of certain Racah polynomials.

We study and classify all the conjugacy classes of the genus zero congruence subgroups of PSL₂(ℝ) with no elliptic elements. We show that it suffices to classify those inside the modular group and determine them completely. We also discuss an application to modular curves.