For any Riemannian foliation $\cal F$ on a closed manifold M with an arbitrary bundle-like metric, leafwise heat flow of differential forms is proved to preserve smoothness on M at infinite time. This result and its proof have consequences about the space of bundle-like metrics on M, about the dimension of the space of leafwise harmonic forms, and mainly about the second term of the differentiable spectral sequence of $\cal F$.

If G is a semisimple Lie group and (π, $\cal H$) an irreducible unitary representation of G with square integrable matrix coefficients, then there exists a number d(π) such that ($\forall$v, v′, w, w′ ∈ $\cal H$) 1/d(π) 〈v, v′〉 〈w′, w〉 = ∫G 〈π(g).v,w〉 $\overline$ 〈π(g).v′.w′〉 dμG(g). The constant d(π) is called the formal dimension of (π, $\cal H$) and was computed by Harish-Chandra in [HC56, 66]. If now H\G is a semisimple symmetric space and (π, $\cal H$) an irreducible H-spherical unitary (π, $\cal H$) belonging to the holomorphic discrete series of H\G, then one can define a formal dimension d(π) in an analogous manner. In this paper we compute d(π) for these classes of representations.

Let R be a homogeneous ring over an infinite field, I⊂R a homogeneous ideal, and $\frak a$⊂I an ideal generated by s forms of degrees d1,…,ds so that codim($\frak a$:I)[ges ]s. We give broad conditions for when the Hilbert function of R/$\frak a$ or of R/($\frak a$:I) is determined by I and the degrees d1,…,ds. These conditions are expressed in terms of residual intersections of I, culminating in the notion of residually S2 ideals. We prove that the residually S2 property is implied by the vanishing of certain Ext modules and deduce that generic projections tend to produce ideals with this property.

We obtain a necessary condition for a cohomology class on a compact locally symmetric space S(Γ)=Γ\X (a quotient of a symmetric space X of the non-compact type by a cocompact arithmetic subgroup Γ of isometries of X) to restrict non-trivially to a compact locally symmetric subspace SH(Γ)=Δ\Y of Γ\X. The restriction is in a ‘virtual’ sense, i.e. it is the restriction of possibly a translate of the cohomology class under a Hecke correspondence. As a consequence we deduce that when X and Y are the unit balls in $\Bbb C$n and $\Bbb C$m, then low degree cohomology classes on the variety S(Γ) restrict non-trivially to the subvariety SH(Γ); this proves a conjecture of M. Harris and J-S. Li. We also deduce the non-vanishing of cup-products of cohomology classes for the variety S(Γ).