Let $\mathcal{K}$=ℂ((t)) be the field of formal Laurent series, and let $\mathcal{O}$=ℂ[[t]] be the ring of formal power series. In this paper we present a version of the Matsuki correspondence for the affine Grassmannian Gr=G($\mathcal{K}$)/G($\mathcal{O}$) of a connected reductive complex algebraic group G. Our main statement is an anti-isomorphism between the orbit posets of two subgroups of G($\mathcal{K}$) acting on Gr. The first subgroup is the polynomial loop group LG_ℝ of a real form G_ℝ of G; the second is the loop group K($\mathcal{K}$) of the complexification K of a maximal compact subgroup K_c of G_ℝ. The orbit poset itself turns out to be simple to describe.

We consider locally symmetric manifolds with a fixed universal covering, and we construct for each such manifold M a simplicial complex ℝ whose size is proportional to the volume of M. When M is noncompact, $\mathcal{R}$ is homotopically equivalent to M, while when M is compact, $\mathcal{R}$ is homotopically equivalent to M\N, where N is a finite union of submanifolds of relatively small dimension. This reflects how the volume controls the topological structure of M, and yields concrete bounds for various finiteness statements that previously had no quantitative proofs. For example, it gives an explicit upper bound for the possible number of locally symmetric manifolds of volume bounded by v>0, and it yields an estimate for the size of a minimal presentation for the fundamental group of a manifold in terms of its volume. It also yields a number of new finiteness results.

Let $M$ be a compact manifold without boundary. Associated to a metric $g$ on $M$ there are various Laplace operators, for example, the de Rham Laplacian on $p$-forms and the conformal Laplacian on functions. For a general geometric differential operator of Laplace type with eigenvalues 0≤λ₁≤λ₂≤⋯, we consider the spectral zeta function $Z(s)=\sum_{\lambda_j\neq0} \lambda_j^{-s}$. The modified zeta function $\mathcal{Z}(s)=Γ(s)Z(s)/Γ(s−n/2)$ is an entire function of $s$. For a fixed value of $s$, we calculate the Hessian of $\mathcal{Z}(s)$ with respect to the metric and show that it is given by a pseudodifferential operator T_s=U_s+V_s, where $U_s$ is polyhomogeneous of degree n−2s and V_s is polyhomogeneous of degree 2. The operators $U_s/Γ(n/2+1−s)$ and $V_s /Γ(n/2+1−s)$ are entire in $s$. The symbol expansion of $U_s$ is computable from the symbol of the Laplacian. Our analysis extends to describing the Hessian of $(d / ds)^k \mathcal{Z}(s)$ for any value of $k$.

We prove a precise bound on the number of sections of a pure sheaf on a projective scheme. Our result strengthens the Le Potier-Simpson estimate and generalizes it to schemes defined over a field of any characteristic. This is used to construct the moduli space of semistable sheaves in mixed characteristic via Simpson's method.

We show that if two rings have equivalent derived categories, then they have the same algebraic K-theory. Similar results are given for G-theory and for a large class of abelian categories.