We give an analogue of triangle comparison for Kähler manifolds with a lower bound on the holomorphic bisectional curvature. We show that the condition passes to noncollapsed Gromov–Hausdorff limits. We discuss tangent cones and singular Kähler spaces.

We prove that any separable ${\mathrm{II}}_1$ factor M admits a coarse decomposition over the hyperfinite ${\mathrm{II}}_1$ factor R—that is, there exists an embedding $R\hookrightarrow M$ such that $L^{2}M\ominus L^{2}R$ is a multiple of the coarse Hilbert R-bimodule $L^{2}R\bar{\otimes }{L}^2{R}^{\mathrm{op}}$. Equivalently, the von Neumann algebra generated by left and right multiplication by R on $L^{2}M\ominus L^{2}R$ is isomorphic to $R\bar{\otimes }{R}^{\mathrm{op}}$. Moreover, if $Q\subset M$ is an infinite-index irreducible subfactor, then $R\hookrightarrow M$ can be constructed to be coarse with respect to Q as well. This implies the existence of maximal abelian ${}^{\ast }$-subalgebras that are mixing, strongly malnormal, and with infinite multiplicity, in any given separable ${\mathrm{II}}_1$ factor.

We study the geometry of reduction modulo p of the Kisin–Pappas integral models for certain Shimura varieties of abelian type with parahoric level structure. We give some direct and geometric constructions for the EKOR (Ekedahl–Kottwitz–Oort–Rapoport) strata on these Shimura varieties, using the theories of G-zips and mixed characteristic local $\mathcal{G}$-Shtukas. We establish several basic properties of these strata, including the smoothness, dimension formula, and closure relation. Moreover, we apply our results to the study of Newton strata and central leaves on these Shimura varieties.