In this paper we study the Newton stratification on the reduction of Shimura varieties of PEL type with hyperspecial level structure. Our main result is a formula for the dimension of Newton strata and the description of their closure, where the dimension formula was conjectured by Chai. As a key ingredient of its proof we calculate the dimension of some Rapoport–Zink spaces. Our result yields a dimension formula, which was conjectured by Rapoport (up to a minor correction).

As an interesting application to deformation theory, we determine the dimension and closure of Newton strata on the algebraization of the deformation space of a Barsotti–Tate group with (P)EL structure. Our result on the closure of a Newton stratum generalizes conjectures of Grothendieck and Koblitz.

We study asymptotic behavior for the determinants of $n\times n$ Toeplitz matrices corresponding to symbols with two Fisher–Hartwig singularities at the distance $2t\ge 0$ from each other on the unit circle. We obtain large $n$ asymptotics which are uniform for $0<t<t_0$, where $t_0$ is fixed. They describe the transition as $t\to 0$ between the asymptotic regimes of two singularities and one singularity. The asymptotics involve a particular solution to the Painlevé V equation. We obtain small and large argument expansions of this solution. As applications of our results, we prove a conjecture of Dyson on the largest occupation number in the ground state of a one-dimensional Bose gas, and a conjecture of Fyodorov and Keating on the second moment of powers of the characteristic polynomials of random matrices.

We show that every operator system (and hence every unital operator algebra) has sufficiently many boundary representations to generate the C∗-envelope.