Let $\Pi$ be a translation-invariant point process on the complex plane $\mathbb{C}$, and let $\mathcal{D}\subset \mathbb{C}$ be a bounded open set. We ask the following: What does the point configuration ${\Pi }_{\mathrm{out}}$ obtained by taking the points of $\Pi$ outside $\mathcal{D}$ tell us about the point configuration ${\Pi }_{\mathrm{in}}$ of $\Pi$ inside $\mathcal{D}$? We show that, for the Ginibre ensemble, ${\Pi }_{\mathrm{out}}$ determines the number of points in ${\Pi }_{\mathrm{in}}$. For the translation-invariant zero process of a planar Gaussian analytic function, we show that ${\Pi }_{\mathrm{out}}$ determines the number as well as the center of mass of the points in ${\Pi }_{\mathrm{in}}$. Further, in both models we prove that the outside says “nothing more” about the inside, in the sense that the conditional distribution of the inside points, given the outside, is mutually absolutely continuous with respect to the Lebesgue measure on its supporting submanifold.

We study the geometry and topology of (filtered) algebra bundles ${\mathbf{\Psi }}^{\mathbb{Z}}$ over a smooth manifold $X$ with typical fiber ${\Psi }^{\mathbb{Z}}(Z;V)$, the algebra of classical pseudodifferential operators acting on smooth sections of a vector bundle $V$ over the compact manifold $Z$ and of integral order. First, a theorem of Duistermaat and Singer is generalized to the assertion that the group of projective invertible Fourier integral operators $PG\left({\mathcal{F}}^{\mathbb{C}}\right(Z;V\left)\right)$ is precisely the automorphism group of the filtered algebra of pseudodifferential operators. We replace some of the arguments in their work by microlocal ones, thereby removing the topological assumption. We define a natural class of connections and $B$-fields on the principal bundle to which ${\mathbf{\Psi }}^{\mathbb{Z}}$ is associated and obtain a de Rham representative of the Dixmier–Douady class in terms of the outer derivation on the Lie algebra and the residue trace of Guillemin and Wodzicki. The resulting formula only depends on the formal symbol algebra ${\mathbf{\Psi }}^{\mathbb{Z}}/{\mathbf{\Psi }}^{-\infty }$. Examples of pseudodifferential algebra bundles are given that are not associated to a finite-dimensional fiber bundle over $X$.

We establish a general “affine representability” result in ${\mathbb{A}}^1$-homotopy theory over a general base. We apply this result to obtain representability results for vector bundles in ${\mathbb{A}}^1$-homotopy theory. Our results simplify and significantly generalize Morel’s ${\mathbb{A}}^1$-representability theorem for vector bundles.

Let $F$ be a free group (pro-$p$ group), and let $U$ and $W$ be two finitely generated subgroups (closed subgroups) of $F$. The Strengthened Hanna Neumann conjecture says that

$${\sum }_{x\in U\backslash F/W}\overline{rk}(U\cap xW{x}^{-1})\le \overline{rk}\left(U\right)\overline{\mathrm{rk}}\left(W\right),\text{where}\overline{rk}\left(U\right)=max\hspace{0.17em}\{rk(U)-1,0\}.$$ This conjecture was proved independently in the case of abstract groups by J. Friedman and I. Mineyev in 2011.

In this paper we give the proof of the conjecture in the pro-$p$ context, and we present a new proof in the abstract case. We also show that the Lück approximation conjecture holds for free groups.