Known examples in plane symmetry or Gowdy symmetry show that, given a $1$-parameter family of solutions to the vacuum Einstein equations, it may have a weak limit which does not satisfy the vacuum equations, but instead has a nontrivial stress-energy-momentum tensor. We consider this phenomenon under polarized $\mathbb{U}\left(1\right)$-symmetry—a much weaker symmetry than most of the known examples—such that the stress-energy-momentum tensor can be identified with that of multiple families of null dust propagating in distinct directions. We prove that any generic local-in-time small-data polarized $\mathbb{U}\left(1\right)$-symmetric solution to the Einstein–multiple null dust system can be achieved as a weak limit of vacuum solutions. Our construction allows the number of families to be arbitrarily large and appears to be the first construction of such examples with more than two families.

Let $B(g,p)$ denote the number of isomorphism classes of $g$-dimensional Abelian varieties over the finite field of size $p$. Let $A(g,p)$ denote the number of isomorphism classes of principally polarized $g$-dimensional Abelian varieties over the finite field of size $p$. We derive upper bounds for $B(g,p)$ and lower bounds for $A(g,p)$ for $p$ fixed and $g$ increasing. The extremely large gap between the lower bound for $A(g,p)$ and the upper bound for $B(g,p)$ implies some statistically counterintuitive behavior for Abelian varieties of large dimension over a fixed finite field.

Nous donnons une nouvelle définition, plus simple mais équivalente, de la catégorie abélienne des espaces de Banach–Colmez introduite par Colmez, et nous expliquons la relation précise de cette catégorie avec celle des faisceaux cohérents sur la courbe de Fargues–Fontaine. On passe d’une catégorie à l’autre en modifiant la $t$-structure sur la catégorie dérivée. Chemin faisant, nous obtenons une description de la cohomologie pro-étale du disque ouvert et de l’espace affine d’intérêt indépendant.

We give a new definition, simpler but equivalent, of the abelian category of Banach–Colmez spaces introduced by Colmez, and we explain the precise relationship with the category of coherent sheaves on the Fargues–Fontaine curve. One goes from one category to the other by changing the $t$-structure on the derived category. Along the way we obtain a description of the proétale cohomology of the open disk and the affine space, which is of independent interest.

A hypersymplectic structure on a 4-manifold $X$ is a triple $\underline{\omega }$ of symplectic forms which at every point span a maximal positive definite subspace of ${\Lambda }^2$ for the wedge product. This article is motivated by a conjecture by Donaldson: when $X$ is compact, $\underline{\omega }$ can be deformed through cohomologous hypersymplectic structures to a hyper-Kähler triple. We approach this via a link with $G_2$-geometry. A hypersymplectic structure $\underline{\omega }$ on a compact manifold $X$ defines a natural $G_2$-structure $\varphi$ on $X\times {\mathbb{T}}^3$ which has vanishing torsion precisely when $\underline{\omega }$ is a hyper-Kähler triple. We study the $G_2$-Laplacian flow starting from $\varphi$, which we interpret as a flow of hypersymplectic structures. Our main result is that the flow extends as long as the scalar curvature of the corresponding $G_2$-structure remains bounded. An application of our result is a lower bound for the maximal existence time of the flow in terms of weak bounds on the initial data (and with no assumption that scalar curvature is bounded along the flow).