In this paper we solve a contact nonsqueezing conjecture proposed by Eliashberg, Kim, and Polterovich. Let $B_R$ be the open ball of radius $R$ in ${\mathbb{R}}^{2n}$, and let ${\mathbb{R}}^{2n}\times {\mathbb{S}}^1$ be the prequantization space equipped with the standard contact structure. Following Tamarkin’s idea, we apply microlocal category methods to prove that if $R$ and $r$ satisfy $1\le \pi r^2<\pi R^2$, then it is impossible to squeeze the contact ball $B_R\times {\mathbb{S}}^1$ into $B_r\times {\mathbb{S}}^1$ via compactly supported contact isotopies.

We describe all the situations in which the Kontsevich–Zorich (KZ) cocycle has zero Lyapunov exponents. Confirming a conjecture of Forni, Matheus, and Zorich, we find this only occurs when the cocycle satisfies additional geometric constraints. We also show that the connected components of the Zariski closure of the monodromy must be from a specific list, and the representations in which they can occur are described. The number of zero exponents of the KZ cocycle is then as small as possible, given its monodromy.

We investigate the cocompact action of Higman’s group on a $CAT\left(0\right)$ square complex associated to its standard presentation. We show that this action is in a sense intrinsic, which allows for the use of geometric techniques to study the endomorphisms of the group, and we show striking similarities with mapping class groups of hyperbolic surfaces, outer automorphism groups of free groups, and linear groups over the integers. We compute explicitly the automorphism group and outer automorphism group of Higman’s group and show that the group is both Hopfian and co-Hopfian. We actually prove a stronger rigidity result about the endomorphisms of Higman’s group: every nontrivial morphism from the group to itself is an automorphism. We also study the geometry of the action and prove a surprising result: although the $CAT\left(0\right)$ square complex acted upon contains uncountably many flats, the Higman group does not contain subgroups isomorphic to ${\mathbb{Z}}^2$. Finally, we show that this action possesses features reminiscent of negative curvature, which we use to prove a refined version of the Tits alternative for Higman’s group.

We prove the Tate conjecture for divisor classes and the Mumford–Tate conjecture for the cohomology in degree $2$ for varieties with $h^{2,0}=1$ over a finitely generated field of characteristic $0$, under a mild assumption on their moduli. As an application of this general result, we prove the Tate and Mumford–Tate conjectures for several classes of algebraic surfaces with $p_g=1$.