The continuity of the inverse Klain map is investigated and the class of centrally symmetric convex bodies at which every valuation depends continuously on its Klain function is characterized. Among several applications, it is shown that McMullen’s decomposition is not possible in the class of translation-invariant, continuous, positive valuations. This implies that there exists no McMullen decomposition for translation-invariant, continuous Minkowski valuations, which solves a problem first posed by Schneider and Schuster.

Using an approach emerging from the theory of closable derivations on von Neumann algebras, we exhibit a class of groups $\mathcal{CR}$ satisfying the following property: given any groups ${\Gamma }_1,{\Gamma }_2\in \mathcal{CR}$, then any free, ergodic, measure-preserving action on a probability space ${\Gamma }_1\times {\Gamma }_2\curvearrowright X$ gives rise to a von Neumann algebra with a unique group-measure space Cartan subalgebra. Pairing this result with Popa’s orbit equivalence superrigidity theorem we obtain new examples of $W^{\ast }$-superrigid actions.

We consider equivariant solutions for the Schrödinger map problem from ${\mathbb{R}}^{2+1}$ to ${\mathbb{S}}^2$ with energy less than $4\pi$ and show that they are global in time and scatter.

Let $E$ be an elliptic curve defined over $\mathbf{Q}$ without complex multiplication. The field $F$ generated over $\mathbf{Q}$ by all torsion points of $E$ is an infinite, nonabelian Galois extension of the rationals which has unbounded, wild ramification above all primes. We prove that the absolute logarithmic Weil height of an element of $F$ is either zero or bounded from below by a positive constant depending only on $E$. We also show that the Néron–Tate height has a similar gap on $E\left(F\right)$ and use this to determine the structure of the group $E\left(F\right)$.