Heuristics based on the Sato–Tate conjecture and the Lang–Trotter philosophy suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We prove this result for abelian surfaces with real multiplication. As in previous work by Charles and Elkies, this shows that a density $0$ set of primes pertaining to the reduction of abelian varieties is infinite. The proof relies on the Arakelov intersection theory on Hilbert modular surfaces.

We show that the finite time blowup solutions for the corotational wave-map problem constructed by the first author along with Gao, Schlag, and Tataru are stable under suitably small perturbations within the corotational class, provided that the scaling parameter $\lambda (t)=t^{-1-\nu }$ is sufficiently close to $t^{-1}$; that is, the constant $\nu$ is sufficiently small and positive. The method of proof is inspired by recent work by the first author and Burzio, but takes advantage of geometric structures of the wave-map problem, already used in previous work by the first author, Bejenaru, Tataru, Raphaël, and Rodnianski, to simplify the analysis. In particular, we heavily exploit the fact that the resonance at zero satisfies a natural first-order differential equation.

For a function $W\in \mathbb{C}\left[X\right]$ on a smooth algebraic variety $X$ with Morse–Bott critical locus $Y\subset X$, Kapustin, Rozansky, and Saulina suggest that the associated matrix factorization category $MF(X;W)$ should be equivalent to the differential graded category of $2$-periodic coherent complexes on $Y$ (with a topological twist from the normal bundle). We confirm their conjecture in the special case when the first neighborhood of $Y$ in $X$ is split and establish the corrected general statement. The answer involves the full Gerstenhaber structure on Hochschild cochains. This note was inspired by the failure of the conjecture, observed by Pomerleano and Preygel, when $X$ is a general $1$-parameter deformation of a $K3$ surface $Y$.

We say that a finite subset of the unit sphere in ${\mathbf{R}}^d$ is transitive if there is a group of isometries which acts transitively on it. We show that the width of any transitive set is bounded above by a constant times $(logd{)}^{-1/2}$.

This is a consequence of the following result: if $G$ is a finite group and $\rho :G\to U_d\left(\mathbf{C}\right)$ a unitary representation, and if $v\in {\mathbf{C}}^d$ is a unit vector, then there is another unit vector $w\in {\mathbf{C}}^d$ such that $${sup\hspace{0.17em}}_{g\in G}|\langle \rho (g)v,w\rangle |\le (1+clogd{)}^{-1/2}.$$

These results answer a question of Yufei Zhao. An immediate consequence of our result is that the diameter of any quotient $S\left({\mathbf{R}}^d\right)/G$ of the unit sphere by a finite group $G$ of isometries is at least $\pi /2-o_{d\to \infty }\left(1\right)$.