For any right-angled Artin group $A_{\mathrm{\Gamma }}$, we construct a finite-dimensional space ${\mathcal{O}}_{\mathrm{\Gamma }}$ on which the group $Out(A_{\mathrm{\Gamma }})$ of outer automorphisms of $A_{\mathrm{\Gamma }}$ acts with finite point stabilizers. We prove that ${\mathcal{O}}_{\mathrm{\Gamma }}$ is contractible, so that the quotient is a rational classifying space for $Out(A_{\mathrm{\Gamma }})$. The space ${\mathcal{O}}_{\mathrm{\Gamma }}$ blends features of the symmetric space of lattices in ${\mathbb{R}}^n$ with those of outer space for the free group $F_n$. Points in ${\mathcal{O}}_{\mathrm{\Gamma }}$ are locally CAT(0) metric spaces that are homeomorphic (but not isometric) to certain locally CAT(0) cube complexes, marked by an isomorphism of their fundamental group with $A_{\mathrm{\Gamma }}$.

We give a lower bound of the δ-invariants of ample line bundles in terms of Seshadri constants. As applications, we prove the uniform K-stability of infinitely many families of Fano hypersurfaces of arbitrarily large index, as well as the uniform K-stability of most families of smooth Fano threefolds of Picard number one.

We exhibit families of smooth projective threefolds with both stably rational and non stably rational fibers.

Let ${\mathrm{\pi }}_1$, ${\mathrm{\pi }}_2$, ${\mathrm{\pi }}_3$ be three cuspidal automorphic representations for the group $\mathrm{SL}(2,\mathbb{Z})$, where ${\mathrm{\pi }}_1$ and ${\mathrm{\pi }}_2$ are fixed and ${\mathrm{\pi }}_3$ has large analytic conductor. We prove a subconvex bound for $L(1∕2,{\mathrm{\pi }}_1\otimes {\mathrm{\pi }}_2\otimes {\mathrm{\pi }}_3)$ of Weyl-type quality. Allowing ${\mathrm{\pi }}_3$ to be an Eisenstein series, we also obtain a Weyl-type subconvex bound for $L(1∕2+it,{\mathrm{\pi }}_1\otimes {\mathrm{\pi }}_2)$.