We construct infinitely many complete Calabi–Yau metrics on ${\mathbf{C}}^n$ for $n\ge 3$, with maximal volume growth and singular tangent cones at infinity. In addition, we construct Calabi–Yau metrics in neighborhoods of certain isolated singularities whose tangent cones have singular cross section, generalizing work of Hein and Naber.

We construct examples of finite covers of punctured surfaces where the first rational homology is not spanned by lifts of simple closed curves. More generally, for any set $\mathcal{O}\subset F_n$ which is contained in the union of finitely many $Aut\left(F_n\right)$-orbits, we construct finite-index normal subgroups of $F_n$ whose first rational homology is not spanned by powers of elements of $\mathcal{O}$. These examples answer questions of Farb and Hensel, Kent, Looijenga, and Marché. We also show that the quotient of $Out\left(F_n\right)$ by the subgroup generated by $k^{th}$ powers of transvections often contains infinite-order elements, strengthening a result of Bridson and Vogtmann that it is often infinite. Finally, for any set $\mathcal{O}\subset F_n$ which is contained in the union of finitely many $Aut\left(F_n\right)$-orbits, we construct integral linear representations of free groups that have infinite image and that map all elements of $\mathcal{O}$ to torsion elements.

We establish the Hasse principle for smooth projective quartic hypersurfaces of dimension greater than or equal to $28$ defined over $\mathbb{Q}$.