A version of the Hardy–Littlewood circle method is developed for number fields $K/\mathbb{Q}$ and is used to show that nonsingular projective cubic hypersurfaces over $K$ always have a $K$-rational point when they have dimension at least $8$.

Cubic fourfolds behave in many ways like $K3$ surfaces. Certain cubics—conjecturally, the ones that are rational—have specific $K3$ surfaces associated to them geometrically. Hassett has studied cubics with $K3$ surfaces associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with $K3$ surfaces associated to them at the level of derived categories.

These two notions of having an associated $K3$ surface should coincide. We prove that they coincide generically: Hassett’s cubics form a countable union of irreducible Noether–Lefschetz divisors in moduli space, and we show that Kuznetsov’s cubics are a dense subset of these, forming a nonempty, Zariski-open subset in each divisor.

We introduce some definitions of uniruledness for affine varieties and use these ideas to show symplectic invariance of various algebraic invariants of affine varieties. For instance we show that if $A$ and $B$ are symplectomorphic smooth affine varieties, then any compactification of $A$ by a projective variety is uniruled if and only if any such compactification of $B$ is uniruled. If $A$ is acylic of dimension $2$, then we show that $B$ has the same log Kodaira dimension as $A$. If $A$ has dimension $3$, has log Kodaira dimension $2$, and satisfies some other conditions, then $B$ cannot be of log general type.

The sphere packing problem asks for the greatest density of a packing of congruent balls in Euclidean space. The current best upper bound in all sufficiently high dimensions is due to Kabatiansky and Levenshtein in 1978. We revisit their argument and improve their bound by a constant factor using a simple geometric argument, and we extend the argument to packings in hyperbolic space, for which it gives an exponential improvement over the previously known bounds. Additionally, we show that the Cohn–Elkies linear programming bound is always at least as strong as the Kabatiansky–Levenshtein bound; this result is analogous to Rodemich’s theorem in coding theory. Finally, we develop hyperbolic linear programming bounds and prove the analogue of Rodemich’s theorem there as well.