Let $P_1,\dots ,P_m\in \mathbb{Z}[y]$ be any linearly independent polynomials with zero constant term. We show that there exists $\gamma >0$ such that any subset of ${\mathbb{F}}_q$ of size at least $q^{1-\gamma }$ contains a nontrivial polynomial progression $x,x+P_1(y),\dots ,x+P_m(y)$, provided that the characteristic of ${\mathbb{F}}_q$ is large enough.

We consider the class of measurable functions defined in all of ${\mathbb{R}}^n$ that give rise to a nonlocal minimal graph over a ball of ${\mathbb{R}}^n$. We establish that the gradient of any such function is bounded in the interior of the ball by a power of its oscillation. This estimate, together with previously known results, leads to the $C^{\infty }$ regularity of the function in the ball. While the smoothness of nonlocal minimal graphs was known for $n=1,2$—but without a quantitative bound—in higher dimensions only their continuity had been established. To prove the gradient bound, we show that the normal to a nonlocal minimal graph is a supersolution of a truncated fractional Jacobi operator, for which we prove a weak Harnack inequality. To this end, we establish a new universal fractional Sobolev inequality on nonlocal minimal surfaces. Our estimate provides an extension to the fractional setting of the celebrated gradient bounds of Finn and of Bombieri, De Giorgi, and Miranda for solutions of the classical mean curvature equation.

The works of Hassett and Kuznetsov identify countably many divisors $C_d$ in the open subset of ${\mathbb{P}}^{55}=\mathbb{P}(H^0({\mathcal{O}}_{{\mathbb{P}}^5}(3)))$ parameterizing all cubic fourfolds and conjecture that the cubics corresponding to these divisors are precisely the rational ones. Rationality has been known classically for the first family $C_{14}$. We use congruences of $5$-secant conics to prove rationality for the first three of the families $C_d$, corresponding to $d=14,26,38$ in Hassett’s notation.

To a complex projective structure $\Sigma$ on a surface, Thurston associates a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms $\Vert {\varphi }_{\Sigma }{\Vert }_{\infty }$ and $\Vert {\varphi }_{\Sigma }{\Vert }_2$ of the quadratic differential ${\varphi }_{\Sigma }$ of $\Sigma$ given by the Schwarzian derivative of the associated locally univalent map. We show that these give a unifying approach that generalizes a number of important, well-known results for convex cocompact hyperbolic structures on $3$-manifolds, including bounds on the Lipschitz constant for the nearest-point retraction and the length of the bending lamination. We then use these bounds to begin a study of the Weil–Petersson gradient flow of renormalized volume on the space $CC(N)$ of convex cocompact hyperbolic structures on a compact manifold $N$ with incompressible boundary, leading to a proof of the conjecture that the renormalized volume has infimum given by one half the simplicial volume of $\mathit{DN}$, the double of $N$.

We prove a global approximation theorem for a general parabolic operator $L$, which asserts that if $v$ satisfies the equation $Lv=0$ in a space-time region $\Omega \subset {\mathbb{R}}^{n+1}$ satisfying a certain necessary topological condition, then it can be approximated in a Hölder norm by a global solution $u$ to the equation. If $\Omega$ is compact and $L$ is the usual heat operator, then one can instead approximate the local solution $v$ by the unique solution that falls off at infinity to the Cauchy problem with a suitably chosen smooth, compactly supported initial datum. We then apply these results to prove the existence of global solutions to the equation $Lu=0$ with a local hot spot that moves along a prescribed curve for all time, up to a uniformly small error. Global solutions that exhibit isothermic hypersurfaces of prescribed topologies for all times and applications to the heat equation on the flat torus are also discussed.