Let $F\in \mathbb{Z}[x_1,\dots ,x_n]$ be a homogeneous form of degree $d\ge 2$, and let $V_F^{\ast }$ denote the singular locus of the affine variety $V(F)=\{\mathbf{z}\in {\mathbb{C}}^n:F(\mathbf{z})=0\}$. In this paper, we prove the existence of integer solutions with prime coordinates to the equation $F(x_1,\dots ,x_n)=0$ provided that F satisfies suitable local conditions and $n-dim{V}_F^{\ast }\ge 2^8{3}^4{5}^2{d}^3{(2d-1)}^2{4}^d$. Our result improves on what was known previously due to Cook and Magyar, which required $n-dim{V}_F^{\ast }$ to be an exponential tower in d.

We prove that Khovanov homology detects the trefoils. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the instanton Floer setting; a bypass exact triangle in sutured instanton homology, proved here; and Kronheimer and Mrowka’s spectral sequence relating Khovanov homology with singular instanton knot homology. As a byproduct, we also strengthen a result of Kronheimer and Mrowka on $\mathrm{SU}(2)$ representations of the knot group.

In a seminal paper “Volumen und Oberfläche” (1903), Minkowski introduced the basic notion of mixed volumes and the corresponding inequalities that lie at the heart of convex geometry. The fundamental importance of characterizing the extremals of these inequalities was already emphasized by Minkowski himself, but has to date only been resolved in special cases. In this paper, we completely settle the extremals of Minkowski’s quadratic inequality, confirming a conjecture of R. Schneider. Our proof is based on the representation of mixed volumes of arbitrary convex bodies as Dirichlet forms associated to certain highly degenerate elliptic operators. A key ingredient of the proof is a quantitative rigidity property associated to these operators.