9 problems

Consider the two-dimensional incompressible isotropic Hookean elastodynamics system for a smooth flow map $X(t,y)$ satisfying … and … For small initial displacements, let the highe…

Let $A$ be a finite set with control $ba$, meaning that it satisfies the control hypotheses introduced in the paper. For $b5>0$, write $X\gg_b5 Y$ to mean that $X$ is bounded below…

Scalar-field estimate conjecture. The scalar matter field obeys a similar estimate with integrand $P^2$. The scalar matter field decouples from the metric perturbations in the line…

Energy Balog–Wooley variant. For every finite $A\subseteq\mathbb R$, there exists a subset $A'\subseteq A$ with

Let $A$ be a finite set of transformations in the affine group $\operatorname{Aff}(\mathbb{F})$, and suppose that no line contains more than $M$ points of $A$. Write $E(A)$ for the…

Sharpness assertion. The estimate above is sharp with respect to the parameters $\omega$ and $\mu$. This concerns the dependence of the localized bound on the parameters in the sta…

Refined bisector energy bound conjecture.

Let $P$ be a set of $n$ points in the Euclidean plane, let $K$ be an upper bound for the number of points of $P$ on any line or circle, and let $\mathcal Q$ be the set of quadruple…

Let $\Sigma_0$ and $\Sigma_1$ be the initial and final spacelike hypersurfaces, let $n_{\Sigma_1}$ denote the future-directed unit normal to $\Sigma_1$, and consider the correspond…