Given $\nu >1/2$ and $\delta >0$ arbitrary, we prove the existence of energy solutions of ${\displaystyle {\partial }_{\mathrm{tt}}u-\Delta u-u^5=0(0.1)}$ in ${\mathbb{R}}^{3+1}$ which blow up exactly at $r=t=0$ as $t\to 0-$. These solutions are radial and of the form $u=\lambda (t{)}^{1/2}W(\lambda (t)r)+\eta (r,t)$ inside the cone $r\le t$, where $\lambda (t)=t^{-1-\nu }$, $W(r)=(1+r^2/3{)}^{-1/2}$ is the stationary solution of (0.1), and $\eta$ is a radiation term with ${\displaystyle {\int }_{[r\le t]}(|\nabla \eta (x,t){|}^2+|{\eta }_t(x,t){|}^2+|\eta (x,t){|}^6)\mathrm{dx}\to 0,t\to 0.}$ Outside of the light-cone, there is the energy bound ${\displaystyle {\int }_{[r>t]}(|\nabla u(x,t){|}^2+|u_t(x,t){|}^2+|u(x,t){|}^6)\mathrm{dx}<\delta }$ for all small $t>0$. The regularity of $u$ increases with $\nu$. As in our accompanying article on wave maps [10], the argument is based on a renormalization method for the “soliton profile” $W(r)$

We completely characterize the boundedness of planar directional maximal operators on $L^p$. More precisely, if $\Omega$ is a set of directions, we show that $M_{\Omega }$, the maximal operator associated to line segments in the directions $\Omega$, is unbounded on $L^p$ for all $p<\infty$ precisely when $\Omega$ admits Kakeya-type sets. In fact, we show that if $\Omega$ does not admit Kakeya sets, then $\Omega$ is a generalized lacunary set, and hence, $M_{\Omega }$ is bounded on $L^p$ for $p>1$

We construct and study the conformal loop ensembles $\mathrm{CLE}(\kappa )$, defined for $8/3\le \kappa \le 8$, using branching variants of $\mathrm{SLE}(\kappa )$ called exploration trees. The $\mathrm{CLE}(\kappa )$ are random collections of countably many loops in a planar domain that are characterized by certain conformal invariance and Markov properties. We conjecture that they are the scaling limits of various random loop models from statistical physics, including the $O(n)$ loop models

In this article we investigate almost-universal positive-definite integral quaternary quadratic forms, that is, those representing sufficiently large positive integers. In particular, we provide an effective characterization of all such forms. In this way we obtain the final solution to a problem first addressed by Ramanujan in [12]. Special attention is given to $2$-anisotropic almost-universal quaternaries

A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean space and sphere. We show that the mean curvature flow preserves the isoparametric condition, develops singularities in finite time, and converges in finite time to a smooth submanifold of lower dimension. We also give a precise description of the collapsing

Given a correspondence between a modular curve $S$ and an elliptic curve $A$, we prove that the intersection of any finite-rank subgroup of $A$ with the set of points on $A$ corresponding to CM points on $S$ is finite. We prove also a version in which $S$ is replaced by a Shimura curve and $A$ is replaced by a higher-dimensional abelian variety