We develop a global twistor correspondence for pseudo-Riemannian conformal structures of signature $(++--)$ with self-dual Weyl curvature. Near the conformal class of the standard indefinite product metric on $S^2\times S^2$, there is an infinite-dimensional moduli space of such conformal structures, and each of these has the surprising global property that its null geodesics are all periodic. Each such conformal structure arises from a family of holomorphic disks in $\mathbb{C}{\mathbb{P}}_3$ with boundary on some totally real embedding of $\mathbb{R}{\mathbb{P}}^3$ into $\mathbb{C}{\mathbb{P}}_3$. Some of these conformal classes are represented by scalar-flat indefinite Kähler metrics, and our methods give particularly sharp results in connection with this special case

We prove a companion forms theorem for mod $l$ Hilbert modular forms. This work generalises results of Gross [Gr] and Coleman and Voloch [CV] for modular forms over $\mathbb{Q}$ and gives a new proof of their results in many cases. The methods used are completely different to previous work in this area and rely on modularity lifting theorems and the general theory of deformations of Galois representations

We establish local Calderón-Zygmund-type estimates for a class of parabolic problems whose model is the nonhomogeneous, degenerate/singular parabolic $p$-Laplacian system $u_t-\mathrm{div}(|\mathrm{Du}|{}^{p-2}\mathrm{Du})=\mathrm{div}(|F|{}^{p-2}F),$ proving that $F\in L_{}{\mathrm{loc}}^q\Rightarrow \mathrm{Du}\in L_{}^q\mathrm{loc},\hspace{1em}\forall \hspace{0.17em}q\ge p.$ We also treat systems with discontinuous coefficients of vanishing mean oscillation (VMO) type

We prove a regularity result for the unstable elliptic free boundary problem ${\displaystyle \Delta u=-{\chi }_{\{u>0\}}}$ related to traveling waves in a problem arising in solid combustion. The maximal solution and every local minimizer of the energy are regular; that is, $\{u=0\}$ is locally an analytic surface, and $u|{}_{\stackrel{̲}{\{u>0\}}},u|{}_{\stackrel{̲}{\{u<0\}}}$ are locally analytic functions. Moreover, we prove a partial regularity result for solutions that are nondegenerate of second order. Here $\{u=0\}$ is analytic up to a closed set of Hausdorff dimension $n-2$. We discuss possible singularities

We discuss in detail the dynamics of maps $z\mapsto {\mathrm{ae}}^z+{\mathrm{be}}^{-z}$ for which both critical orbits are strictly preperiodic. The points that converge to $\infty$ under iteration contain a set $R$ consisting of uncountably many curves called rays, each connecting $\infty$ to a well-defined “landing point” in $\mathbb{C}$, so that every point in $\mathbb{C}$ is either on a unique ray or the landing point of several rays.

The key features of this article are the following:

(1) this is the first example of a transcendental dynamical system, where the Julia set is all of $\mathbb{C}$ and the dynamics is described in detail for every point using symbolic dynamics;

(2) we get the strongest possible version (in the plane) of the “dimension paradox”: the set $R$ of rays has Hausdorff dimension $1$, and each point in $\mathbb{C}\backslash R$ is connected to $\infty$ by one or more disjoint rays in $R$.

As the complement of a $1$-dimensional set, $\mathbb{C}\backslash R$ of course has Hausdorff dimension $2$ and full Lebesgue measure

We show that Haar measures of connected semisimple groups, embedded via a representation into a matrix space, have a homogeneous asymptotic limit when viewed from far away and appropriately rescaled. This is still true if the Haar measure of the semisimple group is replaced by the Haar measure of an irreducible lattice of the group, and the asymptotic measure is the same. In the case of an almost simple group of rank greater than $2$, a remainder term is also obtained. This extends and makes precise anterior results of Duke, Rudnick, and Sarnak [DRS] and Eskin and McMullen [EM] in the case of a group variety