The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice ${\mathbb{Z}}^d$, in which sites with at least $2d$ chips topple, distributing one chip to each of their neighbors in the lattice, until no more topplings are possible. From an initial configuration consisting of $n$ chips placed at a single vertex, the rescaled stable configuration seems to converge to a particular fractal pattern as $n\to \infty$. However, little has been proved about the appearance of the stable configurations. We use partial differential equation techniques to prove that the rescaled stable configurations do indeed converge to a unique limit as $n\to \infty$. We characterize the limit as the Laplacian of the solution to an elliptic obstacle problem.

We obtain Lipschitz regularity results for a fairly general class of nonlinear first-order partial differential equations. These equations arise from the inner variation of certain energy integrals. Even in the simplest model case of the Dirichlet energy the inner-stationary solutions need not be differentiable everywhere; the Lipschitz continuity is the best possible. But the proofs, even in the Dirichlet case, turn out to rely on topological arguments. The appeal to the inner-stationary solutions in this context is motivated by the classical problems of existence and regularity of the energy-minimal deformations in the theory of harmonic mappings and certain mathematical models of nonlinear elasticity, specifically, neo-Hookean-type problems.

We apply the efficient congruencing method to estimate Vinogradov’s integral for moments of order $2s$, with $1⩽s⩽k^2-1$. Thereby, we show that quasi-diagonal behavior holds when $s=o\left(k^2\right)$, and we obtain near-optimal estimates for $1⩽s⩽\frac{1}{4}{k}^2+k$ and optimal estimates for $s⩾k^2-1$. In this way we come halfway to proving the main conjecture in two different directions. There are consequences for estimates of Weyl type and in several allied applications. Thus, for example, the anticipated asymptotic formula in Waring’s problem is established for sums of $s$ $k$th powers of natural numbers whenever $s⩾2k^2-2k-8$ $\left(k⩾6\right)$.

We consider some classical maps from the theory of abelian varieties and their moduli spaces, and we prove their definability on restricted domains in the o-minimal structure ${\mathbb{R}}_{\mathrm{an,exp}}$. In particular, we prove that the projective embedding of the moduli space of the principally polarized abelian variety $Sp(2g,\mathbb{Z})\backslash {\mathbb{H}}_g$ is definable in ${\mathbb{R}}_{\mathrm{an,exp}}$ when restricted to Siegel’s fundamental set ${\mathfrak{F}}_g$. We also prove the definability on appropriate domains of embeddings of families of abelian varieties into projective spaces.

Given an irreducible affine algebraic variety $X$ of dimension $n\ge 2$, we let $SAut\left(X\right)$ denote the special automorphism group of $X$, that is, the subgroup of the full automorphism group $Aut\left(X\right)$ generated by all one-parameter unipotent subgroups. We show that if $SAut\left(X\right)$ is transitive on the smooth locus $X_{\mathrm{reg}}$, then it is infinitely transitive on $X_{\mathrm{reg}}$. In turn, the transitivity is equivalent to the flexibility of $X$. The latter means that for every smooth point $x\in X_{\mathrm{reg}}$ the tangent space $T_{x}X$ is spanned by the velocity vectors at $x$ of one-parameter unipotent subgroups of $Aut\left(X\right)$. We also provide various modifications and applications.