We prove that the set of quadratic growths attainable by integer-valued superharmonic functions on the lattice $\mathbb{Z}^2$ has the structure of an Apollonian circle packing. This completely characterizes the PDE that determines the continuum scaling limit of the Abelian sandpile on the lattice $\mathbb{Z}^2$.

An affine manifold $X$ in the sense of differential geometry is a differentiable manifold admitting an atlas of charts with value in an affine space, with locally constant affine change of coordinates. Equivalently, it is a manifold whose tangent bundle admits a flat torsion free connection. Around 1955 Chern conjectured that the Euler characteristic of any compact affine manifold has to vanish. In this paper we prove Chern's conjecture in the case where $X$ moreover admits a parallel volume form.

We strengthen Gabber's $l$'-alteration theorem by avoiding all primes invertible on a scheme. In particular, we prove that any scheme $X$ of finite type over a quasi-excellent threefold can be desingularized by a $\mathrm{char}(X)$-alteration, i.e., an alteration whose order is only divisible by primes noninvertible on $X$. The main new ingredient in the proof is a tame distillation theorem asserting that, after enlarging, any alteration of $X$ can be split into a composition of a tame Galois alteration and a $\mathrm{char}(X)$-alteration. The proof of the distillation theorem is based on the following tameness theorem that we deduce from a theorem of M. Pank: if a valued field $k$ of residue characteristic $p$ has no nontrivial $p$-extensions, then any algebraic extension $l/k$ is tame.

We prove a homological stability theorem for moduli spaces of manifolds of dimension $2n$, for attaching handles of index at least $n$, after these manifolds have been stabilised by countably many copies of $S^n \times S^n$.

Combined with previous work of the authors, we obtain an analogue of the Madsen--Weiss theorem for any simply-connected manifold of dimension $2n \ge 6$.

Let $X$ be a connected scheme, smooth and separated over an algebraically closed field $k$ of characteristic $p\geq 0$, let $f:Y\rightarrow X$ be a smooth proper morphism and $x$ a geometric point on $X$. We prove that the tensor invariants of bounded length $\leq d$ of $\pi_1(X,x)$ acting on the éetale cohomology groups $H^*(Y_x,\mathbb{F}_\ell)$ are the reduction modulo-$\ell$ of those of $\pi_1(X,x)$ acting on $H^*(Y_x,\mathbb{Z}_\ell)$ for $\ell $ greater than a constant depending only on $f:Y\rightarrow X$, $d$. We apply this result to show that the geometric variant with $\mathbb{F}_\ell$-coefficients of the Grothendieck-Serre semisimplicity conjecture --- namely, that $\pi_1(X,x)$ acts semisimply on $H^*(Y_x,\mathbb{F}_\ell)$ for $\ell\gg 0$ --- is equivalent to the condition that the image of $\pi_1(X,x)$ acting on $H^*(Y_x,\mathbb{Q}_\ell)$ is `almost maximal' (in a precise sense; what we call `almost hyperspecial') with respect to the group of $\mathbb{Q}_\ell$-points of its Zariski closure. Ultimately, we prove the geometric variant with $\mathbb{F}_\ell$-coefficients of the Grothendieck-Serre semisimplicity conjecture.

We consider the structure $\mathbb{R}^{\mathrm{RE}}$ obtained from $(\mathbb{R},\lt ,+,\cdot)$ byadjoining the restricted exponential and sine functions. We proveWilkie's conjecture for sets definable in this structure: the numberof rational points of height $H$ in the transcendental part of anydefinable set is bounded by a polynomial in $\mathrm{log}\ H$. We also provetwo refined conjectures due to Pila concerning the density ofalgebraic points from a fixed number field, or with a fixedalgebraic degree, for $\mathbb{R}^{\mathrm{RE}}$-definable sets.

We show that any sufficiently (finitely) smooth $\mathbb{Z}_2$-symmetric strictly convex domain sufficiently close to a circle is dynamically spectrally rigid; i.e., all deformations among domains in the same class that preserve the length of all periodic orbits of the associated billiard flow must necessarily be isometric deformations. This gives a partial answer to a question of P. Sarnak.

We correct an error in the Multiple Point Formula (7.3) in the paper mentioned in the title. This correction propagates to formulas (7.5), (7.6), (7.23) and (8.18), and it affects minor results in Section 8, where few statements require an extra assumption, but it does not affect the main results of Section 8.