In this paper we consider uniformly random lozenge tilings of arbitrary domains approximating (after suitable normalization) a closed, simply-connected subset of $\mathbb{R}^2$ with piecewise smooth, simple boundary. We show that the local statistics of this model around any point in the liquid region of its limit shape are given by the infinite-volume, translation-invariant, extremal Gibbs measure of the appropriate slope, thereby confirming a prediction of Cohn-Kenyon-Propp from 2001 in the case of lozenge tilings. Our proofs proceed by locally coupling a uniformly random lozenge tiling with a model of Bernoulli random walks conditioned to never intersect, whose convergence of local statistics has been recently understood by the work of Gorin-Petrov. Central to implementing this procedure is to establish a local law for the random tiling, which states that the associated height function is approximately linear on any mesoscopic scale.

Hardt-Simon proved that every area-minimizing hypercone $\mathbf{C}$ having only an isolated singularity fits into a foliation of $\mathbb{R}^{n+1}$ by smooth, area-minimizing hypersurfaces asymptotic to $\mathbf{C}$. In this paper we prove that if a stationary integral $n$-varifold $M$ in the unit ball $B_1 \subset \mathbb{R}^{n+1}$ lies sufficiently close to a minimizing quadratic cone (for example, the Simons' cone $\mathbf{C}^{3,3}$), then $\mathrm{spt}\, M \cap B_{1/2}$ is a $C^{1,\alpha}$ perturbation of either the cone itself, or some leaf of its associated foliation. In particular, we show that singularities modeled on these cones determine the local structure not only of $M$, but of any nearby minimal surface. Our result also implies the Bernstein-type result of Simon-Solomon, which characterizes area-minimizing hypersurfaces asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation.

We present a mathematical theory of dynamical fluctuations for the hard sphere gas in the Boltzmann-Grad limit. We prove that (1) fluctuations of the empirical measure from the solution of the Boltzmann equation, scaled with the square root of the average number of particles, converge to a Gaussian process driven by the fluctuating Boltzmann equation, as predicted by Spohn; (2) large deviations are exponentially small in the average number of particles and are characterized, under regularity assumptions, by a large deviation functional as previously obtained by Rezakhanlou for dynamics with stochastic collisions. The results are valid away from thermal equilibrium, but only for short times. Our strategy is based on uniform a priori bounds on the cumulant generating function, characterizing the fine structure of the small correlations.

We resolve the strong Elementary Equivalence versus Isomorphism Problem for finitely generated fields. That is, we show that for every field in this class, there is a first-order sentence that characterizes this field within the class up to isomorphism. Our solution is conditional on resolution of singularities in characteristic two and unconditional in all other characteristics.

We establish a rather unexpected and simple criterion for the boundedness of Schur multipliers $S_M$ on Schatten $p$-classes which solves a conjecture proposed by Mikael de la Salle. Given $1\lt p\lt \infty$, a simple form of our main result for $\mathbf{R}^n \times \mathbf{R}^n$ matrices reads as follows: $$\big\| S_M \colon S_p \to S_p \big\|_{\mathrm{cb}} \lesssim \frac{p^2}{p-1} \sum_{|\gamma| \le [\frac{n}{2}] +1} \Big\| |x-y|^{|\gamma|} \Big\{ \big| \partial_x^\gamma M(x,y) \big| + \big| \partial_y^\gamma M(x,y) \big| \Big\} \Big\|_\infty.$$In this form, it is a full matrix (nonToeplitz/nontrigonometric) amplification of the Hörmander-Mikhlin multiplier theorem, which admits lower fractional differentiability orders $\sigma > \frac{n}{2}$ as well. It trivially includes Arazy's conjecture for $S_p$-multipliers and extends it to $\alpha$-divided differences. It also leads to new Littlewood-Paley characterizations of $S_p$-norms and strong applications in harmonic analysis for nilpotent and high rank simple Lie group algebras.

We introduce a new class of groups called wreath-like products. These groups are close relatives of the classical wreath products and arise naturally in the context of group theoretic Dehn filling. Unlike ordinary wreath products, many wreath-like products have Kazhdan's property (T). In this paper, we prove that any group $G$ in a natural family of wreath-like products with property (T) is $\mathrm{W}^\ast$-superrigid: the group von Neumann algebra $\mathrm{L}(G)$ remembers the isomorphism class of $G$. This allows us to provide the first examples (in fact, $2^{\aleph_0}$ pairwise non-isomorphic examples) of $\mathrm{W}^\ast$-superrigid groups with property (T).

In his article [1] János Kollár has published a counterexample to the main result Theorem~1 of the paper [2].

The critical point is that in general the positive curvature current of the singular hermitian metric on the constructed holomorphic line bundle does not have vanishing Lelong numbers (cf. Theorem 4 and Theorem 5) as claimed. Consequently the quasi-projectivity criterion of the paper cannot be applied.