In this article, we prove the mean-convex neighborhood conjecture for the mean-curvature flow of surfaces in $\mathbb{R}^3$. Namely, if the flow has a spherical or cylindrical singularity at a space-time point $X=(x, t)$, then there exists a positive $\varepsilon = \varepsilon (X) \gt 0$ such that the flow is mean convex in a space-time neighborhood of size $\varepsilon$ around $X$. The major difficulty is to promote the infinitesimal information about the singularity to a conclusion of macroscopic size. In fact, we prove a more general classification result for all ancient low-entropy flows that arise as potential limit flows near $X$. Namely, we prove that any ancient, unitregular, cyclic, integral Brakke flow in $\mathbb{R}^3$ with entropy at most $\sqrt{2 \pi / e + \delta}$ is either a flat plane, a round shrinking sphere, a round shrinking cylinder, a translating bowl soliton, or an ancient oval. As an application, we prove the uniqueness conjecture for meancurvature flow through spherical or cylindrical singularities. In particular, assuming Ilmanen’s multiplicity‑$1$ conjecture, we conclude that for embedded $2$‑spheres the meancurvature flow through singularities is well-posed.

There are many classical random walk in random environment results that apply to ergodic random planar environments. We extend some of these results to random environments in which the length scale varies from place to place, so that the law of the environment is in a certain sense only translation invariant modulo scaling. For our purposes, an “environment” consists of an infinite random planar map embedded in $\mathbb{C}$, each of whose edges comes with a positive real conductance. Our main result is that under modest constraints (translation invariance modulo scaling together with the finiteness of a type of specific energy) a random walk in this kind of environment converges to Brownian motion modulo time parametrization in the quenched sense.

Environments of the type considered here arise naturally in the study of random planar maps and Liouville quantum gravity. In fact, the results of this paper are used in separate works to prove that certain random planar maps (embedded in the plane via the so-called Tutte embedding) have scaling limits given by SLE-decorated Liouville quantum gravity, and also to provide a more explicit construction of Brownian motion on the Brownian map. However, the results of this paper are much more general and can be read independently of that program.

One general consequence of our main result is that if a translation invariant (modulo scaling) random embedded planar map and its dual have finite energy per area, then they are close on large scales to a minimal energy embedding (the harmonic embedding). To establish Brownian motion convergence for an infinite energy embedding, it suffices to show that one can perturb it to make the energy finite.

A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2 (\Omega)$ has an orthogonal basis of exponential functions. A conjecture due to Fuglede (1974) stated that $\Omega$ is a spectral set if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it has long been known that for a convex body $\Omega \subset \mathbb{R}^d$ the “tiling implies spectral” part of the conjecture is in fact true.

To the contrary, the “spectral implies tiling” direction of the conjecture for convex bodies was proved only in $\mathbb{R}^2$, and also in $\mathbb{R}^3$ under the a priori assumption that $\Omega$ is a convex polytope. In higher dimensions, this direction of the conjecture remained completely open (even in the case when $\Omega$ is a polytope) and could not be treated using the previously developed techniques.

In this paper we fully settle Fuglede’s conjecture for convex bodies affirmatively in all dimensions, i.e. we prove that if a convex body $\Omega \subset \mathbb{R}^d$ is a spectral set, then $\Omega$ is a convex polytope which can tile the space by translations. To prove this we introduce a new technique, involving a construction from crystallographic diffraction theory, which allows us to establish a geometric “weak tiling” condition necessary for a set $\Omega \subset \mathbb{R}^d$ to be spectral.