We consider upper level sets of the Gaussian free field (GFF) on ${\mathbb{Z}}^d$, for $d\ge 3$, above a given real-valued height parameter h. As h varies, this defines a canonical percolation model with strong, algebraically decaying correlations. We prove that three natural critical parameters associated with this model, respectively describing a well-ordered subcritical phase, the emergence of an infinite cluster, and the onset of a local uniqueness regime in the supercritical phase, actually coincide. At the core of our proof lies a new interpolation scheme aimed at integrating out the long-range dependence of the GFF. Due to the strength of correlations, its successful implementation requires that we work in an effectively critical regime. Our analysis relies extensively on certain novel renormalization techniques that bring into play all relevant scales simultaneously. The approach in this article paves the way to a complete understanding of the off-critical phases for strongly correlated disordered systems.

We give general classification and structure theorems for actions of groups of homeomorphisms and diffeomorphisms on manifolds, reminiscent of classical results for actions of (locally) compact groups. This gives a negative answer to Ghys’s extension problem for diffeomorphisms of manifolds with boundary, as well as a classification of all homomorphisms ${Homeo}_0(M)\to {Homeo}_0(N)$ when $dim(M)=dim(N)$ (and related results for diffeomorphisms), and a complete classification of actions of ${Homeo}_0(S^1)$ on surfaces. This resolves many problems in a program initiated by Ghys, and gives definitive answers to conjectures of Militon and Hurtado and a question of Rubin.

We prove that any geometrically irreducible ${\bar{\mathbb{Q}}}_p$-local system on a smooth algebraic variety over a p-adic field K becomes de Rham after a twist by a character of the Galois group of K. In particular, for any geometrically irreducible ${\bar{\mathbb{Q}}}_p$-local system on a smooth variety over a number field the associated projective representation of the fundamental group automatically satisfies the assumptions of the relative Fontaine–Mazur conjecture. The proof uses p-adic Simpson and Riemann–Hilbert correspondences of Diao, Lan, Liu, and Zhu and the Sen operator on the decompletions of those developed by Shimizu. Along the way, we observe that a p-adic local system on a smooth geometrically connected algebraic variety over K is Hodge–Tate if its stalk at one closed point is a Hodge–Tate Galois representation. Moreover, we prove a version of the main theorem for local systems with arbitrary geometric monodromy, which allows us to conclude that the Galois action on the proalgebraic completion of ${\mathrm{\pi }}_1^{\mathrm{ét}}$ is de Rham.

We prove the local Gan–Gross–Prasad conjecture for tempered L-packets of real unitary groups. The proof is based on theta lifts and is very simple.