Given a closed exact Lagrangian in the cotangent bundle of a closed smooth manifold, we prove that the projection to the base is a simple homotopy equivalence.

We study groups of automorphisms and birational transformations of quasi-projective varieties. Two methods are combined; the first one is based on $p$-adic analysis, the second makes use of isoperimetric inequalities and Lang–Weil estimates. For instance, we show that, if $\mathsf{SL}_n(\mathbf{Z})$ acts faithfully on a complex quasi-projective variety $X$ by birational transformations, then $\mathrm{dim}(X) \geqslant n-1$ and $X$ is rational if $\mathrm{dim}(X) = n-1$.

We show that each limiting semiclassical measure obtained from a sequence of eigenfunctions of the Laplacian on a compact hyperbolic surface is supported on the entire cosphere bundle. The key new ingredient for the proof is the fractal uncertainty principle, first formulated in “Spectral gaps, additive energy, and a fractal uncertainty principle” [Dyatlov, S. & Zahl, J. Geom. Funct. Anal., 26 (2016), 1011–1094] and proved for porous sets in “Spectral gaps without the pressure condition” [Bourgain, J. & Dyatlov, S. Ann. of Math., 187 (2018), 825–867].

We study rationality properties of quadric surface bundles over the projective plane. We exhibit families of smooth projective complex fourfolds of this type over connected bases, containing both rational and non-rational fibers.

Let $M \subset \mathbb{C}^N$ be a generic real-analytic submanifold of finite type, $M' \subset \mathbb{C}^{N'}$ be a real-analytic set, and $p \in M$, where we assume that $N, N' \geqslant 2$. Let $H: (\mathbb{C}^N, p) \to \mathbb{C}^{N'}$ be a formal holomorphic mapping sending $M$ into $M'$, and let $\mathcal{E}_{M'}$ denote the set of points in $M'$ through which there passes a complex-analytic subvariety of positive dimension contained in $M'$. We show that, if $H$ does not send $M$ into $\mathcal{E}_{M'}$, then $H$ must be convergent. As a consequence, we derive the convergence of all formal holomorphic mappings when $M'$ does not contain any complex-analytic subvariety of positive dimension, answering by this a long-standing open question in the field. More generally, we establish necessary conditions for the existence of divergent formal maps, even when the target real-analytic set is foliated by complex-analytic subvarieties, allowing us to settle additional convergence problems such as e.g. for transversal formal maps between Levi-non-degenerate hypersurfaces and for formal maps with range in the tube over the light cone.