We propose a new method for constructing partially hyperbolic diffeomorphisms on closed manifolds. As a demonstration of the method we show that there are simply connected closed manifolds that support partially hyperbolic diffeomorphisms. Laying aside many surgery constructions of 3-dimensional Anosov flows, these are the first new examples of manifolds which admit partially hyperbolic diffeomorphisms in the past forty years.

We study mixing properties of commutative groups of automorphisms acting on compact nilmanifolds. Assuming that every non-trivial element acts ergodically, we prove that such actions are mixing of all orders. We further show exponential 2-mixing and 3-mixing. As an application we prove smooth cocycle rigidity for higher-rank abelian groups of nilmanifold automorphisms.

We study Schrödinger operators with a one-frequency analytic potential, focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials. From the dynamical point of view, the transition signals the emergence of non-uniform hyperbolicity, so the dependence of the Lyapunov exponent with respect to parameters plays a central role in the analysis. Though often ill-behaved by conventional measures, we show that the Lyapunov exponent is in fact remarkably regular in a “stratified sense” which we define: the irregularity comes from the matching of nice (analytic or smooth) functions along sets with complicated geometry. This result allows us to establish that the “critical set” for the transition lies within countably many codimension one subvarieties of the (infinite-dimensional) parameter space. A more refined renormalization-based analysis shows that the critical set is rather thin within those subvarieties, and allows us to conclude that a typical potential has no critical energies. Such acritical potentials also form an open set and have several interesting properties: only finitely many “phase transitions” may happen, but never at any specific point in the spectrum, and the Lyapunov exponent is minorated in the region of the spectrum where it is positive. On the other hand, we do show that the number of phase transitions can be arbitrarily large.

We show that given $${n \geqslant 3}$$ , $${q \geqslant 1}$$ , and a finite set $${\{y_1, \ldots, y_q \}}$$ in $${\mathbb{R}^n}$$ there exists a quasiregular mapping $${\mathbb{R}^n\to \mathbb{R}^n}$$ omitting exactly points $${y_1, \ldots, y_q}$$ .

We prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in $${\mathbb{R}^{3}}$$ . More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes in $${\mathbb{R}^{3}}$$ , we show that they can be transformed with a$C$^{$m$}-small diffeomorphism into a set of vortex tubes of a Beltrami field that tends to zero at infinity. The structure of the vortex lines in the tubes is extremely rich, presenting a positive-measure set of invariant tori and infinitely many periodic vortex lines. The problem of the existence of steady knotted thin vortex tubes can be traced back to Lord Kelvin.

We give a method for constructing transcendental entire functions with good control of both the singular values of$f$and the geometry of$f$. Among other applications, we construct a function$f$with bounded singular set, whose Fatou set contains a wandering domain.

We study the critical Ising model on the square lattice in bounded simply connected domains with + and free boundary conditions. We relate the energy density of the model to a discrete fermionic correlator and compute its scaling limit by discrete complex analysis methods. As a consequence, we obtain a simple exact formula for the scaling limit of the energy field one-point function in terms of the hyperbolic metric. This confirms the predictions originating in physics, but also provides a higher precision.

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Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of non-linear echoes; sharp “deflection” estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the non-linear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications. Finally, we extend these results to some Gevrey (non-analytic) distribution functions.

We introduce canonical antisymmetric quasiconformal maps, which minimize the quasiconformality constant among maps sending the unit circle to a given quasicircle. As an application we prove Astala’s conjecture that the Hausdorff dimension of a$k$-quasicircle is at most 1+$k$^{2}.