## Final Report Summary - INVARIANT (Invariant manifolds in dynamical systems and PDE)

There are many problems in mathematical-physics where the natural objects of study are certain geometric structures associated with vector fields or functions that solve a partial differential equation. Some examples of these structures are the level sets of the gravitatory potential, magnetic lines, stream lines of a fluid, or the nodal sets of a wave function. We call these objects “invariant manifolds” of the solutions to a PDE. The goal of this project is to study the invariant manifolds in the context of fluid mechanics and elliptic PDEs, where many important problems and conjectures due to Arnold, Kelvin, Ulam, Yau, etc. have been open for years. The difficulty of these questions is related to the interdisciplinary nature of the problem, which requires a high interaction between methods and technique from several areas of mathematics, including PDEs, dynamical systems and differential geometry. The main achievements obtained during the project life are the following.

-We have proved that any hypersurface of a compact manifold of dimension at least three is the nodal set of the first eigenfunction of the Laplacian for some metric, even if we fix the volume and the conformal class of the metric. For manifolds with boundary and Dirichlet boundary conditions this is a counterexample to Payne’s conjecture in the metric setting, thus solving a problem of Yau.

-We have studied the helicity of divergence-free vector fields showing that it is the only integral invariant under volume-preserving diffeomorphisms. This proves a conjecture of Arnold and Khesin, and invalidates previous attempts to construct invariants that are independent of the helicity.

-We have proved the existence of stationary solutions of the Euler equations with a prescribed set (up to isotopy) of knotted and linked vortex lines and tubes for some compact manifolds, e.g. the 3-torus and the 3-sphere. This tackles Arnold’s conjecture for some compact manifolds, and opens a new line of research consisting in the study of vortices of high-energy Beltrami fields, in analogy with the study of nodal sets of high-energy eigenfunctions of the Laplacian.

-We have proved the existence of global smooth solutions of the Navier-Stokes equations exhibiting a cascade of reconnections of vortex lines. More precisely, we have shown that the knot type of vortex lines and tubes can change as one desires, independently of the value of the viscosity (a positive constant) or the size of the initial condition (so this is not a small data phenomenon). This is the first time reconnections are shown to exist rigorously in Navier-Stokes.

-We have solved a problem posed by Michael Berry in 2001 on the existence of quantum states with knotted and linked nodal lines. More precisely, we have shown that for any link there is an eigenfunction of the harmonic oscillator or the hydrogen atom with a nodal line isotopic to the link. For this we use high-energy eigenfunctions, and techniques close to the ones developed to study high-energy Beltrami fields.

-We have solved Ulam’s problem (stated in the Polish Book in 1935) for a generic set of closed curves. More precisely, we have shown that a generic closed wire creates a closed magnetic line isotopic to the wire. This is one of the very few contributions in the understanding of magnetic fields created by wires. Other results have been obtained in this direction, as the proof of the existence of a wire isotopic to any prescribed knot creating a magnetic line isotopic to another prescribed knot.

-We have developed a global approximation theory for general (linear) parabolic operators. For the case of the usual heat operator we have proved approximation theorems with decay at infinity. These results have been applied to construct solutions to the heat equation (associated with compactly supported initial data) which exhibit a local hot spot moving along a prescribed curve for arbitrarly large times, up to a uniformly small error.

-We have proved that any hypersurface of a compact manifold of dimension at least three is the nodal set of the first eigenfunction of the Laplacian for some metric, even if we fix the volume and the conformal class of the metric. For manifolds with boundary and Dirichlet boundary conditions this is a counterexample to Payne’s conjecture in the metric setting, thus solving a problem of Yau.

-We have studied the helicity of divergence-free vector fields showing that it is the only integral invariant under volume-preserving diffeomorphisms. This proves a conjecture of Arnold and Khesin, and invalidates previous attempts to construct invariants that are independent of the helicity.

-We have proved the existence of stationary solutions of the Euler equations with a prescribed set (up to isotopy) of knotted and linked vortex lines and tubes for some compact manifolds, e.g. the 3-torus and the 3-sphere. This tackles Arnold’s conjecture for some compact manifolds, and opens a new line of research consisting in the study of vortices of high-energy Beltrami fields, in analogy with the study of nodal sets of high-energy eigenfunctions of the Laplacian.

-We have proved the existence of global smooth solutions of the Navier-Stokes equations exhibiting a cascade of reconnections of vortex lines. More precisely, we have shown that the knot type of vortex lines and tubes can change as one desires, independently of the value of the viscosity (a positive constant) or the size of the initial condition (so this is not a small data phenomenon). This is the first time reconnections are shown to exist rigorously in Navier-Stokes.

-We have solved a problem posed by Michael Berry in 2001 on the existence of quantum states with knotted and linked nodal lines. More precisely, we have shown that for any link there is an eigenfunction of the harmonic oscillator or the hydrogen atom with a nodal line isotopic to the link. For this we use high-energy eigenfunctions, and techniques close to the ones developed to study high-energy Beltrami fields.

-We have solved Ulam’s problem (stated in the Polish Book in 1935) for a generic set of closed curves. More precisely, we have shown that a generic closed wire creates a closed magnetic line isotopic to the wire. This is one of the very few contributions in the understanding of magnetic fields created by wires. Other results have been obtained in this direction, as the proof of the existence of a wire isotopic to any prescribed knot creating a magnetic line isotopic to another prescribed knot.

-We have developed a global approximation theory for general (linear) parabolic operators. For the case of the usual heat operator we have proved approximation theorems with decay at infinity. These results have been applied to construct solutions to the heat equation (associated with compactly supported initial data) which exhibit a local hot spot moving along a prescribed curve for arbitrarly large times, up to a uniformly small error.