#### -"Beltrami fields exhibit knots and chaos almost surely", preprint (with D. Peralta-Salas and Á. Romaniega).

We show that, with probability 1, a random Beltrami field exhibits chaotic regions that coexist with invariant tori of complicated topologies. The motivation to consider this question, which arises in the study of stationary Euler flows in dimension 3, is V.I. Arnold's 1965 conjecture that a typical Beltrami field exhibits the same complexity as the restriction to an energy hypersurface of a generic Hamiltonian system with two degrees of freedom. Our results hold both in the case of Beltrami fields on R^{3}and of high-frequency Beltrami fields on the 3-torus.

#### -"Approximation theorems for the Schrödinger equation and quantum vortex reconnection", preprint (with D. Peralta-Salas).

We prove the existence of smooth solutions to the Gross-Pitaevskii equation on R^{3}that feature arbitrarily complex quantum vortex reconnections. We can track the evolution of the vortices during the whole process. This permits to describe the reconnection events in detail and verify that this scenario exhibits the properties observed in experiments and numerics, such as the

*t*

^{1/2}and the change of parity laws. An essential ingredient in the proofs is the development of novel global approximation theorems for the Schrödinger equation on R

^{n}. Specifically, we prove a qualitative approximation result that applies for solutions defined on very general spacetime sets and also a quantitative result for solutions on product sets in spacetime. This hinges on frequency-dependent estimates for the Helmholtz-Yukawa equation that are of independent interest.

#### - "Uniqueness and convexity of Whitham's highest cusped wave", preprint (with J. Gómez-Serrano and B. Vergara).

We prove the existence of a periodic traveling wave of extreme form of the Whitham equation that has a convex profile between consecutive stagnation points, at which it is known to feature a cusp of exactly*C*

^{1/2}regularity. The convexity of Whitham's highest cusped wave had been conjectured by Ehrnström and Wahlén.

#### - "Spectral determination of semi-regular polygons", Journal of Differential Geometry, in press (with J. Gómez-Serrano).

Let us say that an*n*-sided polygon is semi-regular if it is circumscriptible and its angles are all equal but possibly one, which is then larger than the rest. Regular polygons, in particular, are semi-regular. We prove that semi-regular polygons are spectrally determined in the class of convex piecewise smooth domains. Specifically, we show that if Ω is a convex piecewise smooth planar domain, possibly with straight corners, whose Dirichlet or Neumann spectrum coincides with that of an

*n*-sided semi-regular polygon

*P*, then Ω is congruent to

*P*.

#### - "Carleman estimates with sharp weights and boundary observability for wave operators with critically singular potentials", Journal of the European Mathematical Society, in press (with A. Shao and B. Vergara).

We establish a new family of sharp Carleman inequalities for wave operators on a cylindrical spacetime domain containing a potential that diverges as an inverse square on all the boundary of the domain. The proof is based around three key ingredients: the choice of a novel Carleman weight with rather singular derivatives on the boundary, a generalization of the classical Morawetz inequality that allows for inverse-square singularities, and the systematic use of derivative operations adapted to the potential.#### - "Self-intersecting interfaces for stationary solutions of the two-fluid Euler equations", Annals of PDE, in press (with D. Córdoba and N. Grubic).

We prove that there are stationary solutions to the 2D incompressible free boundary Euler equations with two fluids, with or without gravity, that feature a splash singularity. More precisely, in the solutions we construct the interface is a*C*

^{2,α}smooth curve that intersects itself at one point, and the vorticity density on the interface is of class

*C*

^{α}. The proof consists in perturbing Crapper’s family of formal stationary solutions with one fluid, so the crux is to introduce a small but positive second-fluid density. To do so, we use a novel set of weighted estimates for self-intersecting interfaces that squeeze an incompressible fluid.

#### - "Lorentzian Einstein metrics with prescribed conformal infinity", Journal of Differential Geometry 112 (2019) 505-554 (with N. Kamran).

We prove a local well-posedness theorem for the (*n*+ 1)-dimensional Einstein equations in Lorentzian signature, with initial data (

*g*,

*K*) whose asymptotic geometry at infinity is similar to that anti-de Sitter (AdS) space, and compatible boundary data

*G*prescribed at the time-like conformal boundary of space-time the satisfy the Einstein constraint equations. Under certain polyhomogeneous regularity conditions, we prove a local well-posedness result for the Einstein equations that yields a unique (up to a diffeomorphism) solution

*g*that is of class

*C*

^{n-1}up to the boundary and smooth in the polyhomogeneous sense.

#### - "A problem of Ulam about magnetic fields generated by knotted wires", Ergodic Theory & Dynamical Systems 39 (2019) 2048-2070. (with D. Peralta-Salas).

In the context of magnetic fields generated by wires, we study the connection between the topology of the wire and the topology of the magnetic lines. We show that a generic knotted wire has a magnetic line of the same knot type, but that given any pair of knots there is a wire isotopic to the first knot having a magnetic line isotopic to the second. These questions can be traced back to Ulam in 1935.#### - "Approximation theorems for parabolic equations and movement of hot spots", Duke Mathematical Journal 168 (2019) 897-939 (with M.A. García-Ferrero and D. Peralta-Salas).

We prove a global approximation theorem for a general parabolic operator*L*, which asserts that if

*v*satisfies the equation

*Lv*= 0 in a spacetime region Ω ⊂

**R**

^{n+1}satisfying certain necessary topological condition, then it can be approximated in a Hölder norm by a global solution

*u*to the equation. If Ω is compact and the operator

*L*satisfies certain technical conditions (e.g., when it is the usual heat equation), the global solution

*u*can be shown to fall off in space and time. This result is next applied to prove the existence of global solutions to the equation

*Lu*= 0 with a local hot spot that moves along a prescribed curve for all time, up to a uniformly small error. Global solutions that exhibit isothermic hypersurfaces of prescribed topologies for all times and applications to the heat equation on the flat torus are discussed too.

#### - "A problem of Berry and knotted zeros in the eigenfunctions of the harmonic oscillator", Journal of the European Mathematical Society 20 (2018) 301-314 (with D. Hartley and D. Peralta-Salas).

We prove that, given any finite link*L*in

**R**

^{3}, there is a high-energy complex-valued eigenfunction of the harmonic oscillator such that its nodal set contains a union of connected components diffeomorphic to

*L*. This solves a problem of Berry on the existence of knotted zeros in bound states of a quantum system.

#### - "Knotted structures in high-energy Beltrami fields on the torus and the sphere", Annales Scientifiques de l'École Normale Supérieure 50 (2017) 995-1016 (with D. Peralta-Salas and F. Torres de Lizaur).

Let*S*be a finite union of (pairwise disjoint but possibly knotted and linked) closed curves and tubes in the round sphere

**S**

^{3 }or in the flat torus

**T**

^{3}. In the case of the torus,

*S*is further assumed to be contained in a contractible subset of

**T**

^{3}. In this paper we show that for any sufficiently large odd integer

*N*here exists a Beltrami field on

**S**

^{3}or

**T**

^{3}satisfying curl

*u*=

*N u*and with a collection of vortex lines and vortex tubes given by

*S*, up to an ambient diffeomorphism.

#### - "Vortex reconnection in the three dimensional Navier-Stokes equations", Advances in Mathematics 309 (2017) 452-486 (with R. Lucà and D. Peralta-Salas).

We prove that the vortex structures of solutions to the 3D Navier– Stokes equations can change their topology without any loss of regularity. More precisely, we construct smooth high-frequency solutions to the Navier–Stokes equations where vortex lines and vortex tubes of arbitrarily complicated topologies are created and destroyed in arbitrarily small times. This instance of vortex reconnection is structurally stable and in perfect agreement with the existing computer simulations and experiments. We also provide a (non-structurally stable) scenario where the destruction of vortex structures is instantaneous.

#### - "Helicity is the only integral invariant of volume-preserving transformations", Proceedings of the National Academy of Sciences 113 (2016) 2035–2040 (with D. Peralta-Salas and F. Torres de Lizaur).

We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional*F*defined on exact divergence-free vector fields of class C

^{1}on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that

*F*is invariant under arbitrary volume-preserving diffeomorphisms if and only if it is a function of the helicity.

#### - "Beltrami fields with a nonconstant proportionality factor are rare", Archive for Rational Mechanics and Analysis 220 (2016) 243–260 (with D. Peralta-Salas).

We consider the existence of Beltrami fields with a nonconstant proportionality factor*f*in an open subset

*U*of

**R**

^{3}. By reformulating this problem as a constrained evolution equation on a surface, we find an explicit differential equation that

*f*must satisfy whenever there is a nontrivial Beltrami field with this factor. This ensures that there are no nontrivial solutions for an open and dense set of factors

*f*in the

*C*

^{k}topology. In particular, there are no nontrivial Beltrami fields whenever

*f*has a regular level set diffeomorphic to the sphere. This provides an explanation of the helical flow paradox of Morgulis, Yudovich and Zaslavsky.

#### - "Splash and almost-splash stationary solutions to the Euler equations", Advances in Mathematics 288 (2016) 922–941 (with D. Córdoba and N. Grubic).

We discuss the existence of stationary incompressible fluids with splash singularities. Specifically, we show that there are stationary solutions to the Euler equations with two fluids whose interfaces are arbitrarily close to a splash, and that there are stationary water waves with splash singularities.#### - "Existence of knotted vortex tubes in steady fluid flows", Acta Mathematica 214 (2015) 61–134 (with D. Peralta-Salas).

We prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation on**R**

^{3}. More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes in

**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 conjecture of the existence of steady knotted thin vortex tubes can be traced back to Lord Kelvin in 1875.

#### - "Eigenfunctions with prescribed nodal sets", Journal of Differential Geometry 101 (2015) 197–211 (with D. Peralta-Salas).

We consider the problem of prescribing the nodal set of low-energy eigenfunctions of the Laplacian. Our main result is that, given any separating closed hypersurface Σ in a compact*n*-manifold

*M*, there is a Riemannian metric on

*M*such that the nodal set of its first nontrivial eigenfunction is Σ. We present a number of variations on this result, which enable us to show, in particular, that the first nontrivial eigenfunction can have as many non-degenerate critical points as one wishes.

#### - "Submanifolds that are level sets of solutions to a second-order elliptic PDE", Advances in Mathematics 249 (2013) 204–249 (with D. Peralta-Salas).

Motivated by a question of Rubel, we consider the problem of characterizing which noncompact hypersurfaces in**R**

^{n}can be regular level sets of a harmonic function modulo a smooth diffeomorphism, as well as certain generalizations to other PDEs. We prove a versatile sufficient condition that shows, in particular, that any (possibly disconnected) algebraic noncompact hypersurface can be transformed onto a union of components of the zero set of a harmonic function via a diffeomorphism of

**R**

^{n}. The technique we use, which is a significant improvement of the basic strategy we recently applied to construct solutions to the Euler equation with knotted stream lines, combines robust but not explicit local constructions with appropriate global approximation theorems. In view of applications to a problem of Berry and Dennis, intersections of level sets are also studied.

#### - "Knots and links in steady solutions of the Euler equation", Annals of Mathematics 175 (2012) 345–367 (with D. Peralta-Salas).

Given any possibly unbounded, locally finite link, we show that there exists a smooth diffeomorphism transforming this link into a set of stream (or vortex) lines of a vector field that solves the steady incompressible Euler equation in**R**

^{3}. Furthermore, the diffeomorphism can be chosen arbitrarily close to the identity in any

*C*

^{r}norm. This proves a conjecture of Arnold from 1965.