#### - "Spectral determination of semi-regular polygons", preprint (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*.

#### - "Approximation theorems for parabolic equations and movement of hot spots", preprint (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.

#### - "Solutions to the overdetermined boundary problem for semilinear equations with position-dependent nonlinearities", preprint (with M. Domínguez-Vázquez and D. Peralta-Salas).

We show that a wide range of overdetermined boundary problems for semilinear equations with position-dependent nonlinearities admits nontrivial solutions. The result holds true both on**R**

^{n}and on compact Riemannian manifolds. As a byproduct of the proofs we also obtain some rigidity, or partial symmetry, results for solutions to overdetermined problems on Riemannian manifolds of nonconstant curvature.

#### - "Lorentzian Einstein metrics with prescribed conformal infinity", Journal of Differential Geometry, in press (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, in press (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.#### - "A problem of Berry and knotted zeros in the eigenfunctions of the harmonic oscillator", Journal of the European Mathematical Society, in press (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).

In this paper 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.