Here you can find a few papers that illustrate my latest research interests. My full publication list is here.
Topological questions in fluid mechanics, knotted vortex structures for 3D Euler
Obstructions to topological relaxation for generic magnetic fields, Arch. Ration. Mech. Anal., in press (with D. Peralta-Salas).
Beltrami fields exhibit knots and chaos almost surely, Forum Math. Sigma 11:e56 (2023) (with D. Peralta-Salas and A. Romaniega).
Knotted structures in high-energy Beltrami fields on the torus and the sphere, Ann. Sci. Éc. Norm. Sup. 50 (2017) 995–1016 (with D. Peralta-Salas and F. Torres de Lizaur).
Vortex reconnection in the three dimensional Navier-Stokes equations, Adv. Math. 309 (2017) 452–486 (with R. Lucà and D. Peralta-Salas).
Existence of knotted vortex tubes in steady fluid flows, Acta Math. 214 (2015) 61–134 (with D. Peralta-Salas).
Knots and links in steady solutions of the Euler equation, Ann. of Math. 175 (2012) 345–367 (with D. Peralta-Salas).
Water waves, non-smooth interfaces, singularity formation
Finite time singularity formation for angle-crested water waves, preprint (with D. Córdoba and N. Grubic).
Convexity of Whitham's highest cusped wave, preliminary version (with J. Gómez-Serrano and B. Vergara).
MHD equilibria with nonconstant pressure in nondegenerate toroidal domains, J. Eur. Math. Soc. 27 (2025) 2251–2291 (with A. Luque and D. Peralta-Salas).
Self-intersecting interfaces for stationary solutions of the two-fluid Euler equations, Ann. of PDE 7 (2021) 12 (with D. Córdoba and N. Grubic).
Convex integration, structure of global solutions
Hölder continuous dissipative solutions of ideal MHD with nonzero helicity, preprint (with J. Peñafiel-Tomás and D. Peralta-Salas).
Steady 3d Euler flows via a topology-preserving convex integration scheme, preprint (with J. Peñafiel-Tomás and D. Peralta-Salas).
Smooth nonradial stationary Euler flows on the plane with compact support, preprint (with A.J. Fernández and D. Ruiz).
An extension theorem for weak solutions of the 3D incompressible Euler equations and applications to singular flows, preprint (with J. Peñafiel-Tomás and D. Peralta-Salas).
Isolated steady solutions of the 3D Euler equations, Proc. Nat. Acad. Sci. 122 (2025) 12 (with W. Kepplinger and D. Peralta-Salas).
Quasi-periodic solutions to the incompressible Euler equations in dimensions two and higher, J. Differential Equations 354 (2023) 170–182 (with D. Peralta-Salas and F. Torres de Lizaur).
Spectral geometry, nodal sets of eigenfunctions
Local limits of high energy eigenfunctions on integrable billiards, preprint (with A. García-Ruiz and D. Peralta-Salas).
Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator, J. Eur. Math. Soc., in press (with A. Pistoia and L. Provenzano).
A Schiffer-type problem for annuli with applications to stationary planar Euler flows, Duke Math. J., in press (with A.J. Fernández, D. Ruiz and P. Sicbaldi).
Spectral determination of semi-regular polygons, J. Differential Geom. 122 (2022) 399–419 (with J. Gómez-Serrano).
A problem of Berry and knotted zeros in the eigenfunctions of the harmonic oscillator, J. Eur. Math. Soc. 20 (2018) 301–314 (with D. Hartley and D. Peralta-Salas).
Eigenfunctions with prescribed nodal sets, J. Differential Geom. 101 (2015) 197–211 (with D. Peralta-Salas).
Wave equations, Carleman estimates, anti-de Sitter spaces
Carleman estimates with sharp weights and boundary observability for wave operators with critically singular potentials, J. Eur. Math. Soc. 23 (2021) 3459–3495 (with A. Shao and B. Vergara).
Lorentzian Einstein metrics with prescribed conformal infinity, J. Differential Geom. 112 (2019) 505–554 (with N. Kamran).
A singular initial-boundary value problem for nonlinear wave equations and holography in asymptotically anti-de Sitter spaces, J. Math. Pures Appl. 103 (2015) 1053–1091 (with N. Kamran).
Critical points and level sets of solutions to PDEs; global approximation theorems
Approximation theorems for the Schrödinger equation and quantum vortex reconnection, Comm. Math. Phys. 387 (2021) 1111–1149 (with D. Peralta-Salas).
Approximation theorems for parabolic equations and movement of hot spots, Duke Math. J. 168 (2019) 897–939 (with M.A. García-Ferrero and D. Peralta-Salas).
Submanifolds that are level sets of a solution to an elliptic PDE, Adv. Math. 249 (2013) 204–249 (with D. Peralta-Salas).
Critical points of Green’s functions on complete manifolds, J. Differential Geom. 92 (2012) 1–29 (with D. Peralta-Salas).
Critical points and level sets in exterior boundary problems, Indiana Univ. Math. J. 58 (2009) 1947–1970 (with D. Peralta-Salas).
Overdetermined elliptic boundary value problems
Positive solutions to general semilinear overdetermined boundary problems, preprint (with P. Hidalgo-Palencia and X. Ros-Oton).
Piecewise smooth stationary Euler flows with compact support via overdetermined boundary problems, Arch. Ration. Mech. Anal. 239 (2021) 1327–1347 (with M. Domínguez-Vázquez and D. Peralta-Salas).
Solutions to the overdetermined boundary problem for semilinear equations with position-dependent nonlinearities, Adv. Math. 351 (2019) 718–760 (with M. Domínguez-Vázquez and D. Peralta-Salas).
Hamiltonian and volume-preserving dynamics
Non-integrability and chaos for natural Hamiltonian systems with a random potential, Adv. Math. 437 (2024) 109448 (with D. Peralta-Salas and A. Romaniega).
A problem of Ulam about magnetic fields generated by knotted wires, Ergodic Th. Dynam. Syst. 39 (2019) 2048–2070 (with D. Peralta-Salas).
Helicity is the only integral invariant of volume-preserving transformations, Proc. Nat. Acad. Sci. 113 (2016) 2035–2040 (with D. Peralta-Salas and F. Torres de Lizaur).