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Acta Mathematica, ISSN 0001-5962, 3/2015, Volume 214, Issue 1, pp. 61 - 134
We prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in $${\mathbb{R}^{3}}$$ R 3 . More... 
Euler equation | 37N10 | Mathematics | 35Q31 | 37J40 | 57M25 | 35J25 | invariant tori | Beltrami fields | KAM theory | 37C55 | Runge-type approximation | Mathematics, general | knots | DIFFEOMORPHISMS | TOPOLOGY | MATHEMATICS | EQUATIONS | ORBITS | Toruses | Infinity | Mathematical analysis | Vortices | Fluid flow | Texts | Belts | Tubes | Invariants
Journal Article
Ergodic theory and dynamical systems, ISSN 0143-3857, 08/2019, Volume 39, Issue 8, pp. 2048 - 2070
... by knotted wires ALBERTO ENCISO and DANIEL PERALTA-SALAS Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Cientı́ficas, 28049 Madrid, Spain (e... 
Original Article | MATHEMATICS | LINKS | MATHEMATICS, APPLIED | Topology | Magnetic fields | Wire | Knots
Journal Article
Annals of Mathematics, ISSN 0003-486X, 1/2012, Volume 175, Issue 1, pp. 345 - 367
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)... 
Tangents | Periodic orbits | Approximation | Vector fields | Coordinate systems | Mathematical functions | Euler equations | Curl | Knots | TOPOLOGY | MATHEMATICS | NUMBER | EQUILIBRIA
Journal Article
Archive for Rational Mechanics and Analysis, ISSN 0003-9527, 4/2016, Volume 220, Issue 1, pp. 243 - 260
Journal Article
Communications in Mathematical Physics, ISSN 0010-3616, 2019, Volume 368, Issue 3, pp. 1297 - 1322
Using stationary phase methods, we provide an explicit formula for the Melnikov function of the one and a half degrees of freedom system given by a Hamiltonian... 
PHYSICS, MATHEMATICAL
Journal Article
International Journal of Geometric Methods in Modern Physics, ISSN 0219-8878, 10/2016, Volume 13
This is a survey of certain geometric aspects of inviscid and incompressible fluid flows, which are described by the solutions to the Euler equations. We will... 
Knots and links | topological structure | Euler equations | stream and vortex lines | EXISTENCE | FIELDS | WEINSTEIN CONJECTURE | 3-MANIFOLDS | ORBITS | PHYSICS, MATHEMATICAL | EULER EQUATION
Journal Article
Proceedings of the National Academy of Sciences - PNAS, ISSN 0027-8424, 2/2016, Volume 113, Issue 8, pp. 2035 - 2040
We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional 𝓘 defined on... 
Volume-preserving transformation | Integral invariant | Helicity | FIELDS | helicity | volume-preserving transformation | EULER EQUATION | integral invariant | MULTIDISCIPLINARY SCIENCES | Transformations (Mathematics) | Research | Fluid dynamics | Integrals | Mathematical research | Physical Sciences
Journal Article
Journal of Nonlinear Science, ISSN 0938-8974, 06/2017, Volume 27, Issue 3, pp. 721 - 774
We prove the existence of diffusing solutions in the motion of a charged particle in the presence of ABC magnetic fields. The equations of motion are modeled... 
Motion of charges in magnetic fields | Hamiltonian dynamical systems | Global instability | Arnold diffusion | Heteroclinic connections | EXISTENCE | UNBOUNDED ENERGY | MATHEMATICS, APPLIED | PHYSICS, MATHEMATICAL | UNSTABLE HAMILTONIAN-SYSTEMS | MECHANICS | EVOLUTION | GROWTH | ORBITS | 3-BODY PROBLEM | FLOWS | TRANSITION TORI | RESONANCES
Journal Article
Physical Review Letters, ISSN 0031-9007, 10/2013, Volume 111, Issue 15, p. 150404
We construct analytically, a new family of null solutions to Maxwell's equations in free space whose field lines encode all torus knots and links. The... 
VORTEX LINES | LINKS | MECHANICS | TURBULENCE | PHYSICS, MULTIDISCIPLINARY | MAXWELL EQUATIONS
Journal Article
PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, ISSN 1364-503X, 11/2019, Volume 377, Issue 2158, p. 20190034
Journal Article
Journal of Differential Geometry, ISSN 0022-040X, 10/2015, Volume 101, Issue 2, pp. 197 - 211
In this paper we consider the problem of prescribing the nodal set of low-energy eigenfunctions of the Laplacian. Our main result is that, given any separating... 
LAPLACIAN | EIGENVALUE | MATHEMATICS | GENERIC PROPERTIES | 2ND EIGENFUNCTION | DOMAINS | CRITICAL-POINTS | LINES | RIEMANNIAN-MANIFOLDS
Journal Article
Annales Scientifiques de l'Ecole Normale Superieure, ISSN 0012-9593, 07/2017, Volume 50, Issue 4, pp. 995 - 1016
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... 
MATHEMATICS
Journal Article
Mathematische Annalen, ISSN 0025-5831, 2/2016, Volume 364, Issue 1, pp. 97 - 124
... · Daniel Peralta-Salas 2 · Shigeru Sakaguchi 3 Received: 10 July 2014 / Revised: 26 January 2015 / Published online: 10 April 2015 © Springer-Verlag Berlin Heidelberg... 
Mathematics, general | Primary 35K05 | Secondary 53A10 | Mathematics | 35K15 | 58J70 | EMBEDDED SURFACES | TOPOLOGY | MATHEMATICS | MINIMAL-SURFACES | UNIQUENESS | GEOMETRY
Journal Article
Advances in Mathematics, ISSN 0001-8708, 07/2019, Volume 351, pp. 718 - 760
We show that a wide range of overdetermined boundary problems for semilinear equations with position-dependent nonlinearities admits nontrivial solutions. The... 
Symmetric spaces | Asymptotically homogeneous spaces | Semilinear elliptic problems | Harmonic spaces | Overdetermined boundary value problems | EXTREMAL DOMAINS | 1ST EIGENVALUE | CLASSIFICATION | PROOF | HYPERPOLAR | MATHEMATICS | SYMMETRY | MASS
Journal Article
Advances in Mathematics, ISSN 0001-8708, 12/2013, Volume 249, pp. 204 - 249
We consider the problem of characterizing which noncompact hypersurfaces in Rn can be regular level sets of a harmonic function modulo a C∞ diffeomorphism, as... 
Global approximation theorem | Diffeomorphism | Stability theorem | Level set | Harmonic function | Second-order elliptic PDE | APPROXIMATION | SINGULARITIES | RIEMANN SURFACES | EQUATIONS | CRITICAL-POINTS | MATHEMATICS | SYMMETRY | HARMONIC-FUNCTIONS | MANIFOLDS | DOMAINS | CLOSED SUBSETS
Journal Article
Annali di Matematica Pura ed Applicata (1923 -), ISSN 0373-3114, 6/2015, Volume 194, Issue 3, pp. 881 - 901
... Enciso · Daniel Peralta-Salas Received: 11 November 2013 / Accepted: 21 January 2014 / Published online: 4 February 2014 © Fondazione Annali di Matematica Pura ed... 
Green’s function | Geometric properties | 35B38 | Elliptic PDE | Critical points | 35J08 | Mathematics, general | 37B25 | Mathematics | EXTERIOR | MATHEMATICS, APPLIED | Green's function | EIGENFUNCTIONS | MATHEMATICS | COMPLETE MANIFOLDS | ELLIPTIC-EQUATIONS | HEAT KERNEL | DOMAINS | NODAL SETS
Journal Article
Transactions of the American Mathematical Society, ISSN 0002-9947, 08/2012, Volume 364, Issue 8, pp. 4207 - 4224
In this paper we analyze the eigenvalues and eigenfunctions of the Hodge Laplacian for generic metrics on a closed 3-manifold M. In particular, we show that... 
Tensors | Riemann manifold | Mathematical theorems | Spectral theory | Eigenvalues | Differentials | Eigenfunctions | Scalars | Laplacians | MATHEMATICS | EIGENFUNCTIONS | SPECTRUM | SETS | RIEMANNIAN-MANIFOLDS
Journal Article
Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, 01/2018, Volume 51, Issue 2, p. 25204
An initially knotted light field will stay knotted if it satisfies a set of nonlinear, geometric constraints, i.e. the null conditions, for all space-time.... 
Topological | Electromagnetism | Shear-free flow | Knotted fields | electromagnetism | PHYSICS, MULTIDISCIPLINARY | NULL ELECTROMAGNETIC-FIELDS | EQUATIONS | LINKS | MAXWELL | topological | PHYSICS, MATHEMATICAL | shear-free flow | knotted fields | GEOMETRY
Journal Article
Journal of the European Mathematical Society, ISSN 1435-9855, 2018, Volume 20, Issue 2, pp. 301 - 314
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... 
Nodal sets | Eigenfunctions | Links | Harmonic oscillator | Knots | MATHEMATICS | MATHEMATICS, APPLIED | nodal sets | knots and links | SETS | harmonic oscillator
Journal Article
Analysis and PDE, ISSN 2157-5045, 2016, Volume 9, Issue 6, pp. 1433 - 1446
We make use of the flexibility of infinite-index solutions to the Allen-Cahn equation to show that, given any compact hypersurface Sigma of R-d with d >= 3,... 
Level sets | Allen-Cahn equation | MATHEMATICS | MATHEMATICS, APPLIED | level sets | PHASE-TRANSITIONS | CURVATURE | R-3 | DIFFUSION | GIORGIS CONJECTURE
Journal Article
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