I work on different problems in Fluid Mechanics, Dynamical Systems, Spectral Geometry and Mathematical Physics. I have published over 100 papers in high profile mathematical journals such as: Annals of Mathematics, Acta Mathematica, Duke Mathematical Journal, Annales Scientifiques de l'Ecole Normale Superieure, Journal of Differential Geometry (x2), American Journal of Mathematics, Journal of the European Mathematical Society (x2) and Advances in Mathematics (x9). My research has also appeared in top multidisciplinary journals, such as PNAS (x3) or Physical Review Letters.
Among the best results that I have obtained with my collaborators, I highlight:
- A new theory to study geometrically complex structures in the equations that model physical phenomena. Main landmarks of this theory include a proof of a 1965 Arnold's conjecture in hydrodynamics, of the centennial Lord Kelvin conjecture on the existence of steady knotted vortex tubes in Euler flows and the proof of a conjecture of Sir Michael Berry on knotted nodal lines in quantum systems.
- Construction of steady Euler flows in Euclidean space exhibiting wild and universal dynamics (which implies the existence of horseshoes of maximal Hausdorff dimension and Newhouse-type phenomena). These results establish a strong version of V.I. Arnold's vision on the complexity of Beltrami fields in Euclidean space.
- First theoretical construction of a 3D steady fluid flow that can simulate a universal Turing machine, both on a Riemannian manifold and on Euclidean space. This implies the existence of undecidable fluid particle paths, a long standing open problem posed by C. Moore in 1991.
- Construction of stepped-pressure magnetohydrostatic equilibria in toroidal domains of arbitrary topology. This provides a first substantial step towards disproving the influential 1967 Grad's conjecture in plasma physics.
- First proof of the existence of vortex reconnections (topological transitions) for smooth solutions of the Navier-Stokes equations, and in the context of quantum fluids (the Gross-Pitaevskii equation). For the latter we developed a new global approximation theory for the Schrodinger equation.
- Development of the global approximation theory for parabolic equations and applications to hot spots, and the inverse localization technique in spectral geometry.
- Development of a KAM theoretic approach to study ergodicity and topological mixing in the 3D Euler equations. This new tool allowed us to prove that the incompressible fluid flows cannot be ergodic nor mixing, and that a typical steady state cannot be approached by a generic initial condition in its vicinity.
- Proof that the helicity is the only regular integral invariant under volume preserving diffeomorphisms, both in low regularity and in the smooth topology. The latter proves a conjecture by Arnold and Khesin on the description of the Casimirs for adjoint and coadjoint actions of the group of volume preserving diffeomorphisms.
- Classification of the unbounded isothermic surfaces that are stationary for the heat equation in Euclidean 3-space.
- Solution of a 1993 problem of S.T. Yau on the nodal set of Laplace eigenfunctions (number 45 on his famous list).
- Development of the theory of integrable embeddings to study foliations in Euclidean space. This allowed us to classify the submanifolds that can be realized as leaves of a regular foliation, this solving a problem formulated by E. Vogt in 1993.
- Analytical construction of a new family of solutions to the Maxwell's equations whose electric and magnetic lines encode all torus knots and links, which persist for all time. The existence of these structures was an elusive open problem since 1990.
Contact data
- Instituto de Ciencias Matemáticas
- E-mail: dperalta()icmat.es
- Office: 206
- Phone: [+34] 91 2999746