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+34 946 567 842
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+34 946 567 842
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lvega@bcamath.org
Information of interest
 Orcid: 0000000150866345
My research is mainly focused in the interplay of Fourier Analysis and Partial Differential Equations of Mathematical Physics. More recently on fluid mechanics and turbulence. Concretely in the so called Localized Induction Approximation, also known as the binormal curvature flow (BF), for the evolution of vortex filaments and the relevance of the presence of corners in the filament. The results concerning regular polygons seem to me quite striking. Motivated by a numerical experiment done by D. Smets, and together with F. De La Hoz, we established in 2014 a numerical connection between the trajectory followed by a corner of, say an equilateral triangle, and a classical analytical problem that goes back at least to Riemann: the existence of continuous functions which are no where differentiable. Very recently (arXiv:2007.07184), and in collaboration with V. Banica, we have proved analytically that this connection is indeed true.
Right now I am also working together with N. Arrizabalaga and A. Mas, on relativistic and nonrelativistic equations with singular electromagnetic potentials. The singularities of the potentials are critical from the point of view of the scaling symmetry. In the relativistic setting we consider perturbations of Dirac equation given by singular measures supported on smooth hypersurfaces. This mathematical problem is closely related to a relevant question in physics, that of the optimal confinement of relativistic quantum particles.
Finally I continuous working on the deep connection between uncertainty principles, that are easily described using the Fourier transform, and lower bounds for solutions of linear and nonlinear dispersive equations. This is a topic that I started with L. Escauriaza, Carlos E. Kenig and G. Ponce more than 10 years ago and from which very fruitful branches have emerged. For example, one of the first consequences we obtained using these lower bounds, was that a compact perturbation of a solitary wave or soliton of the KortewegDe Vries (KdV) equation instantaneously destroys its exponential decay. KdV is a simplified local model about the dynamics of the frontier of a fluid. In particular, it describes with very high accuracy the propagation of a wave along a narrow and sallow channel. However, when the depth is big so that it can be considered close to be infinite the local approximation is too rough and nonlocal models as the BenjaminOno equation has to be considered. It turns out that the answer to the corresponding question requires completely different techniques that are closer to those developed with A. FernándezBertolin for the discrete laplacian and with L. Roncal and D. Stan for the fractional laplacian.

MemoryBased Monte Carlo Integration for Solving Partial Differential Equations Using Neural Networks
(2023)Monte Carlo integration is a widely used quadrature rule to solve Partial Differential Equations with neural networks due to its ability to guarantee overfittingfree solutions and highdimensional scalability. However, ...

The Frisch–Parisi formalism for fluctuations of the Schrödinger equation
(2022)We consider the solution of the Schrödinger equation $u$ in $\mathbb{R}$ when the initial datum tends to the Dirac comb. Let $h_{\text{p}, \delta}(t)$ be the fluctuations in time of $\int\lvert x \rvert^{2\delta}\lvert ...

Static and Dynamical, Fractional Uncertainty Principles
(202103)We study the process of dispersion of lowregularity solutions to the Schrödinger equation using fractional weights (observables). We give another proof of the uncertainty principle for fractional weights and use it to get ...

Unbounded growth of the energy density associated to the Schrödinger map and the binormal flow
(2021)We consider the binormal flow equation, which is a model for the dynamics of vortex filaments in Euler equations. Geometrically it is a flow of curves in three dimensions, explicitly connected to the 1D Schr¨odinger map ...

On the one dimensional cubic NLS in a critical space
(2021)In this note we study the initial value problem in a critical space for the one dimensional Schr¨odinger equation with a cubic nonlinearity and under some smallness conditions. In particular the initial data is given by ...

Eigenvalue Curves for Generalized MIT Bag Models
(2021)We study spectral properties of Dirac operators on bounded domains Ω ⊂ R 3 with boundary conditions of electrostatic and Lorentz scalar type and which depend on a parameter τ ∈ R; the case τ = 0 corresponds to the MIT ...

On the improvement of the Hardy inequality due to singular magnetic fields
(20200901)We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the AharonovBohm field in all dimensions and establish a sharp Hardytype ...

On the unique continuation of solutions to nonlocal nonlinear dispersive equations
(20200802)We prove unique continuation properties of solutions to a large class of nonlinear, nonlocal dispersive equations. The goal is to show that if (Formula presented.) are two suitable solutions of the equation defined in ...

Riemann's nondifferentiable function and the binormal curvature flow
(20200714)We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object ...

Vortex Filament Equation for a regular polygon in the hyperbolic plane
(20200709)The aim of this article is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and ...

On the energy of critical solutions of the binormal flow
(20200702)The binormal flow is a model for the dynamics of a vortex filament in a 3D inviscid incompressible fluid. The flow is also related with the classical continuous Heisenberg model in ferromagnetism, and the 1D cubic ...

Evolution of Polygonal Lines by the Binormal Flow
(20200601)The aim of this paper is threefold. First we display solutions of the cubic nonlinear Schrödinger equation on R in link with initial data a sum of Dirac masses. Secondly we show a Talbot effect for the same equation. Finally ...

Asymptotics in Fourier space of selfsimilar solutions to the modified Kortewegde Vries equation
(20200501)We give the asymptotics of the Fourier transform of selfsimilar solutions for the modified Kortewegde Vries equation. In the defocussing case, the selfsimilar profiles are solutions to the Painlevé II equation; although ...

Uniqueness properties of solutions to the BenjaminOno equation and related models
(20200315)We prove that if u1,u2 are real solutions of the BenjaminOno equation defined in (x,t)∈R×[0,T] which agree in an open set Ω⊂R×[0,T], then u1≡u2. We extend this uniqueness result to a general class of equations of BenjaminOno ...

Evolution of Polygonal Lines by the Binormal Flow
(20200205)The aim of this paper is threefold. First we display solutions of the cubic nonlinear Schr ̈odinger equation on R in link with initial data a sum of Dirac masses. Secondly we show a Talbot effect for the same equation. ...

A Hardytype inequality and some spectral characterizations for the Dirac–Coulomb operator
(20200101)We prove a sharp Hardytype inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrixvalued potentials V of Coulomb type: we characterise ...

Bilinear identities involving the kplane transform and Fourier extension operators
(20191130)We prove certain L2pRnq bilinear estimates for Fourier extension operators associ ated to spheres and hyperboloids under the action of the kplane transform. As the estimates are L2based, they follow from bilinear ...

Carleman type inequalities for fractional relativistic operators
(20190922)In this paper, we derive Carleman estimates for the fractional relativistic operator. Firstly, we consider changingsign solutions to the heat equation for such operators. We prove monotonicity inequalities and convexity ...

On the Evolution of the Vortex Filament Equation for regular Mpolygons with nonzero torsion
(20190903)In this paper, we consider the evolution of the Vortex Filament equa tion (VFE): Xt = Xs ∧ Xss, taking Msided regular polygons with nonzero torsion as initial data. Us ing algebraic techniques, backed by numerical ...

Some lower bounds for solutions of Schrodinger evolutions
(20190821)We present some lower bounds for regular solutions of Schr odinger equations with bounded and time dependent complex potentials. Assuming that the solution has some positive mass at time zero within a ball of certain radius, ...

On the energy of critical solutions of the binormal flow
(20190720)The binormal flow is a model for the dynamics of a vortex filament in a 3D inviscid incompressible fluid. The flow is also related with the classical continuous Heisen berg model in ferromagnetism, and the 1D cubic Schr ...

A Hardytype inequality and some spectral characterizations for the DiracCoulomb operator
(20190702)We prove a sharp Hardytype inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrixvalued potentials V of Coulomb type: we characterise ...

A Hardytype inequality and some spectral characterizations for the Dirac–Coulomb operator
(201906)We prove a sharp Hardytype inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrixvalued potentials $\mathbf V$ of Coulomb type: ...

Selfsimilar dynamics for the modified Kortewegde Vries equation
(20190409)We prove a local well posedness result for the modified Kortewegde Vries equa tion in a critical space designed so that is contains selfsimilar solutions. As a consequence, we can study the flow of this equation around ...

Uniqueness Properties of Solutions to the BenjaminOno equation and related models
(20190131)We prove that if u1, u2 are solutions of the Benjamin Ono equation defined in (x, t) ∈ R × [0, T ] which agree in an open set Ω ⊂ R × [0,T], then u1 ≡ u2. We extend this uniqueness result to a general class of equations ...

Bilinear identities involving the $k$plane transform and Fourier extension operators
(2019)We prove certain $L^2(\mathbb{R}^n)$ bilinear estimates for Fourier extension operators associated to spheres and hyperboloids under the action of the $k$plane transform. As the estimates are $L^2$based, they follow from ...

On the improvement of the Hardy inequality due to singular magnetic fields
(20180712)We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the AharonovBohm field in all dimensions and establish a sharp Hardytype ...

On the improvement of the Hardy inequality due to singular magnetic fields
(20180712)We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the AharonovBohm field in all dimensions and establish a sharp Hardytype ...

Asymptotics in Fourier space of selfsimilar solutions to the modified Kortewegde Vries equation
(20180706)We give the asymptotics of the Fourier transform of selfsimilar solutions to the modified Kortewegde Vries equation, through a fixed point argument in weighted $W^{1,\infty}$ around a carefully chosen, two term ansatz. ...

Asymptotics in Fourier space of selfsimilar solutions to the modified Kortewegde Vries equation
(20180706)We give the asymptotics of the Fourier transform of selfsimilar solutions to the modified Kortewegde Vries equation, through a fixed point argument in weighted W1,8 around a carefully chosen, two term ansatz. Such knowledge ...

On the regularity of solutions to the kgeneralized kortewegde vries equation
(201807)This work is concerned with special regularity properties of solutions to the kgeneralized Kortewegde Vries equation. In [Comm. Partial Differential Equations 40 (2015), 1336–1364] it was established that if the initial ...

On the Relationship between the OneCorner Problem and the $M$Corner Problem for the Vortex Filament Equation
(20180628)In this paper, we give evidence that the evolution of the vortex filament equation (VFE) for a regular Mcorner polygon as initial datum can be explained at infinitesimal times as the superposition of M onecorner initial ...

Spectral stability of Schrödinger operators with subordinated complex potentials
(20180628)We prove that the spectrum of Schroedinger operators in three dimensions is purely continuous and coincides with the nonnegative semiaxis for all potentials satisfying a formsubordinate smallness condition. By developing ...

Absence of eigenvalues of twodimensional magnetic Schroedinger operators
(20180101)By developing the method of multipliers, we establish sufficient conditions on the electric potential and magnetic field which guarantee that the corresponding twodimensional Schroedinger operator possesses no point ...

On the regularity of solutions to the kgeneralized kortewegde vries equation
(20180101)This work is concerned with special regularity properties of solutions to the kgeneralized Kortewegde Vries equation. In [Comm. Partial Differential Equations 40 (2015), 1336–1364] it was established that if the initial ...

Absence of eigenvalues of twodimensional magnetic Schr ̈odinger operators
(20171017)By developing the method of multipliers, we establish sufficient conditions on the electric potential and magnetic field which guarantee that the corresponding twodimensional Schr ̈odinger operator possesses no point ...

Uniqueness properties for discrete equations and Carleman estimates
(20170325)Using Carleman estimates, we give a lower bound for solutions to the discrete Schrödinger equation in both dynamic and stationary settings that allows us to prove uniqueness results, under some assumptions on the decay of ...

Singularity formation for the 1D cubic NLS and the Schrödinger map on $\mathbb{S}^2$
(20170202)In this note we consider the 1D cubic Schrödinger equation with data given as small perturbations of a Dirac$\delta$ function and some other related equations. We first recall that although the problem for this type of ...

A strategy for selfadjointness of Dirac operators: Applications to the MIT bag model and deltashell interactions
(20161221)We develop an approach to prove selfadjointness of Dirac operators with boundary or transmission conditions at a $C^2$compact surface without boundary. To do so we are lead to study the layer potential induced by the ...

Hardy uncertainty principle, convexity and parabolic evolutions
(20160901)We give a new proof of the $L^2$ version of Hardy’s uncertainty principle based on calculus and on its dynamical version for the heat equation. The reasonings rely on new logconvexity properties and the derivation of ...

An IsoperimetricType Inequality for Electrostatic Shell Interactions for Dirac Operators
(20160601)In this article we investigate spectral properties of the coupling $H + V_{\lambda}$, where $H =i\alpha \cdot \nabla + m\beta$ is the free Dirac operator in $\mathbb{R}^3$, $m>0$ and $V_{\lambda}$ is an electrostatic shell ...

The initial value problem for the binormal flow with rough data
(20151231)In this article we consider the initial value problem of the binormal flow with initial data given by curves that are regular except at one point where they have a corner. We prove that under suitable conditions on the ...

Shell interactions for Dirac operators: On the point spectrum and the confinement
(20151231)Spectral properties and the confinement phenomenon for the coupling $H + V$ are studied, where $H =i\alpha \cdot \nabla + m\beta$ is the free Dirac operator in $\mathbb{R}^3$ and $V$ is a measurevalued potential. The ...

Erratum to: Relativistic Hardy Inequalities in Magnetic Fields [J Stat Phys, 154, (2014), 866876, DOI 10.1007/s1095501409150]
(20151231)[No abstract available]

The Vortex Filament Equation as a Pseudorandom Generator
(20150801)In this paper, we consider the evolution of the socalled vortex filament equation (VFE), $$ X_t = X_s \wedge X_{ss},$$ taking a planar regular polygon of M sides as initial datum. We study VFE from a completely novel ...

The dynamics of vortex filaments with corners
(20150701)This paper focuses on surveying some recent results obtained by the author together with V. Banica on the evolution of a vortex filament with one corner according to the socalled binormal flow. The case of a regular polygon ...

Relativistic Hardy Inequalities in Magnetic Fields
(20141231)We deal with Dirac operators with external homogeneous magnetic fields. Hardytype inequalities related to these operators are investigated: for a suitable class of transversal magnetic fields, we prove a Hardy inequality ...

Vortex filament equation for a regular polygon
(20141231)In this paper, we study the evolution of the vortex filament equation,$$ X_t = X_s \wedge X_{ss},$$with $X(s, 0)$ being a regular planar polygon. Using algebraic techniques, supported by full numerical simulations, we give ...
 CEX2021001142S acredited as "Centro de Excelencia Severo Ochoa". MINECO, Spain. 01/01/2023  31/12/2026. Total amount: 4.000.000€
 SEV20170718 acredited as "Centro de Excelencia Severo Ochoa". MINECO, Spain. 01/07/2018  30/06/2022. Total amount: 4.000.000€
 IT124719 Análisis matemático y numérico de algunas ecuaciones en derivadas parciales y sus aplicaciones. Basque Government, Spain. 01/01/2019  31/12/2021. Total amount: 326.500€
 PGC2018094522BI00, Análisis matemático y numérico de algunas ecuaciones en derivadas parciales y sus aplicaciones. Ministry of Science and Innovation, Spain.01/01/2019  31/12/2021. Total amount: 125.780€
 669689HADEHarmonic Analysis and Differential Equations: new challenges. ERCEA European Research Council Executive Agency (H2020). 01/12/201530/11/2021. Total amount: 1.672.103€
 SEV20130323 accredited as "Centro de de Excelencia Severo OchoaSevero Ochoa Excellence Center". MINECO, Spain 01/07/201430/06/2018. Total amount: 4.000.000€
 IT64113 Mathematical Physics, mathematical Analysis and PDEs and Numerical Analysis. Basque Government, Spain. 01/01/201331/12/2018. Total amount: 302.000€
 MTM201453145P Analisis teórico y numérico de ecuaciones de evolución. MINECO, Spain 01/01/201531/122017. Total amount: 102.000€
 BERC.20142017 Centros de Investigación Básica y de Excelencia. Basque Government 01/01/201401/12/2017
Numerical study of the time evolution of the Talbot effect for infinite periodic diffraction grating
This video shows the propagation of the intensity of a plane wave after travelling through an infinite and periodic diffraction grating. The simulation is done using a finite elements method. It's aim is to show the time evolution and shaping of the Talbot effect.
News
 (ES) El Análisis de Fourier y las Ecuaciones Diferenciales, Nuevos Retos  20210428
 (EN) Fourier Analysis and Differential Equations, New Challenges  20210428
Dissemination activities
Date  Title  Place 

20230616  Desingularization of the BiotSavart integral and the Localised Induction Approximation (LIA)  Bilbao 
20230524  Blowup for the 1d cubic NLS and related systems  IHES, Conference in honor of FMerles. Bures sure Yvette, France 
20230313  New Conservations Laws and Energy Cascade for 1d Cubic NLS  University of Cambridge, Depart. of Mathematics 
20220317  Matemáticas y Turbulencia  Universidad Politécnica de Madrid 
20220808  Intermittency and the Talbot effect  Université Lyon 
20211101  Lower Bounds fro Oscilatory Integrals  Depart. of Mathematics, Brown University, RI, USA 
20211018  Fluctuations of ∂moments free Schrödinger equation  ICERM, Brown University, RI, USA 
20210917  Riemmann's nondifferentable function and the binormal curvature flow  ICERM, Brown University, RI, USA 
20210802  Riemmann's nondifferentable function and the binormal curvature flow  Virtual  Inst. Matem Pura e Appl (IMPA), Rio de Janeiro, Brasil 
20210505  Riemmann's nondifferentable function and the binormal curvature flow  Virtual  Ecole Normale Superieure de Lyon 
20210412  Riemmann's nondifferentable function and the binormal curvature flow  Virtual  Math. Science Res Int. (MSRI), Berkeley, USA 
20210108  Riemmann's nondifferentable function and the binormal curvature flow  Virtual  NYU Abu Dhabi 
20201208  Riemmann's nondifferentable function and the binormal curvature flow  Virtual  One World PDE Seminar, Portugal 
20201028  Riemmann's nondifferentable function and the binormal curvature flow  Virtual  Fields Institute, Toronto 
20201024  Riemmann's nondifferentable function and the binormal curvature flow  Toronto 
20201020  Riemmann's nondifferentable function and the binormal curvature flow  Virtual  Building Bridges  Academia Europea Barcelona Knowledge Hub 
20200910  The binormal flow, the Talbot effect, and noncircular jets  PDE Zoomseminar, Shanghai 
20190208  Lower bounds for Schrödinger evolutions  UCSB 
20170127  Shell interactions for dirac operators. an isoperimetrictype Inequality  Zaragoza 
20161214  Singular solutions of the Binormal Flow: transfer of energy and momentum  Valdivia, Chile 
20161020  The Talbot Effect and The Ecolution of Vortex Filaments  MSRI, USA 
20160823  Shell interactions for Dirac operators: An isoperimetrictype inequality  Sapporo, Japan 
20160509  Las matemáticas en la hora de la innovación  Periódico "El Correo" 
20150625  Some Remarks about the Uncertainty Principle  UAM, Spain 
20150617  The Vortex filament equation for a regular polygon and the Talbot effect  https://www.youtube.com/watch?v=Z1gXJKLw2Fo 
20141113  El tablero de Galton  Facultad de Ciencia y Tecnología de la UPV/EHU 
20140921  Shell interactions for Dirac operators: selfadjointeness, point spectrum, and confinement  Chicago, USA 
20121102  Las matemáticas están para usarlas  UPV/EHU, Spain 
20111019  Asymptotic Lower Bounds for Solutions to Dispersive Equations  Mexico DF 
20110901  A New Approach to Hardy's Uncertainty Principle  Lexington 
20110614  Unidad de Biomedicina Cuantitativa  BioCruces, Barakaldo 
20110610  A Geometric Description of the Formation of Singularities in the Binormal Flow  ETH Zurich 
20110201  ¿Matemáticas? Sí, gracias.  Periódico "El Correo" 
20100601  Oscilatory Integrals and Euler Equations  CSIC Madrid 
20060103  The initial value problem for nonlinear Shrödinger equations  ICM Madrid 
 Ministry of Science and Technology, Spain  PI of the Strategic Network of Mathematics  Scientific Committee 2020
 International Congress of Industrial and Applied Mathematics (ICIAM)  Officer  Scientific Committees  October 2019  December 2019
 European Academy of Sciences, Division of Mathematics  Officer  Scientific Committee  March 2019  March 2023
 Journal Serie A Matemáticas, Spanish Academy of Sciences  General Editor  2018
 L'institut Universitaire de France (IUF)  Member of the Comission  Scientific Committees  20162017
 Asociación ICIAM Valencia 2019  Vicepresicent  Scientific Committees  20142019
 Basque Center for Applied Mathematics (BCAM)  Scientific Director  Scientific Committee 20132019
 Spanish Candidacy for ICIAM Valencia 2019  Defensor of the bid  Scientific Committee  2013
 Journal Serie A Matemáticas, Spanish Academy of Sciences  Member of the Editorial Boards  2013
 Training and Research Unit of Mathematics and Applications (UPV/EHU)  Director  Scientific Committee  20122013
 Spanish Candidacy for ICIAM Valencia 2019  Defensor of the prebid  Scientific Committee  2012
 Mathematical European Weekend, Bilbao  Chair of the Program Committee  Scientific Committee  2011
 Linköping University, Sweden  Jury Member  Scientific Committee  2011
 Department of Quantitative Biomedicine, Biocruces Research Institute  Director  Scientific Committee  2011
 Academy of Finland  Panel member of the research projects  Scientific Committee  2010
 Spanish Royal Mathematical Society (RSME)  Vicepresident  Scientific Committee  20092012
 ICIAM  Member of the Executive Board  Scientific Committee  20082013
 University of the Basque Country/Universidad del País Vasco/Euskal Herriko Unibertsitatea (UPV/EHU)  Director of the PhD Programme in Mathematics  Scientific Committee  2007  2013
 Interuniversity master program in Applied Mathematics (UPV/EHU and another four universities)  Director  Scientific Activities  20062012
 Habilitation and PhD theses at several foreign universities  Jury Member  Scientific Committee  2004
 Medalla Real Sociedad Matemática Española (RSME) 2023  Spanish Royal Mathematical Society (RSME) Medal 2023, Spain, 2023
 Severo Ochoa Excellence Accreditation, BCAM, 20232026
 National Research Award "Julio Rey Pastor" 2021, in Mathematics and Information and Communication Technologies
 Severo Ochoa Excellence Accreditation, BCAM, 20182022
 Severo Ochoa Excellence Accreditation, BCAM, 20142018
 ERCAdvance Grant, European Research Council, Brussels, Belgium, 20152021
 Elected member of the Real Academia de Ciencias, Spain, 2018
 Member of the European Academy of Sciences; European Academy of Sciences, Liege, Belgium, 2014
 Blaise Pascal Medal, European Academy of Sciences, Liege, Belgium, 2014
 Le Prix La Recherche, Mathematiques, France, 2014
 Member of the "Real Academia de Ciencias Exactas, Físicas y NaturalesRAC", Spain, 2013
 Fellow of the American Mathematical Society, Inaugural Class, 2012
 Premio Euskadi de Investigación  Euskadi Research Prize, Basque Government, Spain, 2012
 Highly Cited Researcher, ISI Web os Science, 2004
 Iberdrola grant for Visiting Fellows, Spain, 1997
 National Science Foundation grant, USA, 198990