Marcel Guardia

About me

Contact

Postal address:

Departament de Matemàtiques
Universitat Politècnica de Catalunya
Escola Tècnica Superior d'Enginyeria Industrial de Barcelona
Av. Diagonal 647, 08028 Barcelona.

Office: 3.19

Phone: (+34) 93 401 74 27

marcel dot guardia at upc dot edu.

Since 2017, I am Associate Professor (Professor Agregat) at UPC, where I am a member of the Dynamical Systems group at UPC and the Lab of Geometry and Dynamical Systems. I am also member of the BGSMath. In 2018 I have obtained a Distinció ICREA Acadèmia.

Information about my ERC Starting grant can be found here.

Before being at UPC, I held post-doctoral positions at Penn State University, Fields Insitute (University of Toronto), University of Maryland at College Park, Institute for Advanced Study (Princeton) and Université de Paris 7 - Diderot.

Research Interests

Hamiltonian Systems, Celestial Mechanics, Hamiltonian Partial Differential Equations, Arnol'd diffusion, Exponentially small phenomena. Here is my CV.

Research

Preprints

Marcel Guardia, Zaher Hani, Emanuele Haus, Alberto Maspero, Michela Procesi
Strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation
45 pages. Available at arXiv.
Otavio M. L. Gomide, Marcel Guardia, Tere M. Seara
Critical velocity in kink-defect interaction models: rigorous results
47 pages. Available at arXiv.
Oriol Castejon, Marcel Guardia, Vadim Kaloshin
Random Iteration of Cylinder Maps and diffusive behavior away from resonances
57 pages. [PDF].
Marcel Guardia, Vadim Kaloshin
Orbits of nearly integrable systems accumulating to KAM tori
115 pages. arXiv.

Publications

Marcel Guardia, Zaher Hani, Emanuele Haus, Alberto Maspero, Michela Procesi
A note on growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation
17 pages. Accepted for publication in Rendiconti Lincei - Matematica e Applicazioni. See the preprint version here.
Marcel Guardia, Vadim Kaloshin, Jianlu Zhang
Asymptotic density of collision orbits in the Restricted Circular Planar 3 Body Problem
Archive for Rational Mechanics and Analysis, 233(2): 799-836 (2019). See the preprint version on arXiv.
Marcel Guardia, Pau Martin, Lara Sabbagh, Tere M. Seara
Oscillatory orbits in the restricted elliptic planar three body problem
Discrete and Continuous Dynamical Systems A, 37(1): 229-256 (2017). See the preprint version on arXiv.
Emanuele Haus, Marcel Guardia, Michela Procesi
Growth of Sobolev norms for the analytic NLS on T2
Advances in Mathematics, 301(1): 615-692 (2016). See the preprint version on the arXiv.
Marcel Guardia, Vadim Kaloshin, Jianlu Zhang
A second order expansion of the separatrix map for trigonometric perturbations of a priori unstable systems
Communications in Mathematical Physics, 348(1): 321-361 (2016). See the preprint version on the arXiv.
Jacques Féjoz, Marcel Guardia, Vadim Kaloshin, Pablo Roldan
Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem
Journal of the European Mathematical Society, 18(10): 2315-2403 (2016). See the preprint version on the arXiv.
Marcel Guardia, Jacques Féjoz
Secular instability in the spatial three-body problem
Archive for Rational Mechanics and Analysis, 221(1): 335-362 (2016). See the preprint version on arXiv.
Marcel Guardia, Pau Martin, Tere M. Seara
Oscillatory motions for the restricted planar circular three body problem
Inventiones Mathematicae, 203(2): 417-492 (2016). See the preprint version on the arXiv.
Marcel Guardia, Vadim Kaloshin
Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation
Journal of the European Mathematical Society, 17 (1): 71–149 (2015). [PDF].
Marcel Guardia
Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation with a convolution potential
Communications in Mathematical Physics, 329 (1): 405–434 (2014). [PDF].
Marcel Guardia
Splitting of separatrices in the resonances of nearly integrable Hamiltonian Systems of one and a half degrees of freedom
Discrete and Continuous Dynamical Systems A, 33 (7): 2829-2859 (2013). [PDF].
Inma Baldomá, Marcel Guardia, Ernest Fontich, Tere M. Seara
Exponentially small splitting of separatrices beyond Melnikov analysis: rigorous results
Journal of Differential Equations, 253 (12): 3304-3439 (2012). [PDF].
Marcel Guardia, Tere M. Seara
Exponentially and non-exponentially small splitting of separatrices for the pendulum with a fast meromorphic perturbation
Nonlinearity, 24 (5): 1367-1412 (2012). [PDF].
Marcel Guardia, Tere M. Seara, Marco Antonio Teixeria
Generic bifurcations of low codimension of planar Filippov Systems
Journal of Differential Equations, 250 (4): 1967-2023 (2011) . [PDF].
Marcel Guardia, S. John Hogan, Tere M. Seara
An analytical approach to codimension 2 sliding bifurcations in the dry friction oscillator
SIAM Journal on Applied Dynamical Systems, 9: 769–798 (2010). [PDF].
Marcel Guardia, Carme Olivé, Tere M. Seara
Exponentially small splitting for the pendulum: a classical problem revisited
Journal of Nonlinear Science, 20: 595-685, (2010) . [PDF].

Advanced courses

Oscillatory motions in the restricted three body problem. Winter School in Conservative Dynamics, Engelberg (Switzerland), February 2017. Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5.

Instabilities in the three body problem through geometric methods. CIMPA Research School - Hamiltonian and Lagrangian Dynamics, Salto (Uruguay), March 2015. You can find the slides here.

The quasi-ergodic hypothesis and Arnold diffusion in nearly integrable Hamiltonian systems. Semester on Dynamics Beyond Uniform Hyperbolicity, Rio de Janeiro (Brazil), September 2013. This course was recorded. Watch the lectures on youtube: Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5.

Growth of Sobolev norms for the cubic defocusing NLS. Ecole d'hiver Dynamics and PDE's , Saint-Etienne de Tineé, February 2013. You can find the slides here.

Given Talks

Here you will find a selection of slides of presentations in seminars and congresses. Please note that these have not been revised and they may have misprints.

Growth of Sobolev norms for the analytic non-linear Schrödinger equation , Hamiltonian Dynamics, PDEs and Waves on the Amalfi coast, Maiori. September 6, 2016.
Oscillatory motions for the restricted planar three body problem , Workshop in Dynamical Systems, Oberwolfach. July 23, 2015.
Nearly integrable systems with orbits accumulating to KAM tori , Courant Institute Dynamical Systems seminar. October 9, 2014.
Growth of Sobolev norms for the cubic defocusing NLS , Orsay PDE seminar. May 15, 2014.
Oscillatory motions for the restricted planar circular three body problem , Conference on New Perspectives on the N body problem, Banff (Canada). January 14, 2013. This talk was recorded. You can watch it here.
Growth of Sobolev norms for the cubic defocusing nonlinear Schrodinger equation in polynomial time , Princeton University Analysis Seminar. April 9, 2012.
An instability mechanism along the mean motion resonances in the restricted three body problem , Nanjing Conference on Hamiltonian Dynamics . Nanjing, China. August 21-28, 2011
Exponentially small splitting of separatrices for one and a half degrees of freedom Hamiltonian Systems close to a resonance , McMaster University PDE/Analysis Seminar. April 15, 2011.
Analytic properties of one and a half degrees of freedom Hamiltonian Systems and exponentially small splitting of separatrices , University of Maryland Dynamical Systems Seminar. November 18, 2010.
Exponentially small splitting of separatrices for the pendulum with fast periodic or quasiperiodic meromorphic perturbation , Equadiff 11. Loughborough, United Kingdom. August 1-5, 2011.
Exponentially small splitting of separatrices of the pendulum: two different examples , Universitat de Barcelona - Universitat Politècnica de Catalunya Dynamical Systems Seminar. July 8, 2009.
Global phenomena in a neighborhood of codimension two local singularities of planar Filipov systems , International Workshop on Resonance Oscillations and Stability of Nonsmooth Systems. London, United Kingdom. June 16-25, 2009.

Teaching

This semester I am teaching

Hamiltonian systems at the Master's degree in Advanced Mathematics and Mathematical Engineering. You can find the exercises list here and the list of proposed projects here.

The spring semester of 2017 I taught Hamiltonian systems at the Master's degree in Advanced Mathematics and Mathematical Engineering. You can find the exercises list here and the list of proposed projects here.

TFG/TFM

If you are interested in doing an undergraduate or Master thesis in dynamical systems and/or PDEs look at the topics I have proposed in the Intranet or send me an email. I have supervised the following undergraduate/Master thesis:

Master thesis supervision: Mar Giralt, The Kepler problem with varying masses, co-advised with Tere Seara.
Master thesis supervision: Guillem Sanchis, Interaction between kinks and deffects in nonlinear wave equations, co-advised with Tere Seara.
Master thesis supervision: Francesc Granell, Instabilities in the Arnold model, co-advised with Tere Seara.
Undergraduate thesis supervision: Francesc Granell, Asymptotic Size of Herman Rings Using Quasiconformal Surgery, co-advised with Pau Martín, 2015-16, Matrícula d'honor 10/10.

Last update February 14, 2017.