Research
Research
I am very interested in the interactions between geometry, topology, and dynamical systems. My research focuses on topological and geometrical methods in Hamiltonian and fluid dynamics, and on contact/symplectic topology. Other topics I enjoy are h-principles, foliations, mathematical physics, computability in dynamical systems, and singular symplectic geometry.
Local construction of knotted screw dislocations in smectic liquid crystals
R. Cardona, A. Vega
To appear in the Proceedings of the Royal Society A (2025) [arXiv]
Fillability obstructions for high-dimensional confoliations
R. Cardona, F. Gironella
Preprint arXiv:2410.06918 (2024) [arXiv]
Non-density results in high dimensional stable Hamiltonian topology
R. Cardona, F. Gironella
Journal of the London Mathematical Society 111.4 (2025): e70143 [arXiv]
Towards a fluid computer,
R. Cardona, E. Miranda, D. Peralta-Salas
Foundations of Computational Mathematics, online first (2025) [arXiv]
Survey, on the occasion of the Foundations of Computational Mathematics 2023 conference.
Topological entropy of Turing complete dynamics
R. Bruera, R. Cardona, E. Miranda, D. Peralta-Salas (appendix by V. Salo)
Preprint arXiv:2404.07288 (2024) [arXiv]
Asymmetry of MHD equilibria for generic adapted metrics
R. Cardona, N. Duignan, D. Perrella
Archive for Rational Mechanics and Analysis 249.2 (2025) [arXiv]
Contact type solutions and non-mixing of the 3D Euler equations
R. Cardona, F. Torres de Lizaur
Preprint arXiv:2312.03514 (2023) [arXiv]
Stability is not open or generic in symplectic four-manifolds
R. Cardona
Preprint arXiv:2305.13158 (2023) [arXiv]
Hydrodynamic and symbolic models of computation with advice
R. Cardona
Revista Matemática Iberoamericana, 41.1 (2025), 313–338 [arXiv]
An h-principle for embeddings transverse to a contact structure
R. Cardona, F. Presas
Journal of Topology, 17: e12326 (2024). [arXiv]
Existence and classification of b-contact structures
R. Cardona, C. Oms
To appear in Publicacions Matemàtiques (2025) [arXiv]
Periodic orbits and Birkhoff sections of stable Hamiltonian structures
R. Cardona, A. Rechtman
Journal de l'École polytechnique - Mathématiques 12 (2025), 235-286 [arXiv]
You can take a look at some more pictures I did of "helix boxes".
Morse functions and contact convex surfaces
R. Cardona, C. Oms
Journal of Geometry and Physics, 191 (2023), 104886 [arXiv]
Computability and Beltrami fields in Euclidean space
R. Cardona, E. Miranda, D. Peralta-Salas
Journal de Mathématiques Pures et Appliquées, 169 (2023), 50-81 [arXiv]
See also a review by Robert Ghrist at the AMR about this and previous work.
Turing universality of the incompressible Euler equations and a conjecture of Moore
R. Cardona, E. Miranda, D. Peralta-Salas
International Mathematics Research Notices, 22 (2022), 18092–18109 [arXiv]
The periodic orbit conjecture for steady Euler flows
R. Cardona
Qualitative Theory of Dynamical Systems, 20.52 (2021) [arXiv]
Constructing Turing complete Euler flows in dimension 3
R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas
Proceedings of the National Academy of Sciences,118.19 (2021) [arXiv]
See our outreach article for the Spanish and German (translation) editions of Scientific American.
The work is mentioned in this Quanta magazine article on undecidability in physics.
Integrable systems on singular symplectic manifolds: From local to global
R. Cardona, E. Miranda
International Mathematics Research Notices, 24 (2022), 9565–19616 [arXiv]
The topology of Bott integrable fluids
R. Cardona
Discrete & Continuous Dynamical Systems, 42.9 (2022), 4321-4345 [arXiv]
Steady Euler flows and Beltrami fields in high dimensions
R. Cardona
Ergodic Theory and Dynamical Systems, 41.12 (2021), 3610-3633 [arXiv]
Universality of Euler flows and flexibility of Reeb embeddings
R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas
Advances in Mathematics, 428 (2023), 109142 [arXiv]
Euler flows and singular geometric structures
R. Cardona, E. Miranda and D. Peralta-Salas
Philosophical Transactions of the Royal Society A, (2019), 37720190034 [arXiv]
On the volume elements of a manifold with transverse zeroes
R. Cardona, E. Miranda
Regular and Chaotic Dynamics, 24.2 (2019), 187-197 [arXiv]
Integrable systems and closed one forms
R. Cardona, E. Miranda
Journal of Geometry and Physics, 131 (2018), 204-209 [arXiv].
Principal Investigator of "SymHyCon", funded by the Spanish Ministry of Science. Title: Interactions between symplectic topology, hydrodynamics, and conservative flows, 09/2024 - 08/2027.
Member of "GiT-UB", funded by the Generalitat de Catalunya. Title: Geometria i Topologia. Years 2021-2025.
Member of "COMPLEX FLUIDS", research project funded by Fundación BBVA. Title: Computational, dynamical and geometrical complexity in fluid dynamics. Years 2022-2025.
Premi IEC de la Secció de Ciències i Tecnologia de Matemàtiques, April 2025, from the Institute of Catalan Studies. Catalan award for the best PhD theses in Mathematics.
Vicent Caselles award, June 2023, from the Real Sociedad Matemática Española. Spanish award for the best PhD theses in Mathematics.
Extraordinary PhD Prize in Basic Science, year 2021, from the Universitat Politecnica de Catalunya.
Évariste Galois prize 2019, from the Societat Catalana de Matemàtiques.
Looking at Euler flows through a contact mirror: universality and undecidability, with E. Miranda, D. Peralta-Salas. Proceedings of the 8th European Congress of Mathematics (Portoroz, Slovenia, 2021) [arXiv]
Survey, with some new results about undecidable properties of Reeb flows.
Ph.D. Thesis, May 2021, under the supervision of Prof. Eva Miranda (UPC). Click here to download an updated version of my thesis.
Reeb embeddings and Universality of Euler flows, with E. Miranda, D. Peralta-Salas and F. Presas. Birkhauser extended abstracts GEOMVAP 2019. Research announcement.
Talk at the Geometry and Physics Spanish Network meeting. 15-17 September 2025, Barcelona. Title: Exploring confoliations beyond dimension three.
Talk at the UB-UPC Dynamical Systems Seminar. 1 October 2025, Barcelona.
Lectures at the specialized course Hyperbolic Geometry with application to dynamics, joint with Gustavo Rodrigues Ferreira. At the Faculty of Mathematics and Computer Science of the University of Barcelona.
BCN-Face(t)s in Symplectic geometry, Barcelona, 28-31 January 2025
Plenary speaker at VII Congreso de Jóvenes Investigadores de la RSME, Bilbao, 13-17 January 2025.
Geometry seminar at Université de Nantes, 12 December 2024.
Symplectic geometry seminar at ETH Zurich, 4 November 2024.
Fluid Dynamics, Geometry, and Computer Science in Interaction, Barcelona, 16-20 September 2024.
International Fall Workshop in Geometry and Physics. Coimbra, 2-5 September 2024.
Talk at the Summer School on h-principle, Madrid,18 June 2024. Title: Flexibility in Symplectic dynamics.
Talk at the Dynamical Systems Oberseminar of Ruhr-Universität Bochum, 21 May 2024. Title: Time-dependent contact hydrodynamics.
Talk at the Geometry seminar of the Universitat de Barcelona, 26 January 2024. Title: Stable hypersurfaces in symplectic four-manifolds.
Talk at the Geometry and Dynamics seminar, Tel-Aviv University (Online), 17 January 2024. Title: Contact topology and time-dependent hydrodynamics: non-mixing and spectral invariants.
Talk at BACH seminar on symplectic and contact geometry, Cologne, 20 October 2023. Title: Stability is not open or generic in symplectic four-manifolds.
Talk at "From smooth to C^0 symplectic geometry: topological aspects and dynamical implications" in CIRM, Luminy, 2-7 July 2023. Title: Dynamics of stable Hamiltonian structures.
Talk at the UPM seminar, Madrid, 27 April 2023. Title: Dynamics of conservative flows in three dimensions: Reeb fields and beyond.
Talk at the UB-UPC dynamical systems seminar, Barcelona, 29 March 2023. Title: Periodic orbits and surfaces of section on Hamiltonian systems: from contact to stable energy level sets.
Organizer of "Symplectic geometry and Hamiltonian dynamics" parallel session, talk at "Differential geometry, mathematical physics and control theory" session. Congreso de Jóvenes de la RSME 2023, León, 6-10 February 2023.
Talk at Geometry and Topology seminar, Institut de Mathématiques Jussieu, Paris, 17 January 2023. Title: Undecidability in conservative dynamics.
Talk at Symplectic Zoominar, Montréal-Princeton/IAS-Tel Aviv-Paris, 9 December 2022. Title: Periodic orbits and Birkhoff sections of stable Hamiltonian structures. Video
Talk at Universidad de Sevilla, 24 November 2022. Title: Periodic orbits and Birkhoff sections of stable Hamiltonian structures.
Talk at ICMAT Geometry seminar, Madrid, 16 November 2022. Title: Periodic orbits and Birkhoff sections of stable Hamiltonian structures.
Talk at Poisson research intensive programme's seminar, Centre de Recerca Matemàtica, 8 June 2022. Title: Periodic orbits and Birkhoff sections of stable Hamiltonian Reeb fields.
Talk at Geometry, Mechanics and Control seminar, ICMAT, 20 May 2022 (Online). Title: Undecidability and universal computation in Euler and Reeb flows.
Talk at Université Lyon 1 séminaire Géométries, 6 May 2022. Title: Indécidabilité et calcul universel dans des flots d'Euler et de Reeb.
Talk at SimBa seminar, Universitat de Barcelona - Universitat Politècnica de Catalunya, 20 April 2022. Title: Undecidability in dynamical systems: from Turing machines to hydrodynamics. Video
Talk at IRMA séminaire interne, Université de Strasboug, 7 April 2022. Title: La philosophie du principe d'homotopie.
Talk at IRMA Geometry seminar, Université de Strasbourg, 17 January 2022. Title: Universal computation in Euler and Reeb flows.
Talk at Einstein Chair Mathematics Seminar at City University of New York (Online), 13 October 2021. Title: Universal Computation and the Euler equations. Video
Talk at Geometry seminar ICMAT (Madrid), Title: Universal computation and the Euler equations. 22 September 2021.
Talk at Symposium 2021: New developments in Momentum Polytope Theory ICMAT (Madrid, online). Title: Integrable systems on singular symplectic manifolds. 27th July 2021.
Talk at Geometric Structures Laboratory Seminar . Fields institute, University of Toronto (online). Title: Universal computation in 3D steady Euler flows. 17th May 2021. Video
Talk at Barcelona-Wurzburg Mathematical Physics Factory Workshop, Title: Action-angle coordinates on singular symplectic manifolds, 23-24 February 2021.
Talk at Hamiltonian Systems Seminar , University of Arizona - University of Toronto joint seminar, Title: The topology of Bott integrable fluids, 2 February 2021.
Talk at 15th International Young Researchers Workshop on Geometry, Mechanics, and Control , Title: The topology of Bott integrable fluids, 4 December 2020. Video
Talk at 3rd BYMAT conference 2020 , Title: New topological constructions of steady Euler flows, 1-3 December 2020.
Talk at Junior Global Poisson Workshop 2020 , Title: The topology of Bott integrable fluids, 16 September 2020.
Talk at Jovenes de la RSME in Castellón , Title: Flexibilidad en topología de contacto y universalidad de fluidos ideales, 30 January 2020.
Talk at SIMBA seminar at Universitat de Barcelona , Title: The philosophy of the h-principle, 4 December 2019. Video (in catalan)
Geometry Seminar at ICMAT in Madrid , Title: Euler flows universality and flexibility of Reeb embeddings, 2 October 2019.