Research
Current research interests
- Symplectic Geometry
- Poisson Geometry
- Symplectic and contact topology
- Integrable systems and group actions
- Foliation theory
- Geometric quantization
- Groupoids and Algebroids
- Hamiltonian Dynamics
- Fluid Dynamics and Contact Geometry
- Euler and Navier-Stokes equations
- Limits of computation
- Complexity theory
- undecidability
- Mathematical Physics in the large
Open calls to join my group:
Our BBVA project COMPLEXFLUIDS:
Summary
I am particularly interested in understanding connections between different areas such as Geometry, Dynamical Systems, Mathematical Physics and, more recently, Fluid Dynamics and Computer Science. I often get a better understanding of some phenomena by using techniques at the intersection of disciplines.
Singularities
My research deals with geometrical and dynamical aspects of singularities. In particular, I am interested in Hamiltonian systems, their singularities and the so-called realm of Hamiltonian Dynamics. I am fascinated by periodic orbits and different approaches to their study (including Floer homology and KAM theory). I study normal forms and equivariant geometric problems arising in Symplectic, Contact, and Poisson manifolds. I am also interested in rigidity problems for group actions on these manifolds. I also work in geometric quantization of real polarizations. Collaboration with my colleagues has recently taken me to explore fascinating new lands, including that of connection between Geometry, Dynamical systems, Fluid dynamics and computer science.
Some years ago, I started to consider geometrical problems on b-manifolds (inspired by Melrose b-calculus). Their symplectic reincarnations are called b-symplectic manifolds which model several problems in Celestial Mechanics. This is a fascinating new subject that I am working on which lies between the Symplectic and Poisson worlds. I am lately trying to understand possible generalizations of b-manifolds such as almost regular foliations and E-symplectic manifolds and finding (unexpected!) applications of b-theory to problems in celestial mechanics. I also like localization theorems, equivariant cohomology and I am a recent fan of Floer homology and the study of periodic orbits which I am trying to understand in connection to problems in Celestial Mechanics such as the three-body problem I am interested in building bridges between different areas of mathematics and lately focusing on intersections between Dynamical Systems, Fluid Dynamics, contact geometry and computer science. The ICREA Academia prize permitted me to focus on research and attain influential results in these areas.
Periodic orbits
The Weinstein conjecture on periodic orbits asserts that the Reeb vector field of a compact contact manifold always have periodic orbits. With my student Cédric Oms we have understood the Weinstein Conjecture if we allow singularities in the contact form. In particular under compactness assumptions on the critical set we have been able to prove the existence of infinite periodic orbits on the critical set for 3-dimensional b^m-contact structure.
This has led us to formulate the singular Weinstein conjecture about existence of singular periodic orbits on b^m-contact manifolds. Those singularities on contact structures model some problems of Beltrami flows on manifolds with boundary. This variant of the Weinstein conjecture is very revealing: The singular orbits are indeed periodic orbits which are no longer smooth but have points as marked singularities. This opens a door to a new world.
In the direction of the singular Weinstein conjecture we are now trying to prove that the set of b^m-contact structures admitting singular Reeb orbits is generic in the set of b^m-contact forms. So far we could prove the existence of escape orbits and generalized singular periodic orbits in a number of cases where genericity occurs in the class of Melrose contact forms.
I also extended the Floer apparatus to the b-world.
See more in this talk I gave at a workshop in Zurich in January 2021.
Update: We have recently disproved the singular Weinstein conjecture. Check it out here: here.
Fluid Dynamics: Universality of Euler flows, h-principle for contact geometry and Turing completeness in dimension 3
I have been recently interested in Fluid Dynamics where I entered driven by singularity theory. With Daniel Peralta Salas and Robert Cardona, we had been working on b-contact forms appearing in Fluid Dynamics using the correspondence between contact forms and Beltrami vector fields (see our paper on Phylosophical Transactions of the Royal Society below). In Febrary 17, 2019 I came across this entry (thank you Twitter!) in the blog of Terry Tao. This was a source of inspiration to work on h-principles for Reeb embeddings and proving universality properties of Euler flows and Turing completeness of Euler flows. This is the content of our paper arXiv:1911.01963. Just for the New Year's Eve of 2020 we finished our article on constructing Turing complete Euler flows in dimension 3, closing up an open problem since the 90's by Moore and also Tao recently.
You can read it here: arXiv:here. It has been published at PNAS.
This result captured the attention of mass media. You may want to consult, for instance, the article at El Pais or at Pour la Science (complete list of articles in the Outreach section of the webpage). See also this article in Quanta Magazine: Quanta Magazine Later we obtained a generalization for t-dependent Euler flows (now published at IMRN) by compactifying a former result by Graça et al of Turing complete polynomials and embedding them as Euler flows, see below.
In all the constructions above, the metric is seen as an additional "variable" and thus the method of proof does not work if the metric is prescribed.
You can learn more about this in this video of my invited talk at the 8th European congress of mathematics.
Is it still possible to construct a Turing complete Euler flow on a 3-dimensional space with the standard metric?
The answer is YES. You can check our article with Robert Cardona and Daniel Peralta-Salas published at JMPA.
We have recently got a good understanding of the entropy of a Turing complete system. Check our recent article here.
Navier-Stokes, undecidability and Turing completeness:
In this work. , I have shown that the Navier–Stokes equations can simulate universal Turing machines. This means that fluid flows can encode undecidable problems, revealing a profound link between fluid dynamics, computation, and the limits of predictability.
New computational model? The hybrid computer:
One of my recent endeavours is to extend the construction of the Fluid computer (Turing complete Euler flows) to a more generalized abstract design.
We have defined a new computational model. In our article "Topological Kleene Field Theories: A New Model of Computation” we introduce Topological Kleene Field Theories (TKFTs), a new computational framework that directly realizes any computable function through the flow of a vector field on a bordism. This provides a rigorous and original alternative to Turing machines: instead of encoding computation as an iterative process, TKFTs achieve it in a single dynamical pass. Our results establish TKFTs as a full model of computation equivalent to Turing machines, relate computational complexity to topological complexity, and open perspectives towards models that may surpass both classical and quantum computation.
You can read the article : here
You can check these ideas in a talk of mine at International Congress of Basic Sciences 2025 here:
Poisson topology: Poisson manifolds as towers of E-symplectic manifolds:
One of my recent project concerns understanding Poisson manifold and different approximations from the symplectic perspectives. I am currently working jointly with Ryszard Nest on understanding the possible desingularization of Poisson manifolds as families of E-symplectic manifolds. This will allow to extend to Poisson manifolds the techniques from Symplectic manifolds.
The shape of the undecidable:
For dynamical systems, the horseshoe stands as a metaphor for chaos. A dynamical system that is Turing complete exhibits undecidable trajectories—that is, no algorithm can determine whether certain trajectories will enter a given open set within finite time. The natural question then arises: can we detect such behaviour? What is the shape of the undecidable? What plays the role of Smale’s horseshoe map in this context?
This is a joint project with Ángel González-Prieto and Daniel Peralta-Salas.
Undecidability in Billiards and Physics:
Together with Isaac Ramos we have proved that the 2D billiards are Turing complete. This yields interesting consequences about existence of Undecidable trajectories for physical systems which mimic billiards.
This includes collisions in Celestial mechanics.
You can check the article in the list of publications.
Below, a billiard table associated to a Turing machine.
Undecidability in Celestial mechanics and the three body problem:
With Ángel González-Prieto and Daniel Peralta-Salas we are striving to prove that there exist Turing complete dynamics in the three-body problem. This would indeed yield existence of undecidable phenomena in the three-body problem.
Floer theory, spectral theory and quantum mechanics:
With Alberto Ibort we are working in Floer theory using Witten's approach to Morse theory. The interesting analytic phenomena are guided by spectral theory and quantum mechanics. More soon!
