Research
Current research interests
- Symplectic Geometry
- Poisson Geometry
- Integrable systems and group actions
- Foliation theory
- Geometric quantization
- Groupoids and Algebroids
- Hamiltonian Dynamics
- Fluid Dynamics and Contact Geometry
- 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.
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 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.
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. We are also extending 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. More soon!
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:2012.12828. 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). 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.