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The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao [38], [39] launched a programme to address the global existence problem for the Euler and Navier Stokes equations based on the concept of universality. Inspired by this proposal, in this article we prove that the stationary Euler equations exhibit several universality features. More precisely, we show that any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension. The solutions we construct are of Beltrami type, and being stationary they exist for all time. Using this result, we establish the Turing completeness of the steady Euler flows, i.e., there exist solutions that encode a universal Turing machine and, in particular, these solutions have undecidable trajectories. Our proofs deepen the correspondence between contact topology and hydrodynamics, which is key to establish the universality of the Reeb flows and their Beltrami counterparts. An essential ingredient in the proofs, of interest in itself, is a novel flexibility theorem for embeddings in Reeb dynamics in terms of an h-principle in contact geometry, which unveils the flexible behavior of the steady Euler flows. These results can be viewed as lending support to the intuition that solutions to the Euler equations can be extremely complicated in nature., Robert Cardona acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R& D (MDM-2014- 0445) via an FPI grant. Robert Cardona and Eva Miranda are supported by the grant PID2019-103849GBI00 of MCIN/AEI/10.13039/501100011033 and AGAUR grant 2021 SGR 00603. Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via the ICREA Academia Prize 2016 and ICREA Academia Prize 2021 and by a Friedrich Wilhelm Bessel Research Award of the Alexander von Humboldt Foundation. She is also partially supported by the Spanish State Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (project CEX2020-001084-M). Daniel Peralta-Salas is supported by the grant PID2019-106715GB GB-C21 funded by MCIN/AEI/10.13039/501100011033. Robert Cardona, Eva Miranda and Daniel Peralta-Salas acknowledge partial support from the grant “Computational, dynamical and geometrical complexity in fluid dynamics”, Ayudas Fundación BBVA a Proyectos de Investigación Científica 2021. Francisco Presas is supported by the grant PID2019-108936GB-C21. Daniel Peralta-Salas and Francisco Presas also acknowledge partial support from the ICMAT-Severo Ochoa grant CEX2019-000904-S., Peer Reviewed, Postprint (published version)

A Lefschetz hyperplane theorem for the determinantal loci of a morphism, between two holomorphic vector bundles E and F over a complex manifold is proved, under the condition that E* x F is Griffiths k-positive. This result is applied to find some homotopy groups of the Brill-Noether loci for a generic curve., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

The loop Space associated to a Riemannian manifold admits a quasi-symplectic structure (that is, a closed 2-form which is non-degenerate up to a finite-dimensional kernel). We show how to construct a compatible almost-complex structure. Finally conditions to have contact structures on loop spaces are studied., MEC, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We define the concept of symplectic foliation on a symplectic manifold and provide a method of constructing many examples, by using asymptotically holomorphic techniques., Research Training Network, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We prove that the vanishing spheres of the Lefschetz pen- cils constructed by Donaldson on symplectic manifolds of any dimension are conjugated under the action of the symplec- tomorphism group of the fiber. More precisely, a number of generalized Dehn twists may be used to conjugate those spheres. This implies the non-existence of homologically triv-ial vanishing spheres in these pencils. To develop the proof,we discuss some basic topological properties of the space of asymptotically holomorphic transverse sections., CICYT, Research Training Network, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

In this paper we use Donaldson's approximately holomorphic techniques to build embeddings of a closed symplectic manifold with symplectic form of integer class in the Grassmannians Gr(r, N). We assure that these embeddings are asymptotically holomorphic in a precise sense. We study first the particular case of CPN obtaining control on N and we improve in a sense a classical result about symplectic embeddings. The main reason of our Study is the construction of singular determinantal submanifolds as the intersection of the embedding with certain "generalized Schubert cycles" defined on a product of Grassmannians. It is shown that the symplectic type of these submanifolds is quite more general that the ones obtained by Donaldson and Auroux,as zeroes of "very ample" vector bundles., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021, We use a new geometrical approach to the universality of Euler flows. By proving flexibility results on embeddings for Reeb flows in contact topology, we deduce some new universal properties for Euler flows. As a byproduct, we deduce the Turing completeness of stationary Euler flows, answering an open question for steady solutions. The results contained in this article are an announcement and short version of [2], where the complete list of results and proofs can be found., This work was partially supported by the ICMAT–Severo Ochoa grant SEV-2015-0554, Peer Reviewed, Postprint (author's final draft)

Published under the PNAS license, Can every physical system simulate any Turing machine? This is a classical problem which is intimately connected with the undecidability of certain physical phenomena. Concerning fluid flows, Moore asked in [15] if hydrodynamics is capable of performing computations. More recently, Tao launched a programme based on the Turing completeness of the Euler equations to address the blow up problem in the Navier-Stokes equations. In this direction, the undecidability of some physical systems has been studied in recent years, from the quantum gap problem [7] to quantum field theories [11]. To the best of our knowledge, the existence of undecidable particle paths of 3D fluid flows has remained an elusive open problem since Moore's works in the early 1990's. In this article we construct a Turing complete stationary Euler flow on a Riemannian S3 and speculate on its implications concerning Tao's approach to the blow up problem in the Navier-Stokes equations., This work was partially supported by ICMAT–Severo OchoaGrant CEX2019-000904-S., Peer Reviewed, Postprint (author's final draft)

Can every physical system simulate any Turing machine? This is a classical problem that is intimately connected with the undecidability of certain physical phenomena. Concerning fluid flows, Moore [C. Moore, Nonlinearity 4, 199 (1991)] asked if hydrodynamics is capable of performing computations. More recently, Tao launched a program based on the Turing completeness of the Euler equations to address the blow-up problem in the Navier¿Stokes equations. In this direction, the undecidability of some physical systems has been studied in recent years, from the quantum gap problem to quantum-field theories. To the best of our knowledge, the existence of undecidable particle paths of three-dimensional fluid flows has remained an elusive open problem since Moore¿s works in the early 1990s. In this article, we construct a Turing complete stationary Euler flow on a Riemannian S3 and speculate on its implications concerning Tao¿s approach to the blow-up problem in the Navier¿Stokes equations.

This paper contributes to the study of Engel structures and their classification. The main result introduces the notion of a loose family of Engel structures and shows that two such families are Engel homotopic if and only if they are formally homotopic. This implies a complete $h$ -principle when auxiliary data is fixed. As a corollary, we show that Lorentz and orientable Cartan prolongations are classified up to homotopy by their formal data.

We compute the fundamental group of each connected component of the space of formal Legendrian embeddings in R3. We use it to show that previous examples in the literature of nontrivial loops of Legendrian embeddings are already nontrivial at the formal level. Likewise, we compute the fundamental group of the different connected components of the space of formal horizontal embeddings into the standard Engel R4. We check that the natural inclusion of the space of horizontal embeddings into the space of formal horizontal embeddings induces an isomorphism at [Pi]1 –level., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Can every physical system simulate any Turing machine? This is a classical problem which is intimately connected with the undecidability of certain physical phenomena. Concerning fluid flows, Moore asked in [15] if hydrodynamics is capable of performing computations. More recently, Tao launched a programme based on the Turing completeness of the Euler equations to address the blow up problem in the Navier-Stokes equations. In this direction, the undecidability of some physical systems has been studied in recent years, from the quantum gap problem [7] to quantum field theories [11]. To the best of our knowledge, the existence of undecidable particle paths of 3D fluid flows has remained an elusive open problem since Moore's works in the early 1990's. In this article we construct a Turing complete stationary Euler flow on a Riemannian $S^3$ and speculate on its implications concerning Tao's approach to the blow up problem in the Navier-Stokes equations., Comment: minor changes, 16 pages, 2 figures

Kostant gave a model for the geometric quantization via the coho-mology associated to the sheaf of flat sections of a pre-quantum line bundle. This model is well-adapted for real polarizations given by integrable systems and toric manifolds. In the latter case, the co-homology can be computed by counting integral points inside the associated Delzant polytope. In this article we extend Kostant’s geometric quantization to semitoric integrable systems and almost toric manifolds. In these cases the dimension of the acting torus is smaller than half of the dimension of the manifold. In particular, we compute the cohomology groups associated to the geometric quantization if the real polarization is the one induced by an in-tegrable system with focus-focus type singularities in dimension four. As an application we determine a model for the geometric quantization of K3 surfaces under this scheme.

Kostant gave a model for the geometric quantization via the cohomology associated to the sheaf of flat sections of a pre-quantum line bundle. This model is well-adapted for real polarizations given by integrable systems and toric manifolds. In the latter case, the cohomology can be computed by counting integral points inside the associated Delzant polytope. In this article we extend Kostant’s geometric quantization to semitoric integrable systems and almost toric manifolds. In these cases the dimension of the acting torus is smaller than half of the dimension of the manifold. In particular, we compute the cohomology groups associated to the geometric quantization if the real polarization is the one induced by an integrable system with focus-focus type singularities in dimension four. As an application we determine a model for the geometric quantization of K3 surfaces under this scheme, Peer Reviewed, Postprint (author's final draft)

In this article we establish efficient geometric criteria to decide whether a contact manifold is overtwisted. Starting with the original definition, we first relate overtwisted disks in different dimensions and show that a manifold is overtwisted if and only if the Legendrian unknot admits a loose chart. Then we characterize overtwistedness in terms of the monodromy of open book decompositions and contact surgeries. Finally, we provide several applications of these geometric criteria.

We study the space of immersions of S that are tangent to an Engel structure D. We show that the full h-principle holds as soon as one excludes the closed orbits of W, the characteristic foliation of D. This is sharp: we elaborate on work of Bryant and Hsu to show that curves tangent to W sometimes form additional isolated components that cannot be detected at a formal level. We then show that this is an exceptional phenomenon: if D is C-generic, curves tangent to W are not isolated anymore. These results, in conjunction with an argument due to M. Gromov, prove that a full h-principle holds for immersions transverse to the Engel structure.

This paper contributes to the study of Engel structures and their classification. The main result introduces the notion of a loose family of Engel structures and shows that two such families are Engel homotopic if and only if they are formally homotopic. This implies a complete -principle when auxiliary data is fixed. As a corollary, we show that Lorentz and orientable Cartan prolongations are classified up to homotopy by their formal data.

We prove the existence of minimal symplectomorphisms and strictly ergodic contactomorphisms on manifolds which admit a locally free S -action by symplectomorphisms and contactomorphisms, respectively. The proof adapts the conjugation method, introduced by Anosov and Katok, to the contact and symplectic setting.

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao launched a programme to address the global existence problem for the Euler and Navier Stokes equations based on the concept of universality. In this article we prove that the Euler equations exhibit universality features. More precisely, we show that any non autonomous flow on a compact manifold can be extended to a smooth solution of the Euler equations on some Riemannian manifold of possibly higher dimension. The solutions we construct are stationary of Beltrami type, so they exist for all time. Using this result, we establish the Turing completeness of the Euler flows, i.e. that there exist solutions that encode a universal Turing machine. The proofs exploit the correspondence between contact topology and hydrodynamics, which allows us to import the flexibility principles from the contact realm, in the form of holonomic h-principles for isocontact embeddings., Preprint

In this article we prove that the inclusion of the space of Engel structures of a smooth 4-manifold into the space of full flags of its tangent bundle induces surjections in all homotopy groups. In particular, we construct Engel structures representing any given full flag.

We construct positive loops of Legendrian submanifolds in several instances. In particular, we partially recover G. Liu’s result stating that any loose Legendrian admits a positive loop, under some mild topological assumptions on the Legendrian. Moreover, we show contractibility of the constructed loops under an extra topological assumption.

In this article we prove that the inclusion of the space of Engel structures of a smooth 4-manifold into the space of full flags of its tangent bundle induces surjections in all homotopy groups. In particular, we construct Engel structures representing any given full flag.

Kostant gave a model for the real geometric quantization associated to polarizations via the cohomology associated to the sheaf of flat sections of a pre-quantum line bundle. This model is well-adapted for real polarizations given by integrable systems and toric manifolds. In the latter case, the cohomology can be computed counting integral points inside the associated Delzant polytope. In this article we extend Kostant’s geometric quantization to semitoric integrable systems and almost toric manifolds. In these cases the dimension of the acting torus is smaller than half of the dimension of the manifold. In particular, we compute the cohomology groups associated to the geometric quantization if the real polarization is the one associated to an integrable system with focus-focus type singularities in dimension four. As application we determine models for the geometric quantization of K3 surfaces, a spin-spin system, the spherical pendulum, and a spin-oscillator system under this scheme., Preprint

In this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular In this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular Lagrangian foliation via sheaf cohomology and obtain important new applications in the case of real polarizations. The starting point is the definition of representation spaces due to Kostant. Besides the classical examples of Gelfand-Cetlin systems due to Guillemin and Sternberg [13] very few examples of explicit computations of real polarizations are known. The computation of Geometric Quantization in [13] is based on a theorem due to Śniatycki for fibrations [32] which identifies the representation space with the set of Bohr-Sommerfeld leaves determined by the integral action coordinates. In this article we check that the associated sheaf cohomology apparatus of Geometric Quantization satisfies Mayer-Vietoris and Künneth formulae. As a consequence, a new short proof of this classical result for fibrations due to Śniatycki is obtained. We also compute Geometric Quantization with respect to any generic regular Lagrangian foliation on a 2-torus and the case of the irrational flow. In the way, we recover some classical results in the computation of foliated cohomology of these polarizations., Peer Reviewed, Preprint

In this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular Lagrangian foliation via sheaf cohomology and obtain important new applications in the case of real polarizations. The starting point is the definition of representation spaces due to Kostant. Besides the classical examples of Gelfand-Cetlin systems due to Guillemin and Sternberg [13] very few examples of explicit computations of real polarizations are known. The computation of Geometric Quantization in [13] is based on a theorem due to Śniatycki for fibrations [32] which identifies the representation space with the set of Bohr-Sommerfeld leaves determined by the integral action coordinates. In this article we check that the associated sheaf cohomology apparatus of Geometric Quantization satisfies Mayer-Vietoris and Künneth formulae. As a consequence, a new short proof of this classical result for fibrations due to Śniatycki is obtained. We also compute Geometric Quantization with respect to any generic regular Lagrangian foliation on a 2-torus and the case of the irrational flow. In the way, we recover some classical results in the computation of foliated cohomology of these polarizations., Peer Reviewed, Postprint (published version)

In this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular Lagrangian foliation via sheaf cohomology. The starting point is the definition of representation spaces due to Kostant. We check that the associated sheaf cohomology apparatus satisfies Mayer-Vietoris and K\"unneth formulae. As a consequence, new proofs of classical results for fibrations are obtained. In the general case of Lagrangian foliations, we compute Geometric Quantization with respect to almost any generic regular Lagrangian foliation on a 2-torus., Preprint

Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We define the concept of symplectic foliation on a symplectic manifold and provide a method of constructing many examples, by using asymptotically holomorphic techniques.

We prove that the vanishing spheres of the Lefschetz pencils constructed by Donaldson on symplectic manifolds of any dimension are conjugated under the action of the symplectomorphism group of the fiber. More precisely, a number of generalized Dehn twists may be used to conjugate those spheres. This implies the non-existence of homologically trivial vanishing spheres in these pencils. To develop the proof, we discuss some basic topological properties of the space of asymptotically holomorphic transverse sections.

Given a Lagrangian submanifold in a symplectic manifold and a Morse function on the submanifold, we show that there is an isotopic Morse function and a symplectic Lefschetz pencil on the manifold extending the Morse function to the whole manifold. From this construction we define a sequence of symplectic invariants classifying the isotopy classes of Lagrangian spheres in a symplectic 4-manifold.

This paper discusses the construction of asymptotically holomorphic embeddings of a (2n−1) -dimensional contact manifold into the projective space CP 2n+1 such that the image is weakly dominated by the symplectic form. The proof follows the same lines as in [L. A. Ibort, D. Martínez-Torres and F. Presas, J. Differential Geom. 56 (2000), no. 2, 235–283 ] and most details are to be found in that paper. The asymptotically holomorphic techniques were pioneered in [S. K. Donaldson, J. Differential Geom. 44 (1996), no. 4, 666–705].

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