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(English) In this thesis we study several mathematical objects that are essential to formulate and model physical systems. Applying the tools provided by differential geometry, we develop and analyze different mathematical structures that are used in three physical contexts: dissipative dynamics, integrable systems and geometric quantization. To do it, we mainly employ the setting of b-symplectic geometry, a natural extension of symplectic geometry which is specifically designed to address manifolds with boundary. It is based on the concept of b-forms introduced by Melrose and was initiated by Guillemin, Miranda and Pires. Firstly, in the context of dissipative dynamics, we introduce and discuss a variety of twisted b-cotangent models. In these models, defined on the cotangent bundle of a smooth manifold, the fundamental structure is a b-symplectic form that is singular within the fibers of the bundle. Our models give rise to dynamical systems governed by the standard Hamiltonian of a free particle, accompanied by a positiondependent potential. After examining different types of potentials and finding that all of them induce dissipation of energy in the system, we prove that these twisted bcotangent models offer a suitable Hamiltonian formulation for dissipative systems. Consequently, they expand the scope of Hamiltonian dynamics and bring a new approach to the study of non-conservative systems. Secondly, in the context of integrable systems, we introduce and investigate bsemitoric systems, a family of systems that generalizes simultaneously semitoric systems and b-toric systems, and which is tailored for b-symplectic manifolds. We provide a comprehensive definition of b-semitoric systems, that adapts the characteristics of semitoric systems to the framework of b-symplectic manifolds, and we construct three examples of this type of system. The three examples are based on modifications of the coupled angular momenta system, a classical semitoric system that represent, (Català) En aquesta tesi estudiem diversos objectes matemàtics que són essencials per a formular i modelar sistemes físics. Aplicant les eines proporcionades per la geometria diferencial, desenvolupem i analitzem diferents estructures matemàtiques que s’utilitzen en tres contextos físics: la dinàmica dissipativa, els sistemes integrables i la quantització geomètrica. Per a fer-ho, utilitzem principalment el marc de la geometria b-simplèctica, una extensió natural de la geometria simplèctica dissenyada específicament per a varietats amb vora, basada en el concepte de b-formes introduït per Melrose, i iniciada per Guillemin, Miranda i Pires. En primer lloc, en el context de la dinàmica dissipativa, introduïm i estudiem un conjunt de models b-cotangents. En aquests models, definits al fibrat cotangent d’una varietat suau, l’estructura fonamental és una forma b-simplèctica que és singular a les fibres. Aquests models generen sistemes dinàmics governats pel Hamiltonià estàndard d’una partícula lliure, acompanyat d’un potencial que depèn de la posició de la partícula. Després d’analitzar diferents tipus de potencials i de trobar que en tots ells s’observa dissipació de l’energia del sistema, demostrem que els models b-cotangents permeten una formulació Hamiltoniana adequada per a sistemes dissipatius. D’aquesta manera, aquests models amplien l’abast de la dinàmica Hamiltoniana i aporten una nova aproximació a l’estudi de sistemes no conservatius. En segon lloc, en el context dels sistemes integrables, introduïm i investiguem els sistemes b-semitòrics, una família de sistemes que generalitza simultàniament els sistemes semitòrics i els sistemes b-tòrics i que està adaptada per a les varietats b-simplèctiques. Proporcionem una definició completa dels sistemes b-semitòrics, que fa encaixar les característiques dels sistemes semitòrics en el marc de les varietats b-simplèctiques, i construïm tres exemples d’aquest tipus de sistema. Els tres exemples es basen en modificacions, Postprint (published version)

The version of record of this article, first published in Journal of fixed point theory and its applications, is available online at Publisher’s website: http://doi.org/10.1007/s11784-024-01105-y, In this article, we study the Hamiltonian dynamics on singular symplectic manifolds and prove the Arnold conjecture for a large class of bm-symplectic manifolds. Novel techniques are introduced to associate smooth symplectic forms to the original singular symplectic structure, under some mild conditions. These techniques yield the validity of the Arnold conjecture for singular symplectic manifolds across multiple scenarios. More precisely, we prove a lower bound on the number of 1-periodic Hamiltonian orbits for b2m-symplectic manifolds depending only on the topology of the manifold. Moreover, for bm-symplectic surfaces, we improve the lower bound depending on the topology of the pair (M, Z). We then venture into the study of Floer homology to this singular realm and we conclude with a list of open questions., Funding OpenAccess funding provided thanks to the CRUE-CSIC agreement with Springer Nature. All the authors are partially supported by the Spanish State Research Agency grant PID2019-103849GB-I00 of MICIU/AEI/10.13039/50110 00110331934 and by the AGAUR project 2021 SGR 00603. J. Brugu´ es is supported by the FWO-FNRS Excellence of Science project G0H4518N “Symplectic techniques in differential geometry” with UA Antigoon Project-ID 36584.E. Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2021 and by the Alexander Von Humboldt foundation via a Friedrich Wilhelm Bessel Research Award. E. Miranda is also supported by the Spanish State Research Agency, through the 16 Page 52 of 60 J. Brugués et al. Severo Ochoa and Mar´ ıa de Maeztu Program for Centers and Units of Excellence in R&D (Project CEX2020-001084-M). C. Oms acknowledges financial support from the Juan de la Cierva postdoctoral grant FJC2021-046811-I / MICIU/AEI /10.13039/501100011033 and by the European Union NextGenerationEU/PRTR., Peer Reviewed, Postprint (published version)

© 2024 The Authors. Published by the Royal Society under the terms of theCreative Commons Attribution Licensehttp://creativecommons.org/licenses/by/4.0, We prove that the correspondence between Reeb and Beltrami vector fields presented in Etnyre & Ghrist (Etnyre, Ghrist 2000 Nonlinearity 13, 441–458 (doi:10.1088/0951-7715/13/2/306)) can be made equivariant whenever additional symmetries of the underlying geometric structures are considered. As a corollary of this correspondence, we show that energy levels above the maximum of the potential energy of mechanical Hamiltonian systems can be viewed as stationary fluid flows, though the metric is not prescribed. In particular, we showcase the emblematic example of the n-body problem and focus on the Kepler problem. We explicitly construct a compatible Riemannian metric that makes the Kepler problem of celestial mechanics a stationary fluid flow (of Beltrami type) on a suitable manifold, the Kepler–Euler flow., Peer Reviewed, Postprint (author's final draft)

© 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license, The investigation of symmetries of b-symplectic manifolds and folded-symplectic manifolds is well-understood when the group under consideration is a torus (see, for instance, [25,21, 28] for b-symplectic manifolds and [12,14] for folded symplec-tic manifolds). However, reduction theory has not been set in this realm in full generality. This is fundamental, among other reasons, to advance in the “quantization commutes with reduction” programme for these manifolds initiated in [29,30]., The project that gave rise to these results received the support of a fellowship from “la Caixa” Foundation (ID 100010434) under the INPHINIT 2018 program for excellence centers. The grant is attached to the retained excellence project Interactions between symmetries and singularities in Geometry and Physics supervised by Eva Miranda. The fellowship code is LCF/BQ/DI18/11660046. Anastasia Matveeva and Eva Miranda are partially supported by the Spanish State Research Agency, through the grant PID2019-103849GB-I00 of AEI /10.13039/501100011033, Peer Reviewed, Postprint (published version)

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)

In this article, we introduce b-semitoric systems as a generalization of semitoric systems, specifically tailored for b-symplectic manifolds. The objective of this article is to furnish a collection of examples and investigate the distinctive characteristics of these systems. A b-semitoric system is a four-dimensional b-integrable system that satisfies certain conditions: one of its momentum map components is proper and generates an effective global S1-action and all singular points are non-degenerate and devoid of hyperbolic components. To illustrate this concept, we provide five examples of b-semitoric systems by modifying the coupled spin oscillator and the coupled angular momenta, and we also classify their singular points. Additionally, we describe the dynamics of these systems through the image of their respective momentum maps., Peer Reviewed, Postprint (published version)

Un dels problemes oberts més excitants en topologia quàntica es relaciona amb la categorificació de la teoria complexa de Chern-Simons. En aquest context, Gukov i Manolescu introdueixen una sèrie en dues variables $F_K(x,q)$ com la restricció dels invariants de 3-varietats $\hat{Z}$ a complements de nusos, la qual s'ha demostrat que té propietats sorprenents. En particular, aquesta sèrie es pot pensar com una continuació analítica del celebrat polinomi de Jones de teoria de nusos. Subjacent a aquesta construcció s'hi hagi l'elecció d'un grup de Lie $G=SU(2)$. Una pregunta natural concerneix l'estudi d'aquests invariants per a $SU(N)$ de major rang. Fins ara, han hagut intents de definir-los rigorosament, però una definició matemàtica de $F_K^{SU(N)}(x_1,\dots,x_{N-1},q)$ per a nusos no torals encara manca en l'escenari general. Ens centrem en el grup de Lie $SU(3)$. Per a poder estudiar el comportament de l'invariant de nusos associat $F_K^{SU(3)}$, proporcionem una construcció a partir de $R$ matrius dels invariants de nusos cuántidos de $\sl_3$ per a qualsevol nus i representació irreductible finit-dimensional. El resultat principal d'aquesta tesi dona una construcció explícita de $F_K^{SU(3)}(x,y,q)$, que estén la seva prèvia definició a la família de major grandària nus de trena positiva. En particular, es mostra que $F_K^{LA SEVA(3)}(x,y,q)$ recupera el $\mathfrak{sl}_3$ invariant quàntic mitjançant especialitzacions en les variables $x$ e $y$., Uno de los problemas abiertos más excitantes en topología cuántica se relaciona con la categorificación de la teoría compleja de Chern-Simons. En este contexto, Gukov y Manolescu introducen una serie en dos variables $F_K(x,q)$ como la restricción de los invariantes de 3-variedades $\hat{Z}$ a complementos de nudos, la cual se ha demostrado que tiene propiedades sorprendentes. En particular, esta serie se puede pensar como una continuación analítica del celebrado polinomio de Jones de teoría de nudos. Subyacente a esta construcción se haya la elección de un grupo de Lie $G=SU(2)$. Una pregunta natural concierne el estudio de estos invariantes para $SU(N)$ de mayor rango. Hasta ahora, han habido intentos de definirlos rigurosamente, pero una definición matemática de $F_K^{SU(N)}(x_1,\dots,x_{N-1},q)$ para nudos no torales todavía carece en el escenario general. Nos centramos en el grupo de Lie $SU(3)$. Para poder estudiar el comportamiento del invariante de nudos asociado $F_K^{SU(3)}$, proporcionamos una construcción a partir de $R$ matrices de los invariantes de nudos cuántidos de $\sl_3$ para cualquier nudo y representación irreducible finito-dimensional. El resultado principal de esta tesis da una construcción explícita de $F_K^{SU(3)}(x,y,q)$, que extiende su previa definición a la familia de mayor tamaño nudos de trenza positiva. En particular, se muestra que $F_K^{SU(3)}(x,y,q)$ recupera el $\mathfrak{sl}_3$ invariante cuántico mediante especializaciones en las variables $x$ e $y$., One of the most exciting open problems in quantum topology concerns the categorification of the complex Chern-Simons theory. In this context, Gukov and Manolescu introduce a two--variable series $F_K(x,q)$ as the restriction of the 3-manifold invariant $\hat{Z}$ to knot complements, which has been shown to exhibit surprising properties. In particular, this series can be thought as an analytical continuation of the celebrated colored Jones polynomial of knot theory. Underlying this construction there is the choice of a Lie group $G=SU(2)$. A natural question turns to the study of this invariants for higher rank $SU(N)$. So far, there have been attempts to define them rigorously, but a mathematical definition of $F_K^{SU(N)}(x_1,\dots,x_{N-1},q)$ for non-torus knots is still lacking in a general scenario. We focus on the Lie group $SU(3)$. In order to study the behaviour of the associated knot invariant $F_K^{SU(3)}$, we provide a construction via $R$-matrices of the $\mathfrak{sl}_3$ quantum knot invariants for any knot and finite-dimensional irreducible representation. The main result of this thesis gives an explicit construction of $F_K^{SU(3)}(x,y,q)$, which extends its former definition to the bigger family of positive braid knots. In particular, it is shown that $F_K^{SU(3)}(x,y,q)$ recovers the $\mathfrak{sl}_3$ quantum invariants via specializations on the $x$ and $y$ variables., Outgoing

In this article we introduce and analyze in detail singular contact structures, with an emphasis on bm -contact structures, which are tangent to a given smooth hypersurface Z and satisfy certain transversality conditions. These singular contact structures are determined by the kernel of non-smooth differential forms, called bm -contact forms, having an associated critical hypersurface Z. We provide several constructions, prove local normal forms, and study the induced structure on the critical hypersurface. The topology of manifolds endowed with such singular contact forms are related to smooth contact structures via desingularization. The problem of existence of bm -contact structures on a given manifold is also tackled in this paper. We prove that a connected component of a convex hypersurface of a contact manifold can be realized as a connected component of the critical set of a bm -contact structure. In particular, given an almost contact manifold M with a hypersurface Z, this yields the existence of a b2k -contact structure on M realizing Z as a critical set. As a consequence of the desingularization techniques in [21], we prove the existence of folded contact forms on any almost contact manifold., Eva Miranda and Cédric Oms are partially supported by the AEI grant PID2019-103849GB-I00 of MCIN/AEI/10.13039/501100011033, Peer Reviewed, Objectius de Desenvolupament Sostenible::14 - Vida Submarina, Objectius de Desenvolupament Sostenible::13 - Acció per al Clima, Postprint (published version)

—In this short note, we prove that singular Reeb vector fields associated with generic b-contact forms on three dimensional manifolds with compact embedded critical surfaces have either (at least) 2N or an infinite number of escape orbits, where N denotes the number of connected components of the critical set. In case where the first Betti number of a connected component of the critical surface is positive, there exist infinitely many escape orbits. A similar result holds in the case of b-Beltrami vector fields that are not b-Reeb. The proof is based on a more detailed analysis of the main result in [19]., Peer Reviewed, Postprint (author's final draft)

Final, 2022/2023, 2n quadrimestre

Manifolds with boundary, with corners, b-manifolds and foliations model configuration spaces for particles moving under constraints and can be described as E-manifolds. E-manifolds were introduced in Nest and Tsygan (2001 Asian J. Math.5 599–635) and investigated in depth in Miranda and Scott (2021 Rev. Mat. Iberoam.37 1207–24). In this article we explore their physical facets by extending gauge theories to the E-category. Singularities in the configuration space of a classical particle can be described in several new scenarios unveiling their Hamiltonian aspects on an E-symplectic manifold. Following the scheme inaugurated in Weinstein (1978 Lett. Math. Phys.2 417–20), we show the existence of a universal model for a particle interacting with an E-gauge field. In addition, we generalise the description of phase spaces in Yang–Mills theory as Poisson manifolds and their minimal coupling procedure, as shown in Montgomery (1986 PhD Thesis University of California, Berkeley), for base manifolds endowed with an E-structure. In particular, the reduction at coadjoint orbits and the shifting trick are extended to this framework. We show that Wong's equations, which describe the interaction of a particle with a Yang–Mills field, become Hamiltonian in the E-setting. We formulate the electromagnetic gauge in a Minkowski space relating it to the proper time foliation and we see that our main theorem describes the minimal coupling in physical models such as the compactified black hole., Peer Reviewed, Postprint (author's final draft)

En aquest treball de fi de màster provem nous resultats sobre la cohomologia de Poisson d'una classe important de varietats de Poisson: les varietats de $b^m$-Poisson. Per $m = 1$, els resultats obtinguts són coneguts i es corresponen amb els resultats previs deguts a Guillemin–Miranda–Pires i Marcut–Osorno. Per a $m$ més gran que $1$, hi ha una contribució infinito-dimensional que analitzem fent servir la idea de quocient en àlgebra i geometria. La cohomologia de Poisson governa molts problemes de classificació en geometria de Poisson i, en particular, mesura la teoria de deformació d'aquestes estructures (gràcies als resultats de Kontsevich). Partint d'aquests resultats aconseguim noves eines per estudiar aquests problemes en varietats de $b^m$-Poisson. Les varietats de $b^m$-Poisson s'anomenen comunament varietats simplèctiques singulars. Conseqüentment, els resultats provats respecte de la cohomologia de Poisson admeten interpretacions en la correspondència de Rham–Poisson per a varietats simplèctiques regulars. Anticipem que aquesta recent caracterització dels grups de cohomologia de Poisson trobarà aplicació a la teoria de quantització, així com a camps més distants com a teoria KAM., En este trabajo de fin de máster probamos nuevos resultados sobre la cohomología de Poisson de una clase importante de variedades de Poisson: las variedades $b^m$-Poisson. Para $m = 1$, los resultados dados son conocidos y coinciden con los resultados previos de Guillemin–Miranda–Pires y Marcut–Osorno. Para $m$ mayor que $1$, hay una contribución infinito-dimensional que analizamos mediante la idea clásica del cociente en álgebra y geometría. La cohomología de Poisson goberna muchos problemas de clasificación en geometría de Poisson y, en particular, mide la teoría de deformación de las estructuras (gracias a los resultados de Kontsevich). Partiendo de estos resultados obtenemos nuevas herramientas para estudiar dichos problemas en variedades de $b^m$-Poisson. Las variedades $b^m$-Poisson son comúnmente llamadas variedades simplécticas singulares. Consecuentemente, los resultados obtenidos respecto a la cohomología de Poisson tienen interpretaciones en la correspondencia de Rham–Poisson para variedades simplécticas regulares. Anticipamos que estos recientes resultados sobre los grupos de cohomología de Poisson encontrarán aplicaciones prácticas en teoría de cuantización, así como en campos más distantes como en teoría KAM., In this master thesis we give new results about the Poisson cohomology of an important class of Poisson manifolds: bm-Poisson manifolds. For $m = 1$ these results were already known and coincide with former results of Guillemin–Miranda–Pires and Marcut–Osorno. For $m$ greater than $1$, there is an infinite dimensional contribution which we analyze using the classical idea of quotient in algebra and geometry. Poisson cohomology governs many classification problems in Poisson geometry and, in particular, it measures deformation theory of the structures (due to the contributions of Kontsevich). By obtaining these results we have a new toolkit for such problems for $b^m$-Poisson manifolds. $b^m$-Poisson manifolds are often referred to as singular symplectic manifolds. As such, the new results obtained for Poisson cohomology have a new interpretation in the de Rham–Poisson cohomology correspondence for honest symplectic manifolds. We anticipate that these fresh Poisson cohomology new results will have practical applications in quantization theory, as well as more distant applications such as KAM theory.

En aquest treball de fi de grau, investiguem la teoria de formes simplèctiques i de contacte singulars introduïdes per Guillemin-Miranda-Pires [GMP14] i Nest-Tsygan [NT96], a més de les connexions entre la geometria de contacte i la dinàmica de fluids. L'objectiu és explorar les possibles aplicacions d'aquestes idees a la dinàmica celeste. Comencem examinant nous aspectes del mirall Reeb-Beltrami donat per Etnyre i Ghrist [EG00a], conduint a una versió equivariant d'aquesta correspondència i desvetllant una interpretació de diversos sistemes mecànics Hamiltonians com a fluxos de fluids estacionaris. Destaquem especialment el problema d'N cossos en la mecànica celeste, i en particular el problema de Kepler. Utilitzant les tècniques singulars desenvolupades en aquest treball, també donem fites inferiors per al nombre d'òrbites d'escapament en sistemes dinàmics associats a les formes singulars que apareixen en la mecànica celeste. Aquest treball de fi de grau culmina amb un contraexemple a la conjectura singular de Weinstein sobre l'existència d'òrbites periòdiques singulars formulada per Miranda-Oms a [MO21]., In this undergraduate thesis, we delve into the theory of singular symplectic and contact forms originally introduced by Guillemin-Miranda-Pires [GMP14] and Nest-Tsygan [NT96], as well as the connections between contact geometry and fluid dynamics. Our objective is to explore the potential applications of these ideas in the field of celestial mechanics. Our thesis begins by examining novel aspects of the Reeb-Beltrami mirror given by Etnyre and Ghrist [EG00a], leading to an equivariant version of this correspondence and unveiling an interpretation of various mechanical Hamiltonian systems as stationary fluid flows. Notably, we focus on different incarnations of the n-body problem of celestial mechanics, particularly the Kepler problem. By utilizing the singular techniques developed in this thesis, we also provide lower bounds for escape orbits in dynamical systems associated with the singular forms that appear in celestial mechanics. This undergraduate thesis culminates with a counterexample to the singular Weinstein conjecture regarding the existence of singular periodic orbits formulated by Miranda-Oms in [MO21]., Outgoing

In this article we analyze several mathematical models with singularities where the classical cotangent model is replaced by a -cotangent model. We provide physical interpretations of the singular symplectic geometry underlying in -cotangent bundles featuring two models: the canonical (or non-twisted) model and the twisted one. The canonical one models systems on manifolds with boundary and the twisted one represents Hamiltonian systems with a singularity on the fiber. The twisted cotangent model includes (for linear potentials) the case of fluids with dissipation. We prove (non)-existence of cotangent lift dynamics and show the existence of an infinite number of escape orbits in this model. We also discuss more general physical interpretations of the twisted and non-twisted -symplectic models. Twisted -symplectic models yield in a natural way escape orbits that go to the critical set. Under compactness assumptions those escape orbits are continued as singular periodic orbits in the sense of Miranda and Oms (2021) and Miranda (2020). These models offer a Hamiltonian formulation for systems which are dissipative, extending the horizons of Hamiltonian dynamics and opening a new approach to study non-conservative systems., Peer Reviewed, Postprint (published version)

We define the Bohr-Sommerfeld quantization via T -modules for a b-symplectic toric manifold and show that it coincides with the formal geometric quantization of [Victor W. Guillemin, Eva Miranda, and Jonathan Weitsman. On geometric quantization of b-symplectic manifolds. Adv. Math., 331:941–951, 2018]. In particular, we prove that its dimension is given by a signed count of the integral points in the moment polytope of the toric action on the manifold, The first-named author is supported by the Doctoral INPhINIT-RETAINING grant ID 100010434 LCF/BQ/DR21/11880025 of “la Caixa” Foundation; by the AEI grant PID2019-103849GB-I00 of MCIN/ AEI /10.13039/501100011033; by the AGAUR project 2021 SGR 00603 Geometry of Manifolds and Applications, GEOMVAP; and by the AGRUPS 2023 grant by UPC. The second-named author is partially supported by the Spanish State Research Agency AEI/10.13039/501100011033, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (project CEX2020-001084-M); and by the grant PID2019-103849GB-I00. The author is also partially supported by the AGAUR project 2021 SGR 00603 Geometry of Manifolds and Applications, GEOMVAP; and by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2021. The third-named author is supported by a Simons collaboration grant., Peer Reviewed, Preprint

In this article, we pursue our investigation of the connections between the theory of computation and hydrodynamics. We prove the existence of stationary solutions of the Euler equations in Euclidean space, of Beltrami type, that can simulate a universal Turing machine. In particular, these solutions possess undecidable trajectories. Heretofore, the known Turing complete constructions of steady Euler flows in dimension 3 or higher were not associated to a prescribed metric. Our solutions do not have finite energy, and their construction makes crucial use of the non-compactness of R³ , however they can be employed to show that an arbitrary tape-bounded Turing machine can be robustly simulated by a Beltrami flow on T³ (with the standard flat metric). This shows that there exist steady solutions to the Euler equations on the flat torus exhibiting dynamical phenomena of (robust) computational complexity as high as desired. We also quantify the energetic cost for a Beltrami field on T³ to simulate a tape-bounded Turing machine, thus providing additional support for the space-bounded ChurchTuring thesis. Another implication of our construction is that a Gaussian random Beltrami field on Euclidean space exhibits arbitrarily high computational complexity with probability 1. Finally, our proof also yields Turing complete flows and diffeomorphisms on S² with zero topological entropy, thus disclosing a certain degree of independence within different hierarchies of complexity, Postprint (author's final draft)

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. In recent papers [5, 6, 7, 8] several unknown facets of the Euler flows have been discovered, including universality properties of the stationary solutions to the Euler equations. The study of these universality features was suggested by Tao as a novel way to address the problem of global existence for Euler and Navier-Stokes [28]. Universality of the Euler equations was proved in [7] for stationary solutions using a contact mirror which reflects a Beltrami flow as a Reeb vector field. This contact mirror permits the use of advanced geometric techniques in fluid dynamics. On the other hand, motivated by Tao’s approach relating Turing machines to Navier-Stokes equations, a Turing complete stationary Euler solution on a Riemannian 3-dimensional sphere was constructed in [8]. Since the Turing completeness of a vector field can be characterized in terms of the halting problem, which is known to be undecidable [30], a striking consequence of this fact is that a Turing complete Euler flow exhibits undecidable particle paths [8]. In this article, we give a panoramic overview of this fascinating subject, and go one step further in investigating the undecidability of different dynamical properties of Turing complete flows. In particular, we show that variations of [8] allow us to construct a stationary Euler flow of Beltrami type (and, via the contact mirror, a Reeb vector field) for which it is undecidable to determine whether its orbits through an explicit set of points are periodic, Robert Cardona acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the Mar´ıa de Maeztu Programme for Units of Excellence in R& D (MDM-2014-0445) via an FPI grant. Robert Cardona and Eva Miranda are partially supported by the grants PID2019-103849GB-I00 / AEI / 10.13039/501100011033 and the AGAUR grant 2017SGR932. Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Daniel Peralta-Salas is supported by the grants MTM PID2019-106715GBC21 (MICINN) and Europa Excelencia EUR2019-103821 (MCIU). This work was partially supported by the ICMAT–Severo Ochoa grant CEX2019-000904-S., Preprint

In this article, motivated by the study of symplectic structures on manifolds with boundary and the systematic study of -symplectic manifolds started in Guillemin, Miranda, and Pires Adv. Math. 264 (2014), 864–896, we prove a slice theorem for Lie group actions on -symplectic manifolds., Postprint (published version)

Manifolds with boundary, with corners, b-manifolds and foliations model configuration spaces for particles moving under constraints and can be described as E-manifolds. E-manifolds were introduced in [NT01] and investigated in depth in [MS20]. In this article we explore their physical facets by extending gauge theories to the E-category. Singularities in the configuration space of a classical particle can be described in several new scenarios unveiling their Hamiltonian aspects on an E-symplectic manifold. Following the scheme inaugurated in [Wei78], we show the existence of a universal model for a particle interacting with an E-gauge field. In addition, we generalize the description of phase spaces in Yang-Mills theory as Poisson manifolds and their minimal coupling procedure, as shown in [Mon86], for base manifolds endowed with an E-structure. In particular, the reduction at coadjoint orbits and the shifting trick are extended to this framework. We show that Wong's equations, which describe the interaction of a particle with a Yang-Mills field, become Hamiltonian in the E-setting. We formulate the electromagnetic gauge in a Minkowski space relating it to the proper time foliation and we see that our main theorem describes the minimal coupling in physical models such as the compactified black hole., Preprint

A física teòrica, la Teoria Quàntica de Camps (TQC) és un marc teòric extremadament reeixit que combina la Relativitat Especial amb la Mecànica Quàntica, permetent el diseny de models físics de les partícules subatómiques i quasipartícules que descriuen els aspectes més fundamentals de la matèria amb una precisió increïblement alta. Entre aquestes teories, la TQC Chern-Simons és una especial, que no només descriu fenòmens topològics a la física com ara l’Efecte Hall Quàntic, sinó que encaixa amb la noció del que es coneix com una Teoria Topològica de Camps Quàntics (TTCQ). Va ser fent servir aquests axiomes de les TTCQs que Edward Witten va mostrar al 1989 com d’estretament relacionada està la teoria de Chern-Simons amb l’àmbit d’invariants polinòmics que apareixen a la Teoria de Nusos, com ara el ben conegut polinomi de Jones. En aquests darrers anys, investigacions en aquest camp han donat lloc a nous i més poderosos invariants d’enllaços i, a través de cirurgies de Dehn sobre ells, de 3-varietats també. Per exemple, la sèrie de Gukov-Manolescu recentment proposta el 2020 —denotada FK(x, q)— és un invariant conjectural de complements de nusos que, en cert sentit, continua analíticament els polinomis de Jones colorejats. Poc després, Sunghyuk Park va introduir l’enfoc de la Matriu R de Gran Color corresponent a sl(2,C) per estudiar FK per trenats positius i calcular FK per a diversos nusos i enllaços. Aquest procediment ha estat així mateix extès per Angus Gruen a totes les altres àlgebres de Lie sl(n+1) més enllà de sl(2). En aquest treball, després d’un extens repàs sobre els anteriorment esmentats conceptes, abordem la família so(2n) d’àlgebres de Lie semisimples sobre els complexos a la classificació de Cartan, centrant-nos principalment en el cas so(8) atrets per la simetria triple al seu diagrama de Dynkin D4., En física teórica, la Teoría Cuántica de Campos (TCC) es un marco teórico extremadamente exitoso que combina la Relatividad Especial con la Mecánica Cuántica, permitiendo el diseño de modelos físicos de las partículas subatómicas y cuasipartículas que describen los aspectos más fundamentales de la materia con una precisión increíblemente alta. Entre dichas teorías, la TCC Chern-Simons es una especial, que no sólo describe fenómenos topológicos en física tales como el Efecto Hall Cuántico, sino que encaja con la noción de lo que se conoce como una Teoría Topológica de Campos Cuánticos (TTCC). Fue utilizando estos axiomas de las TTCCs que Edward Witten mostró en 1989 cómo de estrechamente relacionada está la teoría de Chern-Simons con el ámbito de invariantes polinómicos que aparecen en la Teoría de Nudos, tales como el bien conocido polinomio de Jones. En estos últimos años, investigaciones en este campo han dado lugar a nuevos y más poderosos invariantes de enlaces y, a través de cirugías de Dehn sobre ellos, así mismo de 3-variedades. Por ejemplo, la serie de Gukov-Manolescu recientemente propuesta en 2020 —denotada FK(x, q)— es un invariante conjetural de complementos de nudos que, en cierto sentido, continúa analíticamente los polinomios de Jones coloreados. Poco después, Sunghyuk Park introdujo el enfoque de la Matriz R de Gran Color correspondiente a sl(2,C) para estudiar FK para trenzados positivos y calcular FK para varios nudos y enlaces. Este procedimiento ha sido a su vez extendido por Angus Gruen a todas las otras álgebras de Lie sl(n+1) más allá de sl(2). En la presente obra, tras un extenso repaso sobre los anteriormente mencionados conceptos, abordamos la família so(2n) de álgebras de Lie semisimples sobre los complejos en la clasificación de Cartan, centrándonos principalmente en el caso so(8) atraídos por la simetría triple en su diagrama de Dynkin D4., In theoretical physics, Quantum Field Theory (QFT) is an extremely successful theoretical framework combining both Special Relativity and Quantum Mechanics, enabling to design physical models of subatomic particles and quasiparticles describing the most fundamental aspects of matter with an incredibly high accuracy. Among these theories, the Chern-Simons QFT is a special one, not only describing topological phenomena in physics such as the Quantum Hall Effect, but also fitting the notion of what is known as a Topological Quantum Field Theory (TQFT). It was by using the axioms of TQFTs that Edward Witten showed back in 1989 how closely related the Chern-Simons theory is to the realm of polynomial invariants appearing in Knot Theory, such as the well-known Jones polynomial. In the past years, further research in this field has led to new and more powerful invariants of links and, by means of Dehn surgeries on them, of 3-manifolds as well. For instance, the Gukov-Manolescu series proposed recently in 2020 —denoted FK(x, q)— is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. Shortly after, Sunghyuk Park introduced the Large Color R-matrix approach for sl(2,C) to study FK for some simple links, giving a definition of FK for positive braid knots and computing FK for various knots and links. This procedure has in turn been extended by Angus Gruen to all other Lie algebras sl(n+1) beyond sl(2). In this work, after a broad review on the above mentioned background, we move on to the family so(2n) of complex semisimple Lie algebras in Cartan’s classification, mainly focusing on the so(8) case attracted by the three-fold symmetry in its Dynkin diagram D4., Outgoing

2021/2022, 2n quadrimestre

In this article, we study the Hamiltonian dynamics on singular symplectic manifolds and prove the Arnold conjecture for a large class of bm-symplectic manifolds. More precisely, we prove a lower bound on the number 1-periodic Hamiltonian orbits for b2m-symplectic manifolds depending only on the topology of the manifold. Moreover, for bm-symplectic surfaces, we improve the lower bound depending on the topology of the pair (M, Z). We then venture into the study of Floer homology to this singular realm and we conclude with a list of open questions., Preprint

Les teories gauge descriuen la dinàmica d'una partícula clàssica amb graus interns de llibertat. L'espai de configuracions és un fibrat associat a un fibrat principal $\pi\colon P \longrightarrow M$ amb grup d'estructura $G$, i les equacions de moviment reben el nom d'equacions de Wong. Les teories Yang-Mills, les quals són la base del model estàndard de la física, són una teoria gauge amb grup d'estructura $G = \operatorname{U}(1) \times \operatorname{SU}(2) \times \operatorname{SU}(3)$. Weinstein va provar que l'espai de fase natural d'una teoria gauge es pot descriure mitjançant una reducció simplèctica i que les equacions de Wong són hamiltonianes. Respecte de l'espai de fase en una teoria de Yang-Mills, Montgomery va observar que l'espai de fase de Weinstein és una fulla simplèctica d'una varietat de Poisson i va generalitzar l'isomorfisme de Weinstein amb l'espai de fase de Sternberg a varietats de Poisson. Aquest resultat rep el nom d'acoblament mínim. En aquest treball estenem els resultats de Weinstein i Montgomery a $E$-varietats, on el fibrat tangent se substitueix per un subfibrat anomenat fibrat $E$-tangent. Les varietats amb vora, les varietats amb cantonades i les varietats foliades són exemples d'$E$-varietats. Fent servir resultats coneguts, hem provat que els formalismes de les teories clàssiques de gauge i de Yang-Mills, proposades per Weinstein i Montgomery, respectivament, es poden aplicar quan l'espai base $M$ és una $E$-varietat. Es pot aplicar aquest resultat, per exemple, a l'estudi del model estàndard clàssic en un espai-temps compactificat de Penrose., Las teorías gauge describen la dinámica de una partícula clásica con grados internos de libertad. El espacio de configuraciones es un fibrado asociado a un fibrado principal $\pi\colon P \longrightarrow M$ con grupo de estructura $G$, y las ecuaciones de movimiento reciben el nombre de ecuaciones de Wong. Las teorías Yang-Mills, las cuales son la base del modelo estándar de la física, son una teoría gauge con grupo de estructura $G = \operatorname{U}(1) \times \operatorname{SU}(2) \times \operatorname{SU}(3)$. Weinstein demostró que el espacio de fase natural en una teoría gauge se puede obtener por medio de una reducción simpléctica y que las ecuaciones de Wong son hamiltonianas. Respecto al espacio de fases en una teoría de Yang-Mills, Montgomery observó que el espacio de fases de Weinstein es una hoja simpléctica de un espacio de Poisson y generalizó el isomorfismo de Weinstein con el espacio de fases de Sternberg a variedades de Poisson. Este resultado se conoce como el procedimiento de acoplamiento mínimo. En este trabajo extendemos los resultados conocidos para teorías de gauge clásicas a $E$-variedades, en las que el fibrado tangente se reemplaza por un subfibrado integrable llamado fibrado $E$-tangente. Las variedades con borde, las variedades con esquinas y las variedades foliadas son ejemplos de $E$-variedades. Usando resultados conocidos en la literatura, hemos demostrado que los formalismos de las teorías de gauge y de Yang-Mills clásicas, propuestos por Weinstein y Montgomery, respectivamente, son válidos cuando el espacio base $M$ es una $E$-variedad. Este resultado puede aplicarse, por ejemplo, para el estudio del modelo estándar clásico en un espacio-tiempo compactificado de Penrose., Gauge theories describe the dynamics of a classical particle with internal degrees of freedom. The natural configuration space is associated to a $G$-principal bundle $\pi\colon P \longrightarrow M$ and the equations of motion are called Wong's equations. Yang-Mills theories, which are the basis of the standard model of physics, are a gauge theory with structure group $G = \operatorname{U}(1) \times \operatorname{SU}(2) \times \operatorname{SU}(3)$. Weinstein observed that the natural phase space of gauge theories can be obtained by a symplectic reduction, and that Wong's equations are Hamiltonian. For the natural phase space of Yang-Mills theories, Montgomery observed that the natural phase space of Weinstein is a symplectic leaf of a Poisson manifold, and generalized Weinstein's isomorphism with Sternberg's phase space to Poisson manifolds. This result is called the minimal coupling procedure. In this work, we extend this setting to classical gauge theories over $E$-manifolds, in which the tangent bundle is replace by an integrable subbundle, called the $E$-tangent bundle. Manifolds with boundary, manifolds with corners, and foliated manifolds are examples of $E$-manifolds. Using standard results in the literature, we have proved that the formulation of classical gauge and Yang-Mills theories by Weinstein and Montgomery, respectively, hold when the base manifold is taken to be an $E$-manifold. This result can be applied, for example, to study the classical standard model in a Penrose compactified space-time.

The speaker of the IMTech Colloquium will be Eva Miranda (UPC), a leading figure in Differential Geometry, Mathematical Physics and Dynamical Systems

In this article we analyze several mathematical models with singularities where the classical cotangent model is replaced by a b-cotangent model. We provide physical interpretations of the singular symplectic geometry underlying in b-cotangent bundles featuring two models: the canonical (or non-twisted) model and the twisted one. The first one models systems on manifolds with boundary and the twisted model represents Hamiltonian systems where the singularity of the system is in the fiber of the bundle. The twisted cotangent model includes (for linear potentials) the case of fluids with dissipation. We relate the complexity of the fluids in terms of the Reynolds number and the (non)-existence of cotangent lift dynamics. We also discuss more general physical interpretations of the twisted and non-twisted b-symplectic models. These models offer a Hamiltonian formulation for systems which are dissipative, extending the horizons of Hamiltonian dynamics and opening a new approach to study non-conservative systems., Preprint

Motivated by the group of Galilean transformations and the subgroup of Galilean transformations which fix time zero, we introduce the notion of a b-Lie group as a pair where G is a Lie group and H is a codimension-one Lie subgroup. Such a notion allows us to give a theoretical framework for transformations of space-time where the initial time can be seen as a boundary. In this theoretical framework, we develop the basics of the theory and study the associated canonical b-symplectic structure on the b-cotangent bundle ¿ together with its reduction theory. Namely, we extend the minimal coupling procedure to ¿ and prove that the Poisson reduction under the cotangent lifted action of H by left translations can be described in terms of the Lie Poisson structure on ¿ (where is the Lie algebra of H) and the canonical b-symplectic structure on ¿ , where is viewed as a one-dimensional b-manifold having as critical hypersurface (in the sense of b-manifolds) the identity element., Peer Reviewed, Postprint (published version)

We define the Bohr-Sommerfeld quantization via T-modules for a b-symplectic toric manifold and show that it coincides with the formal geometric quantization of [GMW18b]. In particular, we prove that its dimension is given by a signed count of the integral points in the moment polytope of the toric action on the manifold., We define the Bohr-Sommerfeld quantization via T-modules for a b-symplectic toric manifold and show that it coincides with the formal geometric quantization of [GMW18b]. In particular, we prove that its dimension is given by a signed count of the integral points in the moment polytope of the toric action on the manifold., Finançat pel projecte ICREA ACADEMIA 2016-05 i ICREA ACADEMIA 2021, Preprint

The investigation of symmetries of b-symplectic manifolds and folded-symplectic manifolds is well-understood when the group under consideration is a torus (see, for instance, [GMPS, GLPR,GMW18a]forb-symplecticmanifoldsand [CGP,CM]forfoldedsymplecticmanifolds). However, reduction theory has not been set in this realm in full generality. This is fundamental, among other reasons, to advance in the “quantization commutes with reduction” programme for these manifolds initiated in [GMW18b, GMW21]. In this article, we fill in this gap and investigate the MarsdenWeinstein reductiontheory under generalsymmetriesforgeneralbm-symplecticmanifoldsand other singular symplectic manifolds, including certain folded symplectic manifolds. In this new framework, the set of admissible Hamiltonian functions is larger than the category of smooth functions as it takes the singularities of the differential forms into account. The quasi-Hamiltonian set-up is also considered and brand-new constructions of (singular) quasi-Hamiltonian spaces are obtained via a reduction procedure and the fusion product., MarÍa de Maeztu Program for Centers and Units of Excellence in R&D (proyecto CEX2020-001084-M). La Caixa Incoming 2018 - LCF/BQ/DI18/11660046, Preprint

In this article, we pursue our investigation of the connections between the theory of computation and hydrodynamics. We prove the existence of stationary solutions of the Euler equations in Euclidean space, of Beltrami type, that can simulate a universal Turing machine. In particular, these solutions possess undecidable trajectories. Heretofore, the known Turing complete constructions of steady Euler flows in dimension 3 or higher were not associated to a prescribed metric. Our solutions do not have finite energy, and their construction makes crucial use of the non-compactness of R3, however they can be employed to show that an arbitrary tape-bounded Turing machine can be robustly simulated by a Beltrami flow on T3 (with the standard flat metric). This shows that there exist steady solutions to the Euler equations on the flat torus exhibiting dynamical phenomena of (robust) arbitrarily high computational complexity. We also quantify the energetic cost for a Beltrami field on T3 to simulate a tape-bounded Turing machine, thus providing additional support for the space-bounded Church-Turing thesis. Another implication of our construction is that a Gaussian random Beltrami field on Euclidean space exhibits arbitrarily high computational complexity with probability 1. Finally, our proof also yields Turing complete flows and maps on S2 with zero topological entropy, thus disclosing a certain degree of independence within different hierarchies of complexity., Robert Cardona acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the Mar´ıa de Maeztu Programme for Units of Excellence in R& D (MDM-2014-0445) via an FPI grant. Robert Cardona and Eva Miranda are partially supported by the AEI grant PID2019- 103849GB-I00 of MCIN/ AEI /10.13039/501100011033, and AGAUR grant 2017SGR932. Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016 and by the Spanish State Research Agency, through the Severo Ochoa and Mar´ıa de Maeztu Program for Centers and Units of Excellence in R&D (project CEX2020-001084-M). Daniel Peralta-Salas is supported by the grants CEX2019-000904-S, RED2018- 102650-T, EUR2019-103821 and PID2019-106715GB GB-C21 funded by MCIN/AEI/ 10.13039/501100011033., Preprint

In this article we give a universal model for geometric quantization associated to a real polarization given by an integrable system with non-degenerate singularities. This universal model goes one step further than the cotangent models in [KM17] by both considering singular orbits and adding to the cotangent models a model for the prequantum line bundle. These singularities are generic in the sense that are given by Morse-type functions and include elliptic, hyperbolic and focus-focus singularities. Examples of systems admitting such singularities are toric, semitoric and almost toric manifolds, as well as physical systems such as the coupling of harmonic oscillators, the spherical pendulum or the reduction of the Euler’s equations of the rigid body on T *(SO(3)) to a sphere. Our geometric quantization formulation coincides with the models given in [HM10] and [MPS20] away from the singularities and corrects former models for hyperbolic and focus-focus singularities cancelling out the infinite dimensional contributions obtained by former approaches. The geometric quantization models provided here match the classical physical methods for mechanical systems such as the spherical pendulum as presented in [CS16]. Our cotangent models obey a local-to-global principle and can be glued to determine the geometric quantization of the global systems even if the global symplectic classification of the systems is not known in general., Preprint

Motivated by Poincaré’s orbits going to infinity in the (restricted) three-body problem (see [26] and [6]), we investigate the generic existence of heteroclinic-like orbits in a neighbourhood of the critical set of a b-contact form. This is done by using the singular counterpart [3] of Etnyre– Ghrist’s contact/Beltrami correspondence [9], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [29]. Specifically, we analyze the b-Beltrami vector fields on b-manifolds of dimension 3 and prove that for a generic asymptotically exact b-metric they exhibit escape orbits. We also show that a generic asymptotically symmetric b-Beltrami vector field on an asymptotically flat b-manifold has a generalized singular periodic orbit and at least 4 escape orbits. Generalized singular periodic orbits are trajectories of the vector field whose α- and ω-limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture, Postprint (author's final draft)

In this article we construct a compact Riemannian manifold of high dimension on which the time dependent Euler equations are Turing complete. More precisely, the halting of any Turing machine with a given input is equivalent to a certain global solution of the Euler equations entering a certain open set in the space of divergence-free vector fields. In particular, this implies the undecidability of wether a solution to the Euler equations with an initial datum will reach a certain open set or not in the space of divergence-free fields. This result goes one step further in Tao’s programme to study the blowup problem for the Euler and Navier-Stokes equations using fluid computers. As a remarkable spin-off, our method of proof allows us to give a counterexample to a conjecture of Moore dating back to 1998 on the non-existence of analytic maps on compact manifolds that are Turing complete., 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 partially supported by the grants MTM2015-69135- P/FEDER and PID2019-103849GB-I00 / AEI / 10.13039/501100011033, and AGAUR grant 2017SGR932. Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Daniel Peralta-Salas is supported by the grants MTM PID2019-106715GB-C21 (MICINN) and Europa Excelencia EUR2019-103821 (MCIU). This work was partially supported by the ICMAT– Severo Ochoa grant CEX2019-000904-S., Preprint

In this article, we consider integrable systems on manifolds endowed with symplectic structures with singularities of order one. These structures are symplectic away from a hypersurface where the symplectic volume goes either to infinity or to zero transversally, yielding either a b-symplectic form or a folded symplectic form. The hypersurface where the form degenerates is called critical set. We give a new impulse to the investigation of the existence of action-angle coordinates for these structures initiated in [36] and [37] by proving an action-angle theorem for folded symplectic integrable systems. Contrary to expectations, the action-angle coordinate theorem for folded symplectic manifolds cannot be presented as a cotangent lift as done for symplectic and bsymplectic forms in [36]. Global constructions of integrable systems are provided and obstructions for the global existence of action-angle coordinates are investigated in both scenarios. The new topological obstructions found emanate from the topology of the critical set Z of the singular symplectic manifold. The existence of these obstructions in turn implies the existence of singularities for the integrable system on Z., 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). Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Both authors are supported by the grants reference number 2017SGR932 (AGAUR) and PID2019-103849GB-I00 / AEI / 10.13039/501100011033., Preprint

In this article, we construct a compact Riemannian manifold of high dimension on which the time-dependent Euler equations are Turing complete. More precisely, the halting of any Turing machine with a given input is equivalent to a certain global solution of the Euler equations entering a certain open set in the space of divergence-free vector fields. In particular, this implies the undecidability of whether a solution to the Euler equations with an initial datum will reach a certain open set or not in the space of divergence-free fields. This result goes one step further in Tao’s programme to study the blow-up problem for the Euler and Navier–Stokes equations using fluid computers. As a remarkable spin-off, our method of proof allows us to give a counterexample to a conjecture of Moore dating back to 1998 on the non-existence of analytic maps on compact manifolds that are Turing complete., María de Maeztu Programme (MDM-2014-0445) AGAUR grant 2017SGR932 ICMAT–Severo Ochoa grant CEX2019-000904-S, Robert Cardona acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the Mar´ıa de Maeztu Programme for Units of Excellence in R& D (MDM-2014-0445) via an FPI grant. Robert Cardona and Eva Miranda are partially supported by the grants MTM2015-69135- P/FEDER and PID2019-103849GB-I00 / AEI / 10.13039/501100011033, and AGAUR grant 2017SGR932. Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Daniel Peralta-Salas is supported by the grants MTM PID2019-106715GB-C21 (MICINN) and Europa Excelencia EUR2019-103821 (MCIU). This work was partially supported by the ICMAT– Severo Ochoa grant CEX2019-000904-S., Postprint (author's final draft)

Motivated by the study of symplectic Lie algebroids, we focus on a type of algebroid (called an E-tangent bundle) which is particularly well suited to the study of singular differential forms and their cohomology. This setting generalizes b-symplectic manifolds, foliat ed manifolds, and a wide class of Poisson manifolds. We generalize Moser’s theorem to this setting, and use it to construct symplec tomorphisms between singular symplectic forms. We give applications of this machinery (including the study of Poisson cohomology), and study specificexamples of a few of them in depth., Peer Reviewed, Preprint

© 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)

In this article, we consider integrable systems on manifolds endowed with symplectic structures with singularities of order one. These structures are symplectic away from a hypersurface where the symplectic volume goes either to infinity or to zero transversally, yielding either a b-symplectic form or a folded symplectic form. The hypersurface where the form degenerates is called critical set. We give a new impulse to the investigation of the existence of action-angle coordinates for these structures initiated in [34] and [35] by proving an action-angle theorem for folded symplectic integrable systems. Contrary to expectations, the action-angle coordinate theorem for folded symplectic manifolds cannot be presented as a cotangent lift as done for symplectic and b-symplectic forms in [34]. Global constructions of integrable systems are provided and obstructions for the global existence of action-angle coordinates are investigated in both scenarios. The new topological obstructions found emanate from the topology of the critical set Z of the singular symplectic manifold. The existence of these obstructions in turn implies the existence of singularities for the integrable system on Z¿., Peer Reviewed, Preprint

Motivated by Poincare’s orbits going to infinity in the (restricted) three-body problem ´ (see [29] and [7]), we investigate the generic existence of heteroclinic-like orbits in a neighbourhood of the critical set of a b-contact form. This is done by using a singular counterpart [4] of Etnyre– Ghrist’s contact/Beltrami correspondence [11], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [33]. Specifically, we analyze the b-Beltrami vector fields on b-manifolds of dimension 3 and prove that for a generic asymptotically exact b-metric they exhibit escape orbits. We also show that a generic asymptotically symmetric b-Beltrami vector field on an asymptotically flat b-manifold has a generalized singular periodic orbit and at least 4 escape orbits. Generalized singular periodic orbits are trajectories of the vector field whose a- and ¿-limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture., E. M. is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Eva Miranda and Cedric Oms are supported by the grants reference number MTM2015-69135-P (MINECO/FEDER) ánd reference number 2017SGR932 (AGAUR) and the project PID2019-103849GB-I00 / AEI / 10.13039/501100011033. C. O. has been supported by an FNR-AFR PhD predoctoral grant (project GLADYSS) until October 2nd, 2020 and by a SECTI-Postdoctoral grant financed by Eva Miranda’s ICREA Academia immediately after. D. P.-S. is supported by the grants MTM PID2019-106715GB-C21 (MICINN) and Europa Excelencia EUR2019- 103821 (MCIU). This work is supported in part by the ICMAT–Severo Ochoa grant SEV-2015-0554 and the CSIC grant 20205CEX001., Preprint

In this article, we investigate Reeb dynamics on bm-contact manifolds, previously introduced in [37], which are contact away from a hypersurface Zbut satisfy certain transversality conditions on Z. The study of these contact structures is motivated by that of contact manifolds with boundary. The search of periodic Reeb orbits on those manifolds thereby starts with a generalization of the well-known Weinstein conjecture. Contrary to the initial expectations, examples of compact bm-contact manifolds without periodic Reeb orbits outside Zare provided. Furthermore, we prove that in dimension 3, there are always infinitely many periodic orbits on the critical set if it is compact. We prove that traps for the bm-Reeb flow exist in any dimension. This investigation goes hand-in-hand with the Weinstein conjecture on non-compact manifolds having compact ends of convex, Postprint (published version)

In this article we give a universal model for geometric quantization associated to a real polarization given by an integrable system with non-degenerate singularities. This universal model goes one step further than the cotangent models in [13] by both considering singular orbits and adding to the cotangent models a model for the prequantum line bundle. These singularities are generic in the sense that are given by Morse-type functions and include elliptic, hyperbolic and focus-focus singularities. Examples of systems admitting such singularities are toric, semitoric and almost toric manifolds, as well as physical systems such as the coupling of harmonic oscillators, the spherical pendulum or the reduction of the Euler’s equations of the rigid body on T *(SO(3)) to a sphere. Our geometric quantization formulation coincides with the models given in [11] and [21] away from the singularities and corrects former models for hyperbolic and focus-focus singularities cancelling out the infinite dimensional contributions obtained by former approaches. The geometric quantization models provided here match the classical physical methods for mechanical systems such as the spherical pendulum as presented in [4]. Our cotangent models obey a local-to-global principle and can be glued to determine the geometric quantization of the global systems even if the global symplectic classification of the systems is not known in general., Peer Reviewed, Postprint (published version)

In this thesis, we study the application of symplectic geometry, regular and singular, to symplectic dynamical systems.We start with a motivating case: the relation between symplectic foliations and global transverse Poincaré sections, showing that meaningful dynamical information can be gleaned by simple observations on the geometry of the phase space - in this case, the existence of a symplectic foliation on a hypersurface of the phase space. We then go on study dynamical systems of particular importance in geometry - those given by a group action on a manifold. In particular, we consider a singular symplectic manifold (specifically, a manifold equipped with a symplectic form which blows up in a controlled manner on a hypersurface of that manifold, namely, a b-symplectic form) with a group action preserving the geometry and give a b-symplectic slice theorem which provides an equivariant normal form of the b-symplectic form in the neighbourhood of an orbit. Particular examples of b-symplectic group symmetries are then explored: those given by the cotangent lift of group translation on so-called b-Lie groups. The second part of this thesis focuses on symplectic and b-symplectic dynamical systems coming from celestial mechanics. In particular, the separatrix map of the stable and unstable manifolds of the fixed point at infinity of the planar circular restricted three-body problem is examined and an estimate of the width of the stochastic layer is given - that is the existence of a K.A.M. torus which acts as a boundary to bounded motions is proved. Due to the delicate nature of the problem - namely issues coming from the parabolic nature of the fixed point and exponentially small nature of the splitting, careful control of the errors of the separatrix map is paramount. This is achieved by employing geometric methods, namely, by taking full advantage of generating functions which exist by virtue of the symplectic nature of the system. Finally, motivated by the importa, En esta tesis, estudiamos una aplicación de la geometría simpléctica, regular y singular, a sistemas dinámicos simplécticos. Comenzamos con un caso motivador: la relación entre foliaciones simplécticas y secciones transversales de Poincaré globales, que muestra que se puede obtener información significativa de la dinámica mediante simples observaciones sobre la geometría del espacio de fase, en este caso, la existencia de una foliación simpléctica en una hipersuperficie del espacio de fase. Luego continuamos estudiando sistemas dinámicos de particular importancia en geometría, dados por una acción de grupo sobre una variedad. En particular, consideramos una variedad simpléctica singular (específicamente, una variedad equipada con una forma simpléctica que explota de manera controlada en una hipersuperficie de esa variedad, es decir, una forma $b$-simpléctica ) con una acción de grupo que preserva la geometría y damos un teorema de rebanada $b$-simpléctico que proporciona una forma normal equivariante de la forma b-simpléctica en una vecindad de una órbita. Luego se exploran ejemplos particulares de simetrías de grupo b-simplécticas: las dadas por el levantamiento cotangente de la translación (grupal) en los llamados b-grupos de Lie. La segunda parte de esta tesis se enfoca en sistemas dinámicos simplécticos y b-simplécticos provenientes de la mecánica celeste. En particular, se examina el "separatrix map" de variedades estables e inestables del punto fijo al infinito del problema de los tres cuerpos restringido circular plano, y se da una estimación del ancho de la capa estocástica, es decir, la existencia de un toro K.A.M. que actúa como frontera para movimientos acotados. Debido a la naturaleza delicada del problema, es decir, los problemas que provienen de la naturaleza parabólica del punto fijo y la naturaleza exponencialmente pequeña de la separación, el control cuidadoso de los errores del "separatrix map" es primordial. Esto se logra empleando métodos geométr, Postprint (published version)

We study the formal geometric quantization of bm-symplectic manifolds equipped with Hamiltonian actions of a torus T with nonzero leading modular weight. The resulting virtual T-modules are finite dimensional when m is odd, as in [4]; when m is even, these virtual modules are not finite dimensional, and we compute the asymptotics of the representations for large weight., Peer Reviewed, Preprint

In this article, motivated by the study of symplectic structures on manifolds with bound-ary and the systematic study ofb-symplectic manifolds started in [10], we prove a slice theorem forLie group actions onb-symplectic manifolds, Eva Miranda is supported by the Catalan institution for Research and Advanced Studies via an ICREA AcademiaPrize 2016. Roisin Braddell is supported by a predoctoral grant from UPC with the ICREA Academia project of EvaMiranda. Roisin Braddell and Eva Miranda and partially supported by the grants reference number MTM2015-69135-P (MINECO/FEDER), PID2019-103849GB-I00 / AEI / 10.13039/501100011033 and reference number 2017SGR932(AGAUR). We acknowledge support from the Fondation Sciences Math ́ematiques de Paris via the Chaire d’Excellenceof Eva Miranda supported by a public grant overseen by the French National Research Agency (ANR) as part of the”Investissements d’Avenir” program (reference: ANR-10-LABX-0098) to finance a research stay of Roisin Braddell anda research visit of Anna Kiesenhofer in Paris to start this proje, Preprint

In this article we generalize a theorem by Palais on the rigidity ofcompact group actions to cotangent lifts. We use this result to prove rigidity forintegrable systems on symplectic manifolds including sytems with degeneratesingularities which are invariant under a torus action., Preprint

In this article we introduce and analyze in detail singular contact structures, with an emphasis on $b^m$-contact structures, which are tangent to a given smooth hypersurface $Z$ and satisfy certain transversality conditions. These singular contact structures are determined by the kernel of non-smooth differential forms, called $b^m$-contact forms, having an associated critical hypersurface $Z$. We provide several constructions, prove local normal forms, and study the induced structure on the critical hypersurface. The topology of those are related to smooth contact structures through a desingularization technique. The problem of existence of $b^m$-contact structures on a given manifold is also tackled in this paper. We prove that a connected component of a convex hypersurface of a contact manifold can be realized as a connected component of the critical set of a $b^m$-contact structure., In this article we introduce and analyze in detail singular contact structures, with an emphasis on $b^m$-contact structures, which are tangent to a given smooth hypersurface $Z$ and satisfy certain transversality conditions. These singular contact structures are determined by the kernel of non-smooth differential forms, called $b^m$-contact forms, having an associated critical hypersurface $Z$. We provide several constructions, prove local normal forms, and study the induced structure on the critical hypersurface. The topology of those are related to smooth contact structures through a desingularization technique. The problem of existence of $b^m$-contact structures on a given manifold is also tackled in this paper. We prove that a connected component of a convex hypersurface of a contact manifold can be realized as a connected component of the critical set of a $b^m$-contact structure. In particular, in the $3$-dimensional case, this construction yields the existence of a generic set of surfaces $Z$ such that the pair $(M,Z)$ is a $b^{2k}$-contact manifold and $Z$ is its critical hypersurface. {As a consequence of the desingularization techniques in \cite{gmw1}, we prove the existence of folded contact forms on any almost contact manifold.}, Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA AcademiaPrize 2016. C ́edric Oms is supported by an AFR-Ph.D. grant ofFNR - Luxembourg National Research Fund. Eva Mi-randa and C ́edric Oms are partially supported by the grants reference number MTM2015-69135-P (MINECO/FEDER)and reference number 2017SGR932 (AGAUR). Eva Miranda was supported by aChaire d’Excellenceof theFondationSciences Math ́ematiques de Pariswhen this project started and this work has been supported bya public grant overseenby the French National Research Agency (ANR) as part of the“Investissements d’Avenir”program (reference: ANR-10-LABX-0098). This material is based upon work supported by the National Science Foundation under Grant No.DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, Califor-nia, during the Fall 2018 semester, Peer Reviewed, Preprint

This thesis explores classification and perturbation problems for group actions on a class of Poisson manifolds called $b^m$-Poisson manifolds. $b^m$-Poisson manifolds are manifolds which are symplectic away from a hypersurface along which they satisfy some transversality properties. They often model problems on symplectic manifolds with boundary such as the study of their deformation quantization and celestial mechanics. One of the interesting properties of $b^m$-Poisson manifolds is that their study can be achieved considering the language of $b^m$-forms. That is to say, we can work with forms which are symplectic away from the critical set and admit a smooth extension as a \emph{form over a Lie algebroid} generalizing De Rham forms as form over the standard Lie algebroid of the tangent bundle of the manifold. To consider $b^m$-forms the standard tangent bundle is replaced by the $b^m$-tangent bundle. This thesis starts with the equivariant classification of $b^m$-Poisson structures investigating, in particular, the analogue of Moser's classification theorem for symplectic surfaces and their equivariant analogues. The classification invariants in the case of surfaces are encoded in a cohomology called $b^m$-cohomology which has been deeply studied by \cite{Scott16}. Mazzeo-Melrose type formula for $b^m$-cohomology decomposes it in two pieces which can be read off the De Rham cohomology of both the ambient manifold $M$ and the critical hypersurface. As an outcome of this identification, the Poisson classification of these manifolds is given by the De Rham cohomology of the manifold and the hypersurface. This classification is extended to the equivariant setting if we assume that the singular forms are preserved by the group action of a compact Lie group. These techniques can be extended to the classification of $b^m$-Nambu structures which are also considered in this thesis. Group actions re-appear in the last chapters as integrable systems on these manifolds turn, Esta tesis doctoral explora problemas de clasificación y perturbación para acciones de grupo en una clase particular de variedades de Poisson llamadas variedades de $b^m$-Poisson. Las variedades de $b^m$-Poisson son variedades que son simplécticas fuera de una hipersuperficie en la cual satisfacen ciertas propiedades de transversalidad. A menudo modelan problemas en variedades simplécticas con borde tales como el estudio de la \textcolor{black}{cuantización por deformación} o problemas de mecánica celeste. Una de las propiedades interesantes de las variedades $b^m$-Poisson es que se pueden estudiar usando el lenguage de $b^m$-formas. Es decir, que podemos trabajar con formas que son simplécticas lejos un conjunto crítico y que admiten una extensión suave como forma sobre un algebroide de Lie generalizando formas de De Rham como formas sobre el algebroide de Lie del fibrado tangente de la variedad. Para considerar $b^m$-formas el fibrado tangente estándar debe reemplazarse por el fibrado $b^m$-tangente. Esta tesis empieza con una clasificación equivariante de estructuras $b^m$-Poisson, investigando, en particular, el análogo del teorema de clasificación de Moser para superficies simplécticas y sus análogos equivariantes. La clasificación de invariantes en el caso de superfícies estan codificados en una cohomologia llamada $b^m$-cohomologia que ha sido estudiada en profundiad por \cite{Scott16}. Una fórmula del tipo de Mazzeo-Melrose para la $b^m$-cohomologia descompone en dos partes que pueden interpretarse como las cohomologias de De Rham tanto de la variedad ambiente $M$ como de la hypersuperficie crítica. Como consecuencia de esta identificación, la clasificación de Poisson de estas variedades viene dada por la cohomologia de De Rham de la variedad y de la hipersuperficie. Esta clasificación se extiende al contexto equivariante si asumimos que las formas singulares son preservadas por la acción de un grupo de Lie compacto. Estas técnicas pueden ser extendidas a la, Postprint (published version)

We introduce the notion of b-Lie group as a pair(G, H) where Gis a Lie group and H is a codimension-one Lie subgroup, and study the associated canonical b-symplectic structure on the b-cotangent bundle bT*G together with its reduction theory. Namely, we prove that the Poisson reduction under the cotangent lifted action of H by left translations is globally isomorphic to a product of the minus Lie Poisson structure on h* (where h is the Lie algebra of H) and the canonical b-symplectic structure on bT*(G/H), where G/H is viewed as a one-dimensional b-manifold having as critical hypersurface (in the sense of b-manifolds) the identity element., Preprint

In this article we consider integrable systems on manifolds endowed with singular sym-plectic structures of order one. By singular symplectic structures of order one we mean structureswhich are symplectic away from an hypersurface along which the symplectic volume either goes toinfinity or to zero in a transversal way (singularity of order one) resulting either in ab-symplecticform or a folded symplectic form. The hypersurface where the form degenerates is called critical set.In this article we give a new impulse to the investigation of action-angle coordinates for these struc-tures initiated in [34] and [33] by proving an action-angle theorem for folded symplectic integrablesystems. We also establish new cotangent models for these systems and investigate the duality withb-integrable systems via desingularization. Global constructions of integrable systems are providedand obstructions for global existence of action-angle coordinates are investigated in both scenarios.The new topological obstructions found emanate from the topology of the critical setZof the singularsymplectic manifold. The existence of these obstructions in turn implies the existence of singularitiesfor the integrable system onZ., Preprint

In this article we generalize a theorem by Palais on the rigidity of compact group actions to cotangent lifts. We use this result to prove rigidity for integrable systems on symplectic manifolds including systems with degenerate singularities which are invariant under a torus action, Peer Reviewed, Preprint