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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

For nonnegative even kernels $K$, we consider the $K$-nonlocal perimeter functional acting on sets. Assuming the existence of a foliation of space made of solutions of the associated $K$-nonlocal mean curvature equation in an open set $\Omega\subset\mathbb{R}^n$, we built a calibration for the nonlocal perimeter inside $\Omega\subset\mathbb{R}^n$. The calibrating functional is a nonlocal null-Lagrangian. As a consequence, we conclude the minimality in $\Omega$ of each leaf of the foliation. As an application, we prove the minimality of $K$-nonlocal minimal graphs and that they are the unique minimizers subject to their own exterior data. As a second application of the calibration, we give a simple proof of an important result from the seminal paper of Caffarelli, Roquejoffre, and Savin, stating that minimizers of the fractional perimeter are viscosity solutions., Comment: To appear in Annali di Matematica Pura ed Applicata

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., Robert Cardona was supported by the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Program for Units of Excellence in R&D (MDM-2014-0445) via an FPI grant. R.C. and E.M. are partially supported by Grants MTM2015-69135-P/FEDER, the Spanish Ministry of Science and Innovation PID2019-103849GB-I00/AEI/10.13039/501100011033, and Agència de Gestió d’Ajuts Universitaris i de Recerca Grant 2017SGR932. E.M. is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. D.P.-S. is supported by MICINN Grant MTM PID2019-106715GB-C21 and MCIU Grant Europa Excelencia EUR2019-103821. F.P. is supported by MICINN/FEDER Grants MTM2016-79400-P and PID2019-108936GB-C21. This work was partially supported by ICMAT–Severo Ochoa Grant CEX2019-000904-S.

We develop a new geometric framework suitable for dealing with Hamiltonian field theories with dissipation. To this end we define the notions of $k$-contact structure and $k$-contact Hamiltonian system. This is a generalization of both the contact Hamiltonian systems in mechanics and the $k$-symplectic Hamiltonian systems in field theory. The concepts of symmetries and dissipation laws are introduced and developed. Two relevant examples are analyzed in detail: the damped vibrating string and Burgers' equation., 26 pp. Minor corrections. The bibliography has been updated

Els caleidocicles de Möbius són una família nova de mecanismes amb un grau de llibertat. Presentem cadenes quinemàtiques noves amb propietats similars i busquem casos especials. Donem un argument de plausibilitat per la mobilitat de les cadenes basat en un teorema nou sobre la transposició d'elements adjacents. Finalment, discutim la relació entre els nostres sistemes i el camp emergent de la geometria diferencial discreta. Los caleidociclos de Möbius son una familia nueva de mecanismos con un grado de libertad. Presentamos cadenas quinemáticas nuevas con propiedades similares y buscamos casos especiales. Damos un argumento de plausibilidad para la mobilidad de las cadenas basado en un teorema nuevo sobre la transposición de elementos adyacentes. Finalmente, discutimos la relación entre nuestros sistemas y el campo emergente de la geometría diferencial discreta. Möbius kaleidocycles are a newly discovered family of underconstrained linkages with one degree of freedom. We present many new kinematic chains with similar properties and look for special cases. We give a plausibility argument for the mobility of our chains based on a new theorem on the transposition of adjacent links. Finally, we discuss the relationship between our systems and the emerging field of discrete differential geometry. Outgoing

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.

We construct some ODE solutins for the fractional Yamabe problem in conformal geometry. The fractional curvature, which is a generalization of the usual scalar curvature, is defined from the conformal fractional Laplacian, which is a non-local operator construced on the conformal infinity of a conformally compact Einstein manifold. On one hand, we consider the hyperbolic manifold $\mathbb S^1(L)\times \mathbb R^3$ and study the nonuniqueness of solutions for the fractional Yamabe problem. On the other hand, we look at the existence of radial solutions for the Yamabe problem in Euclidean space with an isolated singularity at the origin. Both equations are fractional order ODE for which new tools need to be developed.. La geometria conforme estudia les transformacions que preserven angles. Això fa que les nocions de curvatura més importants es puguin descriure mitjançant equacions en derivades parcials ellíptiques. El projecte consisteix en aprofondier en aquesta relació, fent servir eines de les EDP per a resoldre problemes geomètrics