The conference will consist in two courses given by renowned researchers, some plenary talks, several contributed short talks, and a poster session.
Title: An introductory course on geometric field theories: the multisymplectic setting
Abstract:This course provides an introduction to the multisymplectic formulation of first-order classical field theories. Multisymplectic geometry offers the most general and powerful framework for a covariant description of such theories. The course begins with a concise overview of geometric field theories and introduces the fundamental concepts of multisymplectic manifolds, together with the geometry of first-order jet bundles and bundles of differential forms that form the mathematical setting of the multisymplectic approach.
Within this framework, the Lagrangian formalism for first-order classical field theories is developed, covering both regular and singular cases, and the corresponding Hamiltonian formalism is constructed for regular and almost-regular theories. The course also addresses the basic notions of symmetries and conservation laws, and illustrates the general theory through a selection of relevant examples drawn from modern physics.
Title: Geometry of infinite-dimensional manifolds and applications to integrable systems
Abstract:This lecture is an introduction to geometric structures on infinite-dimensional manifolds, with applications to the study of integrable systems. We will start with basic notions of manifolds and fiber bundles with model spaces Hilbert, Banach or Fréchet spaces, and then explore geometric structures on infinite-dimensional manifolds, in particular symplectic and Poisson structures. We will show that the Leibniz rule for a Poisson bracket on a Banach manifold does not imply the existence of a Poisson tensor. The existence of a Hamiltonian vector field on a Banach manifold endowed with a Poisson tensor is also not guaranteed. We will review possible definitions of Poisson structures in the infinite-dimensional context that are general enough to include non-trivial examples, avoid possible pitfalls and include all weak symplectic Banach manifolds. Examples of some Banach Poisson-Lie groups and their homogeneous spaces related to the Korteweg-de-Vries hierarchy will be presented. In particular, the construction of a Banach-Poisson-Lie group structure on the unitary group of a Hilbert space will serve as a guide for other unitary groups, like the restricted unitary group. We will show that the restricted Grassmannian inherite a Bruhat-Poisson structure from the restricted unitary group, and that the action of a triangular Banach Lie group on it by "dressing transformations" is a Poisson map which generates the Korteweg-de Vries hierarchy.
Title: Homogeneous Boundaries of Geometric Structures
Abstract:Under appropriate homogeneity conditions, a hypersurface in a symplectic manifold inherits from the ambient a contact or a cosymplectic structure. There are similar statements for Poisson manifolds as well as for complex manifolds. Using ideas from the homogeneous symplectic approach to contact geometry, we present a very general theorem putting all these statements under the same umbrella. This also allows generalizations, e.g., to Kahler Geometry, Dirac Geometry, Generalized Complex Geometry, etc. This is joint work with Alfonso Tortorella.
Title: Hybrid quantum-classical field theory: classical gravity with quantum matter sources
Abstract:Einstein's General Relativity can be formulated as a Hamiltonian field theory with constraints for globally hyperbolic spacetimes using the Poisson structures of the field's phase spaces (gravitational and matter fields). In this talk, we will summarize some recent works of our group where this classical geometrodynamic theory is generalized to include quantum matter fields as sources for the classical gravitational field. In this way, the classical gravitational field and the quantum matter influence each other's evolution, defining a coupled hybrid field theory where the gravitational and quantum degrees of freedom have their dynamics inextricably coupled. The most remarkable physical feature of the construction is the description of a dynamical backreaction of quantum matter on geometry and vice versa.
Title: Weinstein's category for shifted symplectic structures via groupoids
Abstract:Under the motto that “everything is Lagrangian”, Alan Weinstein proposed a category whose objects are symplectic manifolds and whose morphisms are Lagrangian correspondences. As Poisson geometry has developed over the past decades—motivated largely by classical mechanics—additional structures such as Poisson, Dirac, and Courant have emerged. Safronov, drawing on the shifted symplectic structures of Pantev–Toën–Vaquié–Vezzozi (PTVV), places these geometric structures into a unified framework by viewing them as symplectic structures of various shifts.
The quantization of 1-shifted symplectic geometry, as developed through Meinrenken’s work and the Freed–Hopkins–Teleman theorem, naturally takes values in twisted K-theory and K-homology, and in particular in the Verlinde ring. In this sense, shifted symplectic geometry provides not only a conceptual framework for Poisson-type structures, but also a geometric counterpart to K-theoretic and representation-theoretic invariants. Calaque-Haugseng-Scheinbauer constructed a TFT with target a version of such Weinstein category for shifted symplectic structures using higher derived stacks in the sense of Toën–Vezzosi. While the language of stacks is intrinsic, it remains rather implicit for differential geometers and therefore difficult to use for concrete calculations. Pridham approaches higher derived stacks via presentations by groupoids, a setting far more familiar to researchers in differential geometry, dynamical systems, and noncommutative geometry.
Inspired by Pridham’s approach, we work towards a Weinstein category for shifted symplectic structures using higher derived Lie groupoids. However, the topology we use for derived manifolds differs from that of Toën–Vezzosi and Pridham: we work with fibrations admitting local sections, which allow us to treat the odd line and cotangent groupoids—key examples in Poisson geometry.
We prove that when intersections are transversal in a higher derived sense, the composition of Lagrangian correspondences remains Lagrangian. As always, there is technical difficulty when intersection is not transversal. Broadly, two methods are available: (1) perturbation, as in the Fukaya category, which yields explicit results but requires delicate analysis; or (2) (fibrant) replacement, as in PTVV, which is conceptual but requires a homotopical framework and typically leaves explicit fibrant replacements to be worked out.
We take the second approach and build an iCFO (incomplete category of fibrant objects) for derived higher Lie groupoids and provide explicit fibrant replacements using a collection of tubular neighborhood theorems: Weinstein’s Lagrangian tubular neighborhood theorem, and the Hoyo–Fernandes version for Lie groupoids via the Crainic–Fernandes–Torres PMCT program. As applications, we use Calaque–Safronov’s trick, then singular symplectic reduction, quasi-symplectic reduction, and Lu–Weinstein reduction, all appear as shifted symplectic derived Lie groupoids.
Title: Non-lorentzian spacetimes
Abstract:I will review the notion of a kinematical group and summarise the classification of spatially isotropic homogeneous kinematical spacetimes obtained in 2018/2019 with Stefan Prohazka and will report on work in progress with Juanma Lorenzo Naveiro on the much more involved classification of coisotropy-one homogeneous kinematical spacetimes.
Title: Singularity theorems from a mathematical perspective
Abstract:The classical singularity theorems of R. Penrose and S. Hawking from the 1960s are beautiful examples of mathematical results in Lorentzian Geometry with wide physical relevance for General Relativity showing that any spacetime with a smooth Lorentzian metric satisfying certain energy conditions and causality assumptions must be geodesically incomplete. Despite their great success these classical theorems still had and have some drawbacks both in their assumptions and conclusions. Focusing on the assumptions side of the picture I will review the classical theorems and some recent mathematical progress allowing us to relax some of the usual demands placed on the metric regularity as well as the classical pointwise energy conditions, which makes the theorems applicable to a wider class of physically relevant situations.
Title: Geodesics, Electrostatics and Solids
Abstract:Based on recent joint work with Christopher Fillmore and Herbert Edelsbrunner I will explain how to beat the world record for the number of closed geodesics of index one on a K3 surface. Such geodesics are related to some conjectures originating in the Physics literature and their construction is related to an open problem in electrostatics posed by Maxwell in 1873.
Title: Stochastic reduction on Lie groups with Cartan-Schouten connections
Abstract:In this work, we develop a geometric framework for stochastic reduction on Lie groups endowed with Cartan–Schouten connections. We first derive explicit expressions for the mean covariant derivative and stochastic parallel transport. Building on these results, we formulate a connection-dependent stochastic variational principle and reduce both the variational principle and the associated stochastic Euler–Lagrange equations. This leads to stochastic Euler–Poincaré equations that explicitly capture the effects of torsion and curvature.
Title: A k-contact description of finite-dimensional jet geometry
Abstract:In this talk, we demonstrate that jet geometry, typically linked to Cartan distributions rather than strictly geometric contact forms, can be understood through k-contact geometry—a recent extension of contact geometry to field theory. We show that the Cartan distribution of any finite-dimensional jet manifold is a k-contact distribution, thus describing jet geometry and its core tools via k-contact geometry. This framework naturally unifies and expands existing techniques, such as providing a local k-contact Hamiltonian description for Lie symmetries. We also establish a generalised k-contact Darboux theorem that characterises k-contact manifolds locally equivalent to jet manifolds. Our approach allows k-contact geometry to be applied to the study of partial differential equations, Lie symmetries, characteristic, Bäcklund transformations, and general reduction schemes, including lambda-symmetries, extensions for reductions by non-local symmetries, and many other mathematical and physical research areas.
Title: Collective perturbations of Hamiltonian systems
Abstract:We consider perturbations of a G-symmetric Hamiltonian H by collective terms of the form f o J, where J is an equivariant momentum map for the G action. In general this breaks the symmetry, however the perturbed dynamics retain symmetry-related features and in particular can be reconstructed in two stages from the symmetry- reduced unperturbed motion. If the perturbations are stochastic and the original system is deterministic, then the perturbed dynamics are given by a stochastic phase acting on the unperturbed system. For example, in the Kepler problem with angular stochastic perturbations, the perturbed dynamics retain deterministic radial dynamics. On a Lie group, we consider a sum of two collective terms, corresponding to left and right translations. One example is a rigid body with spatial torque. Another is a continuum model with applications to image registration, generalising the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework. This work is joint with Archishman Saha and Cristina Stoica.
Title: Deformation of symmetry and persistence of dynamics
Abstract:We use the deformation of symmetry on cotangent bundles from the Euclidean plane to two-dimensional constant-curvature surfaces to investigate the continuation of local dynamics aspects in Hamiltonian systems. For a fixed curvature sign, the curved problem is set up either on the sphere or on the hyperbolic plane, both with radius R=1/epsilon, recovering flat space in the limit as epsilon tends to 0. The symmetry of these spaces is taken into account by using the contraction of Lie algebras from so(3) or so(2,1) to se(2). We use Riemannian exponential coordinates centred at the North pole together with the pull-back of the associated momentum map and the symplectic form. Within this geometric setting we use a local slice construction and prove the persistence from flat to curved spaces of non-degenerate relative equilibria and relative periodic orbits of general cotangent bundle Hamiltonian systems. We apply the resulting framework to the Newtonian n-body problem.
Title: Einstein Gravity as an Action-Dependent Field Theory
Abstract:There exists a beautiful interplay between symplectic and contact geometry, which arises naturally when a physical theory possesses a particular kind of scaling symmetry, known as a dynamical similarity. The invariants of the symmetry generator form an autonomous subsystem of the symplectic theory, and their evolution is described by a contact Hamiltonian. Physically, we identify the symmetry transformation as a rescaling of a degree of freedom associated with a choice of overall system scale. Such a rescaling does not affect the observable physics, and the corresponding degree of freedom is redundant for the description of the dynamical evolution of the observables. Classical Einstein gravity is an example of a theory which possesses a scaling symmetry. We show how the identification of this symmetry allows general relativity to be cast as an action-dependent field theory. Finally, we discuss some interesting consequences in simple cosmological settings.
Title: Homogeneous Darboux and Frobenius theorem on graded manifolds
Abstract:Gradings appear in geometry and physics in a plethora of scenarios, such as vector bundles, higher-order tangent bundles, Lie algebroids or AKSZ sigma models. A natural and simple way to encode a grading on a manifold is a smooth action of the multiplicative monoid of real numbers, known as a homogeneity structure. Using the homogeneity structure, we can easily prove a Frobenius theorem for graded involutive distributions. As a particular case, we recover the Frobenius theorem for \mathbb{N}-graded manifolds due to Bursztyn, Cueca, and Mehta. This is joint work with Janusz Grabowski.
Title: Stackel Lifts, Symplectic-Haantjes Structures, and Integrable Systems
Abstract:We present the Stackel lift, a geometric framework unifying and extending the Riemannian and Lorentzian Eisenhart lifts. Given an n-dimensional separable Hamiltonian, the Stackel lift constructs an integrable system on a higher-dimensional cotangent bundle by lifting the associated Stackel matrix. The connection between Stackel geometry and the Eisenhart programme was not previously made systematic, and the link to Haantjes geometry had not been established.
We prove that Hamiltonian systems arising from momentum-dependent Stackel matrices are endowed with a non-trivial symplectic-Haantjes structure. The Haantjes operators are constructed explicitly from the lifted Stackel data and account for the separability of the lifted system in a coordinate-free manner, connecting the Stackel-Eisenhart programme to the symplectic-Haantjes manifold theory of Tempesta and Tondo.
We illustrate the framework on magnetic systems separable in cylindrical coordinates, described within the Stackel setting via a modified Stackel basis. Explicitly momentum-dependent lifting matrices produce Hamiltonians associated with Finsler and Hamilton geometries, as well as models interpretable as gravitational waves, with potential applications in modified gravity. We also indicate how the same Haantjes geometry governs separation of variables for the generalized Zernike superintegrable hierarchy, where the lift-form property of the Haantjes operators determines which members of the family admit separation coordinates reachable by extended point transformations.
Title: On the Spencer cohomology and integrability of multisymplectic structures
Abstract:Multisymplectic geometry provides a geometric framework to express the equations of motion of classical field theories, in analogy with how symplectic geometry is used in classical mechanics. A multisymplectic form encodes key structures of a given theory, including symmetries, conserved quantities, and the Poisson bracket.
In the study of these structures, the use of adapted (or Darboux) coordinates is of central importance. However, unlike in symplectic geometry, the closedness of a multisymplectic form is not sufficient to guarantee the existence of flat coordinates. Indeed, there are examples of closed forms of constant linear type that are not flat, such as those arising in $G_2$-structures.
These issues can be understood in a unified way through the theory of G-structures. In this talk, I will introduce the basic notions of G-structures and structure tensors. I will then explain how this framework can be used to construct forms whose integrability depends strictly on conditions of order $k$, for arbitrary $k$. Finally, if time permits, I will discuss some possible directions and ideas related to integrability.
Title: A Geometric Approach to Multilocal Observables on Multiconfiguration Spaces
Abstract:In classical field theory, multilocal observables are naturally indexed by configurations of points on a spacetime $M$ and admit two natural products: a fibrewise (Hadamard) product encoding pointwise multiplication of local observables, and a Cauchy product encoding the combination of observations at separate points. Recently, Frabetti, Kravchenko and Ryvkin organized this structure as a Poisson 2-algebra bundle over open (diagonal-free) configuration spaces. We extend their framework to the multiconfiguration spaces $M^n/S_n$ using the equivarlant language of $S_n$-equivariant vector bundles over $M^n$. The main result is that any skew kernel $k : V \boxtimes V \rightarrow I_{\otimes}$ induces a Poisson 2-algebra structure on the associated symmetric algebra bundle, and we illustrate the construction on $M = \mathbb{R}$ with the canonical symplectic fibre $F = Q \oplus Q^*$, recovering the standard Poisson bracket of classical mechanics. The framework is intended as a foundation for a future treatment of distributional sections and a geometric approach to renormalization.
Title: Infinitesimal symmetries in Newton-Cartan gravity
Abstract:Galilean spacetimes [1] provide a natural geometric framework for the study of non-relativistic physics and have recently attracted renewed attention in connection with geometric mechanics, Newton–Cartan theory, and non-relativistic gravity [2]. In this contribution, we study the geometric structure of Galilean spacetimes admitting conformally Leibnizian vector fields, namely vector fields whose flows generate conformal transformations of the underlying Leibnizian structure [3]. Special emphasis is placed on the interplay between the existence of infinitesimal symmetries and the intrinsic geometry of the spacetime, inducing particular local and global geometric properties.
We establish characterization and decomposition results under several geometric assumptions, focusing on classes of torse-forming, torqued, and concircular vector fields. These conditions lead naturally to decomposition into product structures and provide geometric criteria for identifying Galilean generalized Robertson–Walker and Galilean twisted space-times [4]. Our results highlight the role of conformal Leibnizian symmetries as a unifying mechanism linking symmetry, curvature, and global structure in non-relativistic geometry, thereby contributing to the broader understanding of Galilean geometric models arising in mathematical physics.
[1] A. N. Bernal and M. Sánchez, Leibnizian, Galilean and Newtonian structures of space-time. J. Math. Phys. 44, 1129–1149 (2003).
[2] J. Figueroa-O’Farrill, Non-lorentzian spacetimes. Differ. Geom. Appl. 82, 101894 (2022).
[3] D. de la Fuente, R. M. Rubio and J. Torrente-Teruel, Geometric structure of Galilean spacetimes admitting a conformally Leibnizian vector field. Phys. Scr. 100, 095001 (2025).
[4] D. de la Fuente, R. M. Rubio and J. Torrente-Teruel, On Twisted Spacetimes: a new class of Galilean cosmological models. Adv. Nonlinear Stud. 26, 1–21 (2026).
Title: Harmonic maps into principal bundles and generalized magnetic mappings
Abstract:In this talk, we introduce a geometric framework for studying harmonic maps into principal bundles endowed with Kaluza-Klein metrics. This leads naturally to the notion of generalized magnetic maps, which extend classical magnetic geodesics and Wong’s equations from particles to higher-dimensional objects interacting with arbitrary gauge fields. We show that every Kaluza-Klein metric is completely determined by a principal connection, a metric on the adjoint bundle, and a metric on the base manifold. Using this structure, we derive the Euler–Lagrange equations for harmonic maps into principal bundles and characterize their horizontal and vertical components. A central result of the work is that generalized magnetic maps arise as projections of Kaluza-Klein harmonic maps and that their moduli space is obtained as a quotient by a natural gauge symmetry. We further establish a gauge variation formula, a harmonic gauge-fixing equation, and an existence theorem for generalized magnetic maps. Finally, we investigate the influence of the geometry of the fibers on these equations and construct explicit families of examples based on the complex and quaternionic Hopf fibrations (S^3 \to S^2) and (S^7 \to S^4). Among these families, we identify the unique uncharged solutions, namely the standard Clifford torus and the standard spherical harmonic immersion of (S^3 \times S^3) into (S^7).
DOI: 10.1016/j.geomphys.2026.105780.
Title: Multisymplectic manifolds with boundaries
Abstract:In this talk, I will present a new formalism for studying the field theories on manifolds with boundary. Based on the ideas of relative cohomology [Margalef-Bentabol & Villaseñor, 2021], I will extend the definition of multisymplectic structures to manifolds with boundary. We will see how this structure reproduces the field equations of variational principles with boundary. I will explain how to generalize the observables, the graded Poisson brackets, and the conserved charges to manifolds with boundary. Moreover, I will present the Lagrangian formalism, deriving the Poincaré–Cartan form and the Euler–Lagrange equations. Finally, I will illustrate the formalism with several examples.
Title: GenDis: Transformations Between Lagrangians of Different Order
Abstract:Over the last 25 years, numerous modifications of General Relativity have been proposed to address outstanding problems in cosmology. Disformal transformations and their generalizations (henceforth GenDis) are emerging as powerful tools for the classification of such modified gravity theories. GenDis are characterized by relating Lagrangians of different order. Understanding their properties would enable the systematic construction of higher-order gravity theories while controlling physically relevant features such as symmetries, global stability, and the number of propagating degrees of freedom.
In this talk, I will present the main geometric features of GenDis within the mechanical framework. Focusing on the failure of GenDis to preserve the presymplectic structure of the Lagrangian systems they relate, I will provide an intrinsic characterization of this phenomenon and derive sufficient conditions to ensure dynamical equivalence at the level of the corresponding final constraint submanifolds. Remarkably, dynamical equivalence turns out to be sufficient to recover the preservation of the relevant geometric structures in the GenDis framework.
Title: From Broad-integrability to Euler-Jacobi Theorem and back
Abstract:Integrability has been and still is an active field of research in the Hamiltonian framework (both in finite and infinite dimensions). On the other hand its investigation drove less attention outside the Hamiltonian setup, at least until the last decades, when has attracted the attention of the community interested in nonholonomic mechanics and more recently in plasma. Despite being less studied, there are notions of integrability that extend to non-Hamiltonian framework. In this talk I will focus on two notions of non-Hamiltonian integrable systems: Broad and Euler-Jacobi integrability. We first show that the first notion is stronger. We then investigate which possible ’non-evident’ properties one can add to the Euler-Jacobi Theorem to make the dynamics broadly integrable.
https://arxiv.org/abs/2503.21950
https://iopscience.iop.org/article/10.1088/1361-6544/ae341c
Title: The causal structure of the c-boundary of GRW spacetimes
Abstract:We study the structure of the future causal completion $\hat{M}$ of a globally hyperbolic GRW spacetime $\mathbb{R} \times_\alpha S$ using the novel notion of Lorentzian pre-length spaces. As main result, we prove that the future causal completion of a GRW spacetime is a globally hyperbolic pre-length space provided the (future) chronological topology is Hausdorff. To do this, we distinguish two cases in terms of the warping function $\alpha ∶ (a, b) → (0,\infty)$ is the warping function. In both cases, the chronological topology is Hausdorff, either because we can prove it or because we assume it. This allows us to apply the complete range of tools of the CLT (Beem's topology) and Gromov's compactification. This shows that Hausdorffness serves as a unifying force of these seemingly distant concepts.
Title: Symplectic groupoid multiplication and Poisson integrators
Abstract:In this work, we study the role of local symplectic groupoid multiplication in the construction of approximation methods for Hamiltonian flows on the underlying Poisson manifold. We show it can be used to construct a novel multiplicative type of approximating sequences for the relevant Lagrangian bisection. We describe the corresponding theoretical results, their practical implementation into approximation methods on coordinate charts, and illustrative examples. (joint work with Alejandro Cabrera and Juan Carlos Marrero).
Title: Phase-Space Geometry of Qubits: Deformation Quantization, Star Exponentials, and Quantum Teleportation
Abstract:We develop a phase-space formulation of qubit systems based on the coadjoint orbits of SU(2) and the Stratonovich--Weyl correspondence, providing a deformation quantization of the sphere. The resulting star product reproduces the algebra of complexified quaternions, while its antisymmetric part induces the Lie-Poisson structure associated with the KKS symplectic form. Within this framework, quantum dynamics is formulated through star exponentials of Hamiltonian symbols, yielding an explicit representation of the quantum propagator. Several examples illustrate the construction of star exponentials and the induced Poisson geometry. Finally, this geometric phase-space formalism provides a natural framework for describing quantum teleportation as the teleportation of geometric structures encoded in phase space.
Title: Phase space quantization of anisotropic cosmologies: Taub and Kantowski-Sachs models
Abstract:We introduce an explicit construction of the non-diagonal and diagonal Wigner distributions for the homogeneous but anisotropic Taub and Kantowski-Sachs cosmological models within the framework of phase space deformation quantization. Conventional canonical quantization of these models via the Wheeler-DeWitt equation is inherently plagued by factor ordering ambiguities. To circumvent these issues, we employ the totally symmetric Weyl quantization map and the Moyal star product. By means of a canonical separation of the Hamiltonian constraint, we are able to resolve the formal convergence problems typically associated with the star product. Furthermore, to establish a rigorous connection with conventional quantum cosmology, we calculate the standard wave functions directly from the diagonal Wigner distributions, recovering the exact physical states in terms of modified Bessel functions in both cases.
Title: Hybrid quantum-classical master equations and consistent thermodynamics
Abstract:This work studies the statistical mechanics of hybrid quantum-classical systems within a geometric setting. Modelling the hybrid phase space as a Cartesian product of a (classical) symplectic manifold and a (quantum) Kähler manifold, Ehrenfest dynamics can be endowed with a symplectic Hamiltonian structure [J. Phys. A: Math. Theor. 44, (2011)]. Furthermore, Liouville’s evolution for statistical densities can be recast into an infinite hierarchy of differential equations for the quantum moments of increasing order [Eur. Phys. J. Plus 138, (2023)]. A rigorous definition of hybrid entropy [Phys. Rev. E 102, (2020)] allows us to obtain an effective theory by truncating the infinite hierarchy via a maximum entropy criterion, yielding a closed system of equations for a subset of moments. In this work, we show that neither the full system of coupled equations nor its truncation preserves the hybrid canonical ensemble (HCE), thereby yielding thermodynamic inconsistency. However, we demonstrate that the dynamical deviation of the HCE is suppressed by a factor of ħ compared to other relevant observables, establishing the validity of the effective theory as a physically meaningful approximation. Finally, we discuss how a C*-algebraic Koopman formalism can resolve this issue and lead to a strictly consistent hybrid thermodynamics.
Title: Reduction of the Time-Dependent Elroy's Beanie
Abstract:In a work in progress, we develop a new framework for the reduction of time-dependent Hamiltonian systems that generalizes previous approaches [1]. The proposed method combines the reduction of presymplectic structures of corank 1 and 2, leading to a broader reduction procedure for time-dependent systems. In this poster, we illustrate the theory by applying it to the reduction of a time-dependent Elroy's Beanie system.
[1] I. Lacirasella, J.C. Marrero, and E. Padrón. Reduction of symplectic principal R-bundles. J. Phys. A, 45(32):325202, 29, 2012.
Title: Contact Dynamics and Legendrian submanifolds in infinite dimensions
Abstract:There has recently been a small, but growing, interest in contact dynamics in infinite-dimensional settings. This allows one to define physical systems which are infinite-dimensional but also dissipative in some sense, for example the damped wave equation or the Klein-Gordon equation. It also has some links to multicontact dynamics and non-equilibrium thermodynamics.
In this poster I will outline how contact forms are defined in infinite-dimensions, and give some elementary examples of them. I will then discuss some properties of infinite-dimensional contact manifolds and how they differ from their finite-dimensional cousins. Finally I will discuss some recent results including a class of Legendrian submanifolds, and progress towards a Bambusi-Darboux type theorem for infinite-dimensional contact forms.
Title: Compatible Twisted Multisymplectic Nijenhuis Structures
Abstract:We study the compatibility of twisted multisymplectic Nijenhuis structures on a Lie algebroid $A \to M$, via the correspondence between such structures and Nijenhuis morphisms $J = N + \Omega$ on the shifted exact Courant algebroid $(A \oplus \bigwedge^{p-1}A^*, \Theta = \mu + H)$. We introduce a notion of compatibility for two such morphisms $J = N + \Omega$ and $J' = N'+\Omega'$, and derive the explicit compatibility conditions on $(A,\mu)$ in terms of the geometric data $(\Omega, N, H)$ and $(\Omega', N', H)$.
Title: Applications of $k$-Contact Geometry to Jet Manifolds
Abstract:Jet manifolds provide the geometric framework for the study of partial differential equations and their symmetries. We recall the recently introduced $k$-contact formulation of finite-order jet manifolds, which provides a unified setting for many classical constructions in PDE theory. This poster focuses on applications of the $k$-contact framework to jet bundles introduced by Javier de Lucas, in particular the Hamiltonian description of the characteristics of Lie symmetries and the extension of reduction methods to evolutionary and $\lambda$-symmetries. This is joint work with Javier de Lucas.