Paula Balseiro (UFF, Brazil)

Title:  On conserved measures for nonholonomic systems with symmetries [slides]
Abstract: One of the principal differences between Hamiltonian systems—described on symplectic manifolds—and nonholonomic systems is the existence of an invariant volume for the dynamics. However, certain nonholonomic systems also admit an invariant measure. In this talk, we consider nonholonomic systems reduced by the action of a Lie group of symmetries. Under suitable assumptions, we explore the existence of an invariant volume for the underlying geometric structure. This viewpoint provides a geometric insight into the problem; moreover, as a direct consequence, this volume is preserved by the reduced nonholonomic dynamics.
This is joint work with J.C. Marrero and N. Sansonetto.

Angel Ballesteros (University of Burgos, Spain)

Title: From quantum reference frames to Poisson-Lie groups 

Abstract: The necessity of considering “quantum reference frames”, namely reference frames associated to dynamical material systems obeying the laws of
quantum theory, has been recently stressed both from quantum information and quantum gravity perspectives. In this talk recent results providing a novel group-
theoretical framework for such quantum reference frames will be presented. In particular, we demonstrate a correspondence between certain quantum
reference frame transformations and transformations generated by a quantum group deformation of the centrally extended Galilei group. In this approach,
since Poisson-Lie groups are just the semiclassical counterpart of quantum groups, the classification of Poisson-Lie structures on the Galilei group
becomes essential in order to identify and construct the appropriate quantum Galilei group. Moreover, the relativistic generalization of this result can be envisaged by considering a suitable Poisson-Lie structure on the Poincaré group and by constructing its associated quantum group deformation.

María Barbero (Technical University of Madrid, Spain) 

Title: Discretization maps in optimal control theory {slides]

Abstract: We have recently introduced in the literature a generalization of retraction maps called discretization maps. They are useful to obtain symplectic numerical integrators by adapting the continuous problem to the discretization rule rather than viceversa, as typically done in numerical integration. Consequently, the manifold where the equations of motion live are discretized, in stead of discretizing the equations.

We will describe how discretization maps can be used to define numerical integrators for optimal control problems so that some structure is preserved.

Alessandro Bravetti (University of Camerino, Italy)

Title: A Geometric Perspective on Asymmetric Relaxation Dynamics [slides]

Abstract: Asymmetric relaxation toward equilibrium is a recently-identified phenomenon that has attracted increasing attention in the study of thermodynamic systems. In this talk, we introduce a geometric approach based on gradient flows on appropriate manifolds to describe relaxation dynamics. This framework allows us to formulate a general criterion for determining the emergence of asymmetry in the approach to equilibrium. We conclude by outlining future perspectives and potential extensions.

Alejandro Cabrera (UFRJ, Brazil)

Title: Local symplectic groupoids, multiplication, and Poisson integrators [slides]

Abstract: This talk is based on joint work in progress with D. Iglesias and J.C. Marrero. First, we shall overview the role of local symplectic groupoids in the
construction of Poisson integrators, following O. Cosserat, as well as recent joint work with D. Martin de Diego and M. Vaquero about using approximate
geometric data. We observe that only the "strict bi-realization" data is used in these approaches, but not groupoid multiplication. We then explain how multiplication "m" can be incorporated into this construction and, also, how generating functions for m can be incorporated yielding a practical numerical algorithm. Finally, we show some illustrative examples and discuss future directions.

Alberto Ibort (University Carlos III  of Madrid, Spain)   

Title: The multisymplectic formalism for field theories revisited
Abstract: The multisymplectic formalism for field theories, so relevant in the description of first order Hamiltonian field theories, will be revisited focusing the attention on its extension to higher order field theories.  A novel construction that would provide the dual version of the infinite jet bundle will be presented and discussed.

Eduardo García Río (University of Santiago de Compostela)

Title: Ricci solitons on Lie groups [slides]

Abstract: Ricci solitons, being self-similar solutions of the Ricci flow have received special attention during the last years. Although a complete description
of such metris is far from being complete, they are quite well-understood in the homogeneous setting. For instance, Riemannian Ricci solitons on Lie groups
are homothetic to algebraic ones.
The Lorentzian situation is much richer, allowing the existence of many Ricci solitons without Riemannian counterpart. Hence our main purpose is to present some classification results for Ricci solitons on low-dimensional Lorentzian Lie groups. We show that any four-dimensional Lie group admits a Ricci soliton left-
invariant Lorentzian metric. As a consequence some non-Einstein compact Lorentzian Ricci solitons are constructed in dimensions three and four.
(This is joint work with Rosalía Rodríguez-Gigirey and Ramón Vázquez- Lorenzo)

Francoise Gay-Balmaz (Nanyang Technological University, Singapur)

Title: General Relativistic Fluids: Reduction by Symmetries and Junction Conditions [slides]

Abstract: We introduce a new Lagrangian variational framework for general relativistic fluids, allowing symmetry reduction in the relativistic context. By
including Gibbons-Hawking-York (GHY) boundary terms, this approach naturally produces the Israel-Darmois junction conditions, connecting the fluid’s
internal dynamics to the gravitational field in the exterior spacetime.

Katarzyna Grabowska (University of Warsaw, Poland)

Title: Affine Dirac aglebroids in nonholonomic mechanics [slides]
Abstract: The concept of a Dirac algebroid, a linear almost Dirac structure on a vector bundle, was introduced to generate phase equations for mechanical
systems with linear nonholonomic constraints. It was successfully applied to systems described by the so-called mechanical Lagrangian or Hamiltonian, as well as systems with magnetic-like or gyroscopic potentials. In contrast to other structures reported in the literature in this context, the Dirac algebroid we use is built only from constraints and canonical geometric structures of the underlying bundles and is universal, in the sense that it is independent of the particular Hamiltonian or Lagrangian. This approach can be readily generalized to affine nonholonomic constraints via the concept of an affine Dirac algebroid. During the talk, we will recall the main ideas and provide examples.

References
[1] Grabowska, K., and Grabowski, J. Dirac algebroids in Lagrangian and Hamiltonian mechanics, Journal of Geometry and Physics, 61 (2011), 2233–2253.
[2] García-Naranjo, L. C., Marrero, J. C., de Diego, D. M., and Valdés, E. P. P. Almost-Poisson brackets for nonholonomic systems with gyroscopic terms and Hamiltonisation, J Nonlinear Sci 32 (2024)
[3] Grabowska K, Borczyńska M., Majsak J., and Sobczak, T. Dirac structures in. nonholonomic mechanics, arXiv:2504.18853

Janusz Grabowski (IMPAN, Poland)

Sergio Grillo (Instituto Balseiro, Bariloche, Argentina)

Title: Variational reduction of homogeneous presymplectic Hamiltonian systems [slides]

Abstract: Given a homogeneous (symplectic) Hamiltonian system on a manifold $M$, with respect to a principal action $\mathbb{R}^{+}\times M\rightarrow M$, we have shown in a previous paper that an associated reduced system can be defined (unless locally), and it is given by a contact Hamiltonian system on $M/\mathbb{R}^{+}$. Moreover, we have written a related reconstruction equation and solved it up to quadratures. In this talk we extend our study to the case of homogeneous pre-symplectic Hamiltonian systems, but focusing on the variational formulation of them. We show that (unless locally) a reduced variational principle can be associated to such systems, in such a way that the trajectories of the original system can be reconstructed from the critical points of the mentioned reduced principle. As a by-product, we obtain a variational formulation for all the contact Hamiltonian systems, which offers an alternative to the so-called Herglotz variational principle.

Manuel de León (ICMAT-CSIC and Real academia de Ciencias de España, Spain)

Giuseppe Marmo (University of Napoli)

Title: Differential Geometry of Quantum Evolution [slides]

Abstract: The description of any physical system requires the identification of: Observables, say O; States, say S; A probability map pairing observables and states with values in probability measures on the real line; Evolution equations of motion; A composition/decomposition rule for composing/decomposing systems. 

In this talk we shall consider as primary choice the evolution on the manifold of observables, i.e., we take a dynamical point of view rather than a kinematical one. The additional structures which are required on observables or states because of the physical interpretation shall be compatible with a given dynamics. This statement is usually implemented  by requiring that the Lie derivative, with respect to the vector field describing the evolution ,of the  tensor fields describing the additional geometrical structures identically vanishes. This ideology is fully developed in the book Geometry from Dynamics, Classical and  Quantum, there Quantum Mechanics is considered in the Hilbert Space picture and the evolution  is given by a vector field on the manifold of states. 

Here we consider Quantum Mechanics in the C*-algebraic setting, evolution being described by a vector field on the manifold of observables. We will show that alternative structures of Lie-Jordan  algebras on the manifold of observables are possible. This would be the analog (Dirac’s Analogy  Principle)of alternative compatible Poisson structures invariant under Classical Evolution. 

Reference: J.F.Carinena, A.Ibort, G.Marmo, G.Morandi. Geometry from Dynamics,Classical and Quantum. Springer, 2015

David Martín de Diego  (ICMAT, Spain)

Title: The invariant inverse problem and EDS on Lie algebroids [slides]

Abstract: Given an invariant system of second-order ODEs on a Lie group, when does an invariant Lagrangian exist? This question is the so-called «invariant»; inverse problem. In this talk, we discuss some aspects of the problem, and we explain how it fits within the context of exterior differential systems (EDS), once we have extended EDS theory from manifolds (and their tangent bundles) to Lie algebroids.

Antonio de Nicola (University of Salerno, Italy)

Title: Darboux-Lie Derivatives: a unified calculus for G-structures [slides]

Abstract: In the theory of G-structures on manifolds, geometric structures are often described in terms of gauge equivalence classes of soldering forms.
However, a comprehensive calculus of derivatives for such forms and their gauge transformations has been lacking. In this talk, based on joint work with
Ivan Yudin, I will introduce the concept of the Darboux-Lie derivative, a new tool designed to fill this gap. 

We define the Darboux-Lie derivative for fiber bundle maps from natural bundles to associated fiber bundles. This construction generalizes the metamathematical notion of a derivative, utilizing the Trautman lift as a very general framework. I will demonstrate how this single operator unifies various
classical concepts, showing that the standard Lie derivative, the covariant derivative, and the classical Darboux derivative for group valued maps are all
specific instances of the Darboux-Lie derivative.
Key properties of this new derivative will be discussed, including:

• Its behavior on products, tensor products, and compositions (Leibniz and chain rules).
• Its expression in terms of flows.
• The derivation of a Cartan-type magic formula for the covariant Darboux-Lie derivative.

Finally, I will briefly outline applications to G-structures, specifically how this derivative characterizes infinitesimal automorphisms and torsion-free conditions.

Narciso Román Roy (Technical University of Catalonia, Poland)

Title: Multicontact field theories: Foundations, symmetries and perspectives [slides]

Abstract: A new geometric framework—based on contact and multisymplecticgeometry  and known as multicontact structure— has recently been developed
to model action-dependent field theories. The purpose of this talk is, first, to review the foundations of multicontact geometry and its use in formulating the
Lagrangian and Hamiltonian descriptions of such theories.  Second, we examine their non-conservative features, introducing precise notions ofsymmetries and the  corresponding dissipation laws,  and establishing an adapted version of Noether’s theorem.  Finally, we outline several physical applications and discuss current perspectives and future developments in the field.

References:

1. M. de León, J. Gaset, M. C. Muñoz-Lecanda, X. Rivas, and N. Román- Roy. Multicontact formulation for non-conservative field theories, J. Phys. A: Math.
Theor. 56(2) (2023) 025201. DOI: 10.1088/1751-8121/acb575.
2. M. de León, J. Gaset, M. C. Muñoz-Lecanda, X. Rivas, and N. Román-Roy. Practical introduction to action-dependent field theories, Fortschr. Phys. 7(5)
(2025) e70000. DOI: 10.1002/prop.70000.
3. X. Rivas, N. Román-Roy, and B.M. Zawora. Symmetries and Noether’s theorem for multicontact field theories, Lett. Math. Phys. 115 (2025) 108. DOI:
10.1007/s11005-025-01995-0.
4. M. de León, R. Izquierdo-López, and X Rivas. Brackets in multicontact geometry and multisymplectization, arXiv:2505.13224 (2025).

Arjan van der Schaft (University of Groningen, Netherlands)

Title: On the Geometry and Dynamics of Engineering Systems [slides]

Abstract: Mechanics has always provided an immensely fruitful playing ground for the development of mathematics; from Hamiltonian dynamics, variational
principles, to geometric mechanics. The aim of this talk is to indicate that, next to the mechanical system case, there is a vast area of other physical systems
that, perhaps equally well, gives rise to interesting dynamical and geometric developments. In particular, we will consider classes of systems originating from
engineering. Complex engineering systems are often composed of subsystems stemming from different physical domains (mechanical, electrical, chemical,
etc.). This necessitates the consideration of a unified framework for the modeling and analysis of multi-physics systems, and has given rise to the
geometric theory of port-Hamiltonian systems. While the geometry of mechanical systems is, loosely speaking, based on the geometry of the cotangent bundle of the configuration manifold (with extensions to, e.g., symplectic and Poisson manifolds), the basic geometry of port-Hamiltonian systems is provided by the notion of a Dirac structure. Differently from the mechanical case, this Dirac structure is also determined by the interconnection structure of the system, as exemplified by Kirchhoff’s current and voltage laws.
Furthermore, port-Hamiltonian systems theory admits the inclusion of energy-dissipation (which is essential to most engineering and natural systems), algebraic constraints, and interaction with the environment (through inputs and outputs). Special emphasis in this talk will be given to relaxation systems, which are port-Hamiltonian systems with only one type of energy storage (and hence no oscillatory behavior). It will be shown that such systems admit an alternative geometric description as gradient systems, where the Riemannian metric is given by the inverse of the Hessian matrix of the energy function (Hamiltonian).
Furthermore the dissipation potential qualifies as a candidate Lyapunov function, different from the Hamiltonian. This extends the Brayton-Moser formulation and analysis of nonlinear electrical networks, and provides interesting links to convex analysis. As an appealing example, we will also discuss continuous Hopfield neural networks from this perspective.