Programme

Abstracts

Jinqiao Duan 

Title: A Hamiltonian Framework for  Schrodinger Bridges on the Manifold of Probability Densities

Abstract: The investigation of stochastical dynamical systems often relies on path-wise trajectories within Euclidean space. However, a more profound understanding of complex phenomena—such as critical transitions—emerges when these systems are viewed as evolutions on a manifold of probability densities. This talk explores this geometric perspective on the Schrodinger bridges and discusses a Hamiltonian framework that links calculus of variations, dynamical systems, Riemannian geometry, offering new insights into the "most probable transition " or "most probable transport" behaviours of uncertain systems.

Tudor Ratiu

Title: Stochastic variational principles for systems with advected quantities

Abstract: A stochastic Hamilton variational principle for systems with advected quantities will be presented. The general theorem is discussed for finite dimensional systems. Motivated by this theorem and based on existing work in the literature concerning diffeomorphism groups, an analogous result is obtained showing that the compressible Navier-Stokes equations can be derived from pure stochastic geometric mechanics considerations, without any appeal to thermodynamics.

Qiao Huang

Title: Cartan–Schouten Connections: Geometric Reduction and a Connection-Dependent Variational Principle. 

Abstract: We study the family of Cartan–Schouten connections on Lie groups, parameterized by $\lambda\in[0,1]$, whose geodesics through the identity are one-parameter subgroups. We compute their curvature, torsion, parallel transport, and geodesics, and develop Euler–Poincaré and Lie–Poisson reduction for mechanical systems via these connections, unifying the "minus" and "plus" cases. These inspire us to introduce a connection-dependent variational principle where the Lagrangian is expressed in terms of the parallel-transported velocity, leading to an integro-differential Euler–Lagrange equation that explicitly involves torsion and curvature memory terms. 

Pingyuan Wei

Title: The Stochastic Geometric Hamilton–Jacobi Theorem on Jacobi Structures

Abstract: In recent decades, the Hamilton–Jacobi theorem has been reinterpreted in modern geometric terms and extended to contact systems, LCS systems, and various other contexts. Meanwhile, stochastic versions of Hamilton–Jacobi equations (often referred to as stochastic Hamilton–Jacobi–Bellman equations) have attracted increasing attention as they describe the evolution of optimally controlled systems with random dynamics, and they also serve as useful tools in the study of various classes of stochastic models in probability theory and mathematical physics. In this talk, we first rigorously introduce stochastic Hamiltonian systems on Jacobi manifolds, and then propose a stochastic Hamilton–Jacobi theorem. Part of this talk is based on joint work with Jinqiao Duan (GBU) and Qiao Huang (SEU).

Jianyu Hu

Title: A kernel method for the learning of Wasserstein geometric flows

Abstract: Wasserstein gradient and Hamiltonian flows have emerged as essential tools for modeling complex dynamics in the natural sciences, with applications ranging from partial differential equations (PDEs) and optimal transport to quantum mechanics and information geometry. Despite their significance, the inverse identification of potential functions and interaction kernels underlying these flows remains relatively unexplored. In this work, we tackle this challenge by addressing the inverse problem of simultaneously recovering the potential function and interaction kernel from discretized observations of the density flow. We formulate the problem as an optimization task that minimizes a loss function specifically designed to enforce the underlying variational structure of Wasserstein flows, ensuring consistency with the geometric properties of the density manifold. Our framework employs a kernel-based operator approach using the associated Reproducing Kernel Hilbert Space (RKHS), which provides a closed-form representation of the unknown components. Furthermore, a comprehensive error analysis is conducted, providing convergence rates under adaptive regularization parameters as the temporal and spatial discretization mesh sizes tend to zero. Finally, a stability analysis is presented to bridge the gap between discrete trajectory data and continuous-time flow dynamics for the Wasserstein Hamiltonian flow.

Hongjun Gao

TBA

Álvaro Pelayo 

Title: Integrable Systems: geometry, arithmetic and analysis in interaction.

Abstract: I will give a concise overview of recent progress on symplectic geometry of integrable systems, including analytic, combinatorial and algebraic aspects. I will also mention a new approach  which incorporates the use of p-adic numbers in symplectic geometry and the theory of integrable systems, hence establishing a bridge between arithmetic and symplectic geometry.

Narciso Román Roy

Title: The multisymplectic Lagrangian setting of classical field theories

Abstract: The multisymplectic formulation of classical field theories provides the most general geometric framework for a covariant description of these theories. In particular, the Lagrangian formalism is developed on jet bundles and is formulated using the canonical geometric structures naturally defined on them. In this talk, we present the main features of this framework, ranging from the geometry of jet bundles to the variational principle and the resulting field equations.

Xavi Rivas

Title: Pairs of differential forms in geometric mechanics

Abstract: Many of the geometric structures appearing in mechanics, namely (co)symplectic, contact, locally conformally symplectic, and others, are built from a differential 1-form and a 2-form. This talk presents a general framework for the study of such pairs. The basic invariant of a pair is its class, that is the codimension of the common kernel of the two forms. We classify pairs by their class and show that its parity governs the associated dynamics: an odd class is equivalent to the existence of a Reeb vector field, while an even class corresponds to a Liouville vector field. We then introduce and analyse related geometrical objects, such as the characteristic tensor field, and explain how they encode both the class and these distinguished vector fields. Finally, we study transformations of pairs of forms and characterize when they alter the parity of the class. Throughout, we illustrate the framework with examples from geometric mechanics, including (pre)symplectic, (pre)cosymplectic, (pre)contact, and locally conformally symplectic structures, as well as Hamiltonian systems.

Víctor Jiménez

Title: The natural groupoid structure on symmetries

Abstract: Symmetries play a central role in Hamiltonian mechanics, traditionally encoded by Lie group actions and their associated reduction procedures. However, this paradigm implicitly assumes a global character of the symmetries.

In this work, we propose a groupoid–theoretic approach to symmetries of Hamiltonian systems, aimed at encompassing a broader range of symmetries. Starting from the groupoid of 1–jets of local diffeomorphisms, we introduce several canonical subgroupoids encoding geometric, dynamical, and geometric–dynamical symmetries, both in a pointwise and local sense. A key observation is that, as soon as singularities are present, these symmetry groupoids are generally neither transitive nor Lie groupoids, even in basic examples. This reveals a genuine structural gap between global, local and pointwise symmetries that could look invisible in the classical setting.

To address the lack of smoothness, we replace the notion of associated Lie algebroid by that of the characteristic distribution, which provides the correct infinitesimal object associated with an arbitrary subgroupoid. We show that these distributions admit an explicit description in terms of derivations and connections satisfying natural geometric and Hamiltonian invariance conditions. Their integration yields singular foliations of the phase space, whose leaves classify points that are related by local symmetries.

This talk presents work in progress in collaboration with Prof. Manuel Lainz (CUNEF).

Alberto Ibort

Title: Random perturbations of flows and solutions of the Navier-Stokes equation

Abstract: We show that solutions of the Navier-Stokes equation for an incompressible fluid can be obtained by averaging stochastic perturbations of solutions of Euler's equation.  The average of a Brownian perturbation of a steady solution of the Euler equation defines, for small time, a time-dependent diffeomorphism whose generating vector field satisfies a generalized Navier-Stokes equation.  The deviation of this generalized equation from the standard Navier-Stokes equation is controlled by a viscous residual whose leading term involves a transport--diffusion term.  

We identify a natural class of steady, parallel Euler flows for which this residual vanishes and the perturbed flow satisfies the exact Navier-Stokes equation. These results are placed in the context of recent developments including the stochastic Lagrangian representation of Constantin-Iyer, the SALT formulation of Holm, and regularization-by-noise results of Flandoli and collaborators.

Manuel de León

TBA

Leonardo Colombo

Title: Discrete Dissipative Dynamics on Almost Algebroids 

Abstract: The aim of this talk is to present a relation-based geometric framework for discrete dissipative dynamics on almost algebroids, motivated by the groupoid approach to discrete mechanics. In the symplectic case, discrete Lagrangian mechanics on groupoids is naturally described through source-target matching: a generating object defines a geometric relation, and discrete trajectories are obtained by concatenating consecutive steps. The contact analogue replaces Lagrangian relations in symplectic groupoids by Legendrian relations in contact groupoids.

Starting from this idea, I will explain how one obtains a local almost-algebroid version of the construction. Since a general almost algebroid need not admit an integrating groupoid, the groupoid-level composability condition is replaced by a discrete admissibility relation. A discrete contact Lagrangian then defines a Herglotz update, discrete contact Legendre maps, and a discrete relation on the contact phase space.Discrete trajectories are obtained by concatenating this relation, equivalently by matching the outgoing contact momentum of one step with the incoming contact momentum of the next. This recovers the discrete Herglotz equations and, in the regular tangent-bundle case, the usual contact variational integrators. Its main advantage is that the same formulation remains meaningful for constrained, singular, or almost-algebroid systems, where no discrete phase-space map may be available. 

This is a joint work with Manuel de León. 

Cristina Sardón

TBA

Iván Gutiérrez Sagredo

TBA

Miguel Ángel Berbel

Title: Reconstruction in Lie-Poisson reduction for field theories.

Abstract: Any reduction procedure naturally poses a reconstruction problem: given a solution of the reduced system, can one recover a corresponding solution of the original unreduced problem? While reconstruction is always possible in mechanics, in Euler–Poincaré and Lagrange–Poincaré reduction for field theories, the obstruction to reconstruction is characterised by the curvature of a connection constructed from the reduced solution. Far less is known, however, in the Poisson–covariant formulation of Hamiltonian field theories. In this talk, we present explicit reconstruction conditions for Lie–Poisson systems and extend these results to the case where the symmetry group is a subgroup of the structure group of the configuration space.

Lucía Santamaría-Sanz

Title: Generalized Hamilton Geometry on Cotangent Bundles: Structure, Symmetries, and Quantum-Gravity Applications

Abstract: In this talk, recent advances in the geometry of cotangent bundles with phase-space dependent metrics will be presented, focusing on generalized Hamilton spaces. Starting from a given metric, the main geometric ingredients of the theory, including the Hamiltonian, nonlinear connection, and affine connections, will be constructed in a systematic way. Moreover, symmetries will be analyzed, both in spacetime and momentum space, focusing on the restrictions imposed by demanding consistency with curved spacetime physics. Finally, it will be explained how this framework provides a natural geometric setting for deformed relativistic kinematics and suggests a privileged role for Snyder-type noncommutative spacetimes in quantum-gravity-inspired models.

Rubén Izquierdo

Title: The Jacobi brackets of dissipative field theories 

Abstract: Multicontact geometry has been developed to study action-dependent field theories in a covariant and geometric framework. In this talk, I will present the definition of graded Jacobi brackets in multicontact geometry, emphasizing their connection to brackets in multisymplectic geometry via the so-called process of multisymplectization. This study, together with recent advances in the theory of brackets for classical (non-dissipative) field theories, may be employed to obtain a bracket for dissipative forms of arbitrary degree. This is joint work with Manuel de León and Xavier Rivas. 

Ángel Martínez Muñoz

Title: Dynamics and brackets for a pair of a 1-form and a 2-form

Abstract: We construct a unified geometric framework starting from a manifold equipped with a 1-form and a 2-form under some regularity conditions. The aim of this talk is to systematically study the algebraic and geometric conditions under which these forms induce a well-defined Poisson or Jacobi bracket. We also use this framework to define the relevant dynamics for such pairs of forms and relate them to their associated brackets. 

We will demonstrate how this theory recovers both the classical brackets and the standard dynamics natural to different geometrical structures, including symplectic, contact, locally conformally symplectic, and cosymplectic manifolds.

Paula Alba San Miguel 

Title: Generalized differentiable structure

Abstract: In this talk, we present a method for extending the notion of a differentiable manifold to arbitrary subsets of Rn, as a part of my Phd with Víctor M. Jiménez Morales. Using tools from groupoids and the characteristic distribution, we show how these sets can be decomposed into integrable foliations arising from singular distributions. These tools could allow us to endow a set with a differentiable structure, in the absence of any regularity properties.

In various fields of science, particularly in Hamiltonian and Lagrangian mechanics, it is common to wish to work with sets that do not satisfy the often overly restrictive conditions of a differentiable manifold. This raises the need to develop a more flexible geometric framework that allows for the treatment of such singular structures.