The **Geometry and Applications: Modern Mathematical Approaches (Gamma) Seminar** is a weekly online seminar aimed at discussing basic and advanced research topics conducted by the Polish-Spanish geometric group gathering people from the University of Warsaw, the Universitat Rovira i Virgili, the Polytechnic University of Madrid, ICMAT, and the Polytechnic University of Catalonia.

- Geometric mechanics and field theories
- Geometric structures: symplectic and Poisson geometry, contact geometry and its Generalisations
- Supergeometry
- Integrable systems, superposition rules, Lie systems
- Stochasticity and numerical methods

**e-mail:** gamma.seminar(at)gmail.com

- Javier de Lucas (University of Warsaw)
- Xavier Rivas (Universitat Rovira i Virgili)

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We develop a reduction scheme for the Lie-infinity-algebra of observables on a pre-multisymplectic manifold \((M,\omega)\) in the presence of a compatible Lie algebra action \(\mathfrak{g}\curvearrowright M\) and subset \(N\subset M\). This reduction relates to the geometric multisymplectic reduction recently proposed by Casey Blacker. In particular, when \(M\) is a symplectic manifold and \(N\) the level set of a momentum our approach generalizes Marsden-Weinstein reduction, and has interesting relations to further symplectic reduction schemes.

Based on joint work with Casey Blacker and Antonio Miti.

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It is well known that Lie systems are systems of ordinary differential equations whose general solutions can be described as a function, the superposition rule, of a family of particular solutions and some constants. Importantly, every Lie system is associated with a finite-dimensional Lie algebra of vector fields. Studying these algebras leads to the investigation of foliations, Lie group actions, generalized distributions, and other geometrical issues. In my talk, I will consider the aforementioned vector fields as \(\boldsymbol\eta\)-Hamiltonian vector fields relative to the \(k\)-contact structure. \(k\)-Contact geometry is an extension of the standard contact geometry, and its properties can differ significantly. This approach allows for the analysis of a wider range of Lie systems, which I will demonstrate with some specific examples.

In this talk I will present some recent work with Manuel de León on Multisymplectic Geometry. More particularly, we will study a generalization to Multisymplectic manifolds of some basic results in Symplectic Geometry: An interpretation of (locally) Hamiltonian multivector fields as k-Lagrangian submanifolds and coisotropic reduction when the multisymplectic manifold is a bundle of forms.

In Riemannian geometry, a Jacobi field is a vector field along a geodesic curve arising as the variation field of any one parameter family of geodesics. Since the geodesic curves turned out related to a vector field on the tangent bundle called geodesic spray, the Jacobi fields have several generalizations that allow these fields to play important roles in many areas of differential geometry including Finsler geometry, Lagrangian mechanics, and Lie algebroids. On Lie algebroids, M. Boucetta and later J. Cariñena, E. Martinez, and I. M. Gheorghiu introduced the notion of Jacobi sections generalizing the usual concept from the tangent algebroid. In an ongoing investigation with the Ph. D. Student Jhonny Kama-Mamani, we propose the notion of Jacobi field for sprays on Lie algebroids trying to generalize some properties of the classical Jacobi fields on Riemannian Geometry. In this talk, we present the notion of Jacobi fields and Jacobi sections on Lie algebroids and discuss the relation between these notions which are originated by the same common object. We also present some perspectives on applications of Jacobi fields on Lie algebroids.

Delving into the realm of field theories, \(k\)-contact manifolds represent a sophisticated extension of contact manifolds, akin to how partial differential equations expand upon ordinary differential equations. While contact manifolds are traditionally analysed through contact distributions with the contact form serving as a mere tool for practicality, the same cannot be said for their \(k\)-contact counterparts. This talk introduces a cutting-edge distributional approach tailored specifically for \(k\)-contact manifolds. However, numerous properties inherent to contact manifolds do not seamlessly translate to the \(k\)-contact domain. Nevertheless, we will unveil innovative concepts and findings that open a new way to study \(k\)-contact manifolds.

As it is well known symplectic geometry is the natural arena to develop Classical Mechanics; indeed, a symplectic manifold is locally as a cotangent bundle \(T^*Q\) of a manifold, so that the canonical coordinates \((q^i,p_i)\) can be used as coordinates for the positions \((q^i)\) and the momenta \((p_i)\). The symplectic form is just \(\omega=dq^i\wedge dp_i\), and a simple geometric tool permits to obtain the Hamiltonian vector field \(X_H\) for a Hamiltonian \(H=H(q^i,p_i)\). The integral curves of \(X_H\) are just the solution of Hamilton equations.

In Classical Field Theory, the Hamiltonian function is of the form \(H=H(x^\alpha, q^i, p^\alpha_i)\) where \((x^1, \ldots, x^k)\in \mathbb{R}^k,\, q^i\) represent the components of the fields and \(p^\alpha_i\) are the conjugate momenta.

At the end of the '60s and the beginning of the '70s of the past century, there are some attempts to develop a convenient geometric framework to study classical field theories.

The aim of this talk is to review some of these geometric framework for describing classical field theories. In particular we consider \(k\)-symplectic geometry, \(k\)-cosymplectic geometry and multisymplectic geometry. Finally we analyze the relationships between them.

In this talk I will present an algebraic approach to smooth manifolds in the language of locally ringed spaces. It turns out that this setting allows us to extend the notion of manifold in various ways. Then, I will focus on presenting definition and basic theory on supermanifolds. The main goal of this talk is to explain the Batchelor-Gawędzki theorem that establishes relation between supermanifolds and vector bundles.

Hamiltonian mechanics originates from physics, more precisely from classical mechanics. Examples of such systems can be found in celestial mechanics, describing the motion of a satellite under the gravitational forces of several astronomical objects. One of the main questions in Hamiltonian mechanics is to find periodic orbits on a fixed energy level set. In my talk, I will introduce the notion of Rabinowitz Floer homology and show how this algebraic invariant can be applied to the special class of tentacular Hamiltonians to answer the question of existence of periodic orbits for Hamiltonian systems on a fixed, non-compact energy level set.

In this talk we will review basic properties of groups of contactomorphisms, i.e., groups of diffeomorphisms of a manifold that preserve a given contact structure. In particular we will focus on the notion of orderability: The group of contactomorphisms of a co-oriented contact manifold and its universal cover carry a natural bi-invariant relation. An important open question in contact topology is the classification of co-oriented contact manifolds for which this relation is a partial order. We will discuss several examples of orderable and non-orderable contact manifolds and connections of this question to dynamical and topological properties of the contact manifold.

Many important theories in modern physics can be stated using the tools of differential geometry. It is well known that symplectic geometry is the natural framework to deal with autonomous Hamiltonian mechanics. This admits several generalizations for nonautonomous systems and classical field theories, both regular and singular. In recent years there has been a growing interest in studying dissipative mechanical systems from a geometric perspective by using contact structures. In this talk we will review the main results on contact mechanics and work out some examples. Furthermore, field theories with damping will be described through a modification of the De Donder-Weyl Hamiltonian field theory. This is achieved by combining both contact geometry and \(k\)-symplectic structures, resulting in what we call the \(k\)-contact formalism. Finally we will study some interesting examples: the damped vibrating string, the Burgers' equation and the Maxwell's equations of electromagnetism with dissipation.

In this talk I will introduce the Hamiltonian formalism in multisymplectic geometry and different types of partial multisymplectic Marsden-Weinstein reductions. I will also briefly comment on the applications of multisymplectic geometry and multisymplectic reductions to Lie systems, i.e. systems on t-dependent ordinary differential equations whose dynamics is determined by a t-dependent vector field amounting to a curve in a finite-dimensional Lie algebra of vector fields.

A long-standing problem in the theory of Integrable Systems is how to construct new integrable systems. For systems in Classical Mechanics a very proficient method was introduced by Ballesteros and Ragnisco in mid-90s. Such a method is based on the notion of coalgebra, and roughly speaking allows the extension of integrable systems in one-degree of freedom to an arbitrary number of degrees of freedom. In this talk I will review the fundamental ideas of this approach and present some novel applications also outside its original formulation.

Based on joint works with Danilo Latini, Benjamin K. Tapley.

Morse homology relates the dynamics given by the gradient flow of a function to the topology of the underlying manifold. Knowing the critical points of a Morse function and their Morse indexes, we can infer the topology of the manifold. And vice versa: knowing the topology of the manifold we can establish a lower bound on the number of the critical points of a Morse function. In a similar fashion the Floor homology relates the 1-periodic orbits of the Hamiltonian flow to the topology of the underlying symplectic manifold.

Lie-Hamilton systems are first-order systems of differential equations possessing a (non-linear) superposition rule which are compatible with a certain symplectic structure. They have been widely studied and classified under local diffeomorphisms on the real plane \(\mathbb{R}^{2}\) (see [1] for a general exposition). Nevertheless, there are few examples of higher-dimensional Lie-Hamilton systems which do not come from lower-dimensional systems. For the more general case of Lie systems, it was shown recently that higher-dimensional Lie systems (even nonlinear ones) can be constructed through a representation theory approach [2,3].

In this talk we will propose a new method which allows us to obtain higher-dimensional Lie-Hamilton systems, mainly based on ideas coming from the representation theory of Lie algebras. We will focus on \(4\)-dimensional Lie-Hamilton systems which we will interpret as systems on the configuration space \(T^{*} \mathbb{R}^{2} \simeq \mathbb{R}^{4}\), particularly obtaining \(t\)-dependent Hamiltonian systems with a Lorentzian kinetic energy coming from the Lorentz algebra \(\mathfrak{so}(3, 1)\), from which coupled systems of \(t\)-dependent 1D oscillators will be obtained. As we will show, the symplectic Lie algebra \(\mathfrak{sp}(4, \mathbb{R})\) will play a prominent role when constructing these systems (\(t\)-dependent electromagnetic fields and \(t\)-dependent coupled oscillators), which we will prove that are *intrinsic*, in the sense that they are not locally diffeomorphic to lower-dimensional systems. This last step will be carried out using Cartan's approach to completely integrable distributions [4].

This is a joint work with R. Campoamor-Stursberg and F. J. Herranz.

References:

- J. de Lucas and C. Sardón.
*A Guide to Lie Systems with Compatible Geometric Structures*. (Singapore: World Scientific) 2020. doi:10.1142/q02080 - R. Campoamor-Stursberg. Reduction by invariants and projection of linear representations of Lie algebras applied to the construction of nonlinear realizations.
*J. Math. Phys.*59 (2018) 033502. doi:10.1063/1.4989890 - R. Campoamor-Stursberg. Invariant functions of vector field realizations of Lie algebras and some applications to representation theory and dynamical systems.
*J. Phys.: Conf. Ser.*1071 (2018) 012005. doi:10.1088/1742-6596/1071/1/012005 - É. Cartan.
*Les systèmes différentiels extérieurs et leurs applications géométriques.*(Paris: Hermann) 1945.

The infinite-dimensional jet bundle formulation of symmetries of differential equations is presented and the particular case of non-local symmetries is studied in detail. The theory is applied to study non-local symmetries of certain non-linear oscillators and retrieve previous results. Despite being technically more complicated than standard finite-dimensional jet bundles, the infinite-dimensional jet bundle setting is able to simplify certain mathematical structures while making the application of jets easier in practical cases.

This talk will focus on the construction of canonical lifts of group actions and their generating vector fields to the first-jet and first-jet dual bundles on which the Lagrangian and De Donder-Weyl Hamiltonian formalisms for first-order field theories take place. Our work introduces the notion of a canonical lift in the De Donder-Weyl Hamiltonian formalism, for both regular and singular field theories, while the Lagrangian analog has been well known for many years. The importance of canonical lifts in defining natural symmetries will be emphasized in our Hamiltonian construction. Our work will be illustrated in an example field theory: Einstein-Cartan gravity in 3+1 dimensions.

In the 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.

The aim of the talk is to introduce the scaling symmetries for Hamiltonian systems associated with Kirillov structures. I will begin by revisiting the definition of a Kirillov structure and recalling the notions of \(\mathbb{R}^\times\)-principal line bundle \(M\) and its associated line bundle \(L\). Then, to show how Kirillov structure generalises contact manifolds, I will present theorems establishing a one-to-one correspondence between sections \(h\in \Gamma(L^*)\) and one-homogeneous functions \(H\in C^\infty(M)\). Next, I will introduce a scaling symmetry for a dynamical system on a symplectic manifold that is a triple \((S,\omega, H)\), where \(\omega\) is a one-homogeneous symplectic form on a manifold \(S\) and one-homogeneous Hamiltonian function \(H\in C^\infty(S)\). Finally, I will present a reduction by scaling symmetries on an illustrative example of the 2D harmonic oscillator.

A Kirillov structure on a real line bundle \(L\) refers to a Lie algebra structure \([\cdot,\cdot]\) on the space of sections of the dual bundle, \(\Gamma(L^*)\), where, for a fixed section \(h\in\Gamma(L^*)\), the operator \([\cdot,h]\) is a derivation. I will begin my talk by reviewing the concept of a real linear bundle. Then, I will recall the definitions of contact, Poisson, and Jacobi structures, providing some illustrative examples. Additionally, I will demonstrate how Kirillov structures can be derived from these structures and comment on the triviality of a real line bundle. If time permits, I will introduce scaling symmetries for dynamical systems and present the example of the 2D harmonic oscillator.

The main aim of this talk is to show that *k*-symplectic structures can naturally be employed to investigate ordinary differential equations instead of field theories. This leads to endow *k*-symplectic structures with new geometric useful constructions and to recover notions that were previously ignored because they are not appropriate for field theories.

To illustrate the above claims, I will first survey *k*-symplectic structures and Lie systems, namely systems of ordinary differential equations whose general solutions can be described as a function, the superposition rule, of a family of particular solutions and some constants. Lie systems are equivalent to curves in a finite-dimensional Lie algebra of vector fields. In the so-called *k*-symplectic Lie systems [1], the Lie algebra can be chosen to consist of Hamiltonian vector fields with respect to a *k*-symplectic structure. This suggests us to endow *k*-symplectic structures with a Lie algebra of admissible functions and several related Poisson algebras. These Lie algebras give rise, through a Poisson-coalgebra approach, to methods to derive geometrically superposition rules for *k*-symplectic Lie systems. Our theory will be illustrated with examples from control theory, physics and mathematics [1, 2].

References:

- J. de Lucas and S. Vilarino.
*k*-symplectic Lie systems: theory and appli- cations,*J. Differential Equations***258**(6) (2015) 2221-2255. - J. de Lucas, M. Tobolski and S. Vilariño. A new application of
*k*-symplectic Lie systems,*Int. J. Geom. Methods Mod. Phys.*(2015) 1550071.

Classical field theories can be formalized geometrically through *k*-symplectic geometry. In this presentation, I will introduce the fundamentals of *k*-vector fields, integral sections, and *k*-symplectic geometry. This geometric framework will enable us to provide a geometric description for both Hamiltonian and Lagrangian field theories. The talk will include the discussion of relevant examples, such as the wave equation and Maxwell's equations for electromagnetism.

Over the last 50 years, fiber bundles have slowly but surely claimed their title as the optimal language for discussing field theories. However, many fundamental notions used in contemporary field theory stem from the work of Charles Ehresmann in the 1930s and 40s. Among them are jet bundles, a powerful construction allowing for an elegant generalization of the tangent space of a manifold. Jets describe equivalence classes of sections of a fiber bundle, extending the concept beyond one-parameter curves, typical for classical mechanics. My talk aims to introduce jet bundles and emphasize their extraordinary relationship with Ehresmann connections.

In this talk I am going to introduce the formalism of category theory. In particular, functors, natural transformations, limits and adjoint functors will be defined. Furthermore, I will prove two basic categorical results, namely Yoneda’s lemma and Freyd theorem. All notions will be illustrated with examples from different branches of mathematics.

In this talk I am going to introduce the formalism of category theory. In particular, functors, natural transformations, limits and adjoint functors will be defined. Furthermore, I will prove two basic categorical results, namely Yoneda’s lemma and Freyd theorem. All notions will be illustrated with examples from different branches of mathematics.

This course surveys some of the most relevant geometric structures appearing in modern differential geometric theories: Poisson, symplectic, Dirac, Jacobi, multisymplectic, and *k*-symplectic manifolds. Next, we introduce the notion of Lie system, i.e. a nonautonomous system of first-order differential equations whose general solution can be described as a function, the superposition rule, of a particular generic family of particular solutions and some constants. The use of geometric structures for the calculation of superposition rules and constants of motion for Lie systems is to be analysed. Applications of the theory to systems of ordinary and partial differential equations of physical and mathematical relevance will be studied. Among other applications, we plan to look into nonautonomous frequency Smorodinsky-Winternitz oscillators, types of diffusion equations, and Backlund transformations for sine-Gordon equations.

This course surveys some of the most relevant geometric structures appearing in modern differential geometric theories: Poisson, symplectic, Dirac, Jacobi, multisymplectic, and *k*-symplectic manifolds. Next, we introduce the notion of Lie system, i.e. a nonautonomous system of first-order differential equations whose general solution can be described as a function, the superposition rule, of a particular generic family of particular solutions and some constants. The use of geometric structures for the calculation of superposition rules and constants of motion for Lie systems is to be analysed. Applications of the theory to systems of ordinary and partial differential equations of physical and mathematical relevance will be studied. Among other applications, we plan to look into nonautonomous frequency Smorodinsky-Winternitz oscillators, types of diffusion equations, and Backlund transformations for sine-Gordon equations.

This course surveys some of the most relevant geometric structures appearing in modern differential geometric theories: Poisson, symplectic, Dirac, Jacobi, multisymplectic, and *k*-symplectic manifolds. Next, we introduce the notion of Lie system, i.e. a nonautonomous system of first-order differential equations whose general solution can be described as a function, the superposition rule, of a particular generic family of particular solutions and some constants. The use of geometric structures for the calculation of superposition rules and constants of motion for Lie systems is to be analysed. Applications of the theory to systems of ordinary and partial differential equations of physical and mathematical relevance will be studied. Among other applications, we plan to look into nonautonomous frequency Smorodinsky-Winternitz oscillators, types of diffusion equations, and Backlund transformations for sine-Gordon equations.

In this talk, I will discuss briefly the Hamilton-Jacobi theory. In short, a complete solution to the Hamilton-Jacobi equation gives rise to a family of first-order differential equations on the configuration space, which are sufficient to recover all the solutions to Hamilton’s equation on the respective cotangent space. The Hamilton-Jacobi problem may be analyzed by introducing the concept of slicing vector fields and complete slicings in this context. After introducing all required details, I will discuss Hamiltonian systems defined on symplectic manifolds derived from this general framework.

I will introduce the concept of category and related notions at a basic level. Then, I will use them to present superalgebra structures, in particular, linear ones. More specifically, I will explain the theory of superspaces and their morphisms, superrings, superderivations, supermodules and their supermorphisms, which will lead to supermatrices, etc. I will apply some of these techniques to explain differential geometric structures and their properties.

I will introduce the concept of category and related notions at a basic level. Then, I will use them to present superalgebra structures, in particular, linear ones. More specifically, I will explain the theory of superspaces and their morphisms, superrings, superderivations, supermodules and their supermorphisms, which will lead to supermatrices, etc. I will apply some of these techniques to explain differential geometric structures and their properties.

The world of vanishing vector fields and differential forms with blow-ups is intriguing. In this talk, we will use basic algebraic and geometric tools to enter it. In particular, notions such as *b*-manifolds and *b*-vector fields will be introduced and utilized to construct the *b*-tangent bundle. Once familiarized with the setting, we will inspect the singularities of the so-called *b*-forms. Since understanding their behavior is the main goal of the talk, we will discuss them in detail. Finally, we will utilize this theory to describe a twin sister of symplectic geometry: the *b*-symplectic geometry.

Over the last 50 years, fiber bundles have slowly but surely claimed their title as the optimal language for discussing field theories. However, many fundamental notions used in contemporary field theory stem from the work of Charles Ehresmann in the 1930s and 40s. Among them are jet bundles, a powerful construction allowing for an elegant generalization of the tangent space of a manifold. Jets describe equivalence classes of sections of a fiber bundle, extending the concept beyond one-parameter curves, typical for classical mechanics. My talk aims to introduce jet bundles and emphasize their extraordinary relationship with Ehresmann connections.

In this presentation, I will provide the fundamental overview of Lyapunov stability theory, which has broad applications in control theory and physics. I will begin by defining the equilibrium point of a non-autonomous system of ordinary differential equations and proceed to the classification of the stability of equilibrium points. Throughout my talk, I will show illustrative examples to demonstrate the different types of stability. In the end, I will introduce tools that enable the study of the behaviour of a system close to an equilibrium point without the need to solve, often highly complex, differential equations.

The circular restricted three body problem studies the movement of a free particle subject to the gravitational attraction of two massive bodies moving in circular orbits around their common center of mass. In this talk we will construct the model and analyze some dynamical properties like the existence of fixed points and their stability. Thanks to the dynamical comprehension of the problem, we will be able to explain why we can find Troyan asteroids as well as to justify the location of James Webb Telescope at approximately 1.5 million kilometers from the Earth. We will end the talk explaining how these simplified dynamics can help to save fuel in space travels.

In this talk, I will discuss the concept of completely integrable system. The Liouville-Arnold theorem and the main ideas of its proof will be presented, along with several illustrative examples. Additionally, I will mention the relationship between action-angle coordinates and the Hamilton-Jacobi equation, as well as the Kolmogorov-Arnold-Moser theory.

Field theories, like Maxwell's or Einstein's equations, require a generalization of symplectic and cosymplectic geometry. In this session we will explore the geometry of field theories, showing the differences with respect to mechanics. We will define some of the most common geometric objects in field theories: the jet bundle and multisymplectic forms.

This talk, divided into two parts, is intended to be an introduction to mechanics on symplectic manifolds, both in the Hamiltonian and Lagrangian formalisms. We will start by reviewing some basic results on symplectic geometry and Hamiltonian systems, and introducing some concepts about symmetries of Hamiltonian systems and Noether's theorem. Then, we will see how to obtain a symplectic structure on the tangent bundle of the configuration space of a mechanical system from a Lagrangian function. In the second part of the talk we will cover two geometric formulations for time-dependent systems using cosymplectic and contact geometry. Throughout the talk, some examples will be used to illustrate the theory.

This talk, divided into two parts, is intended to be an introduction to mechanics on symplectic manifolds, both in the Hamiltonian and Lagrangian formalisms. We will start by reviewing some basic results on symplectic geometry and Hamiltonian systems, and introducing some concepts about symmetries of Hamiltonian systems and Noether's theorem. Then, we will see how to obtain a symplectic structure on the tangent bundle of the configuration space of a mechanical system from a Lagrangian function. In the second part of the talk we will cover two geometric formulations for time-dependent systems using cosymplectic and contact geometry. Throughout the talk, some examples will be used to illustrate the theory.

This talk, divided into three parts, is mostly aimed at explaining fundamental facts on symplectic geometry, the Marsden-Weinstein-Meyer reduction theorem, and their applications in Physics. I will try to provide some not very well-known facts to make to talk more interesting to those knowing the theory, while keeping it simple for beginners. Facts about modern research problems concerning the Marsden-Weinstein-Meyer theorem will be detailed.

This talk, divided into three parts, is mostly aimed at explaining fundamental facts on symplectic geometry, the Marsden-Weinstein-Meyer reduction theorem, and their applications in Physics. I will try to provide some not very well-known facts to make to talk more interesting to those knowing the theory, while keeping it simple for beginners. Facts about modern research problems concerning the Marsden-Weinstein-Meyer theorem will be detailed.

This talk, divided into three parts, is mostly aimed at explaining fundamental facts on symplectic geometry, the Marsden-Weinstein-Meyer reduction theorem, and their applications in Physics. I will try to provide some not very well-known facts to make to talk more interesting to those knowing the theory, while keeping it simple for beginners. Facts about modern research problems concerning the Marsden-Weinstein-Meyer theorem will be detailed.