Double Lie groupoids generalize Lie groupoids by the addition of two distinct manifolds of arrows and a manifold of 2-arrows that span them. The classical Ambrose-Singer theorem states that the curvature of a principal bundle connection generates the holonomy. This was later adapted to the setting of Lie groupoids by Mackenzie, who established that the Lie algebroid of a holonomy groupoid is the curvature reduction of the original Lie algebroid. In this talk, we introduce the constructions of a path connection and a path curving on a double Lie groupoid and those of a connection and a curving on the tangent LA-groupoid. We further discuss the relation between the holonomy double groupoid of a path curving and the 3-curvature of the associated LA-groupoid curving. This is joint work in progress with Derek Krepski.

Zapraszamy na spotkanie o godzinie 15:15

Ruben Izquierdo (ICMAT, Spain)

The graded Jacobi bracket in dissipative field theories

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.

Zapraszamy na spotkanie o godzinie 15:15

R. Popovych (Silesian University in Opava, Czech Republic & Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine)

Realizations of Lie algebras on the line and the new group classification of (1+1)-dimensional generalized nonlinear Klein–Gordon equations

We formalize the notion of class of systems of differential equations within the framework of group analysis of differential equations. Point (resp. contact) transformations in certain underlying spaces, which are associated with a class of differential equations, form various algebraic objects. The basic object is the equivalence groupoid of the class, which consists of all admissible point transformations within the class. We discuss classical notions of group analysis, like the point symmetry group of a system of differential equations or the equivalence group of a class of such systems, in the context of groupoid theory. As an example, we essentially generalize some Lie’s results by proving that the contact equivalence groupoid of a class of (1+1)-dimensional generalized nonlinear Klein–Gordon equations is the first-order prolongation of its point equivalence groupoid, and then we carry out the complete group classification of this class. Since it is normalized, the algebraic method of group classification is naturally applied here. Using the specific structure of the equivalence group of the class, we essentially employ the classical Lie theorem on realizations of Lie algebras by vector fields on the line. This approach allows us to enhance previous results on Lie symmetries of equations from the class and substantially simplify the proof. After finding a number of integer characteristics of cases of Lie-symmetry extensions that are invariant under action of the equivalence group of the class under study, we exhaustively describe successive Lie-symmetry extensions within this class.

Zapraszamy do sali 1.01, ul. Pasteura 5 o godzinie 15:15

Rui Loja Fernandes (University of Illinois Urbana-Champaign)

Integrability of Lie Algebroids

This lecture will review the main developments in the theory of integrability of Lie algebroids over the past 25 years and discuss several open problems that remain. Please also consider that the talk can be attended also online via Zoom with the following link https://uw-edu-pl.zoom.us/j/94548599338?pwd=K1NWTkI3czdqZGNNalZMdWJNNHh1UT09#success The password is "GAMMA".

Zapraszamy na spotkanie o godzinie 16:15

Eugene M. Lerman (University of Illinois)

Poisson C^\infty-schemes

We introduce Poisson C∞ -rings and Poisson C∞ -ringed spaces. We show that the spectrum of a Poisson C∞ -ring is an affine Poisson C∞ -scheme. We then discuss applications that include singular symplectic and Poisson reductions.

Zapraszamy na spotkanie o godzinie 15:15

Joanna Gonera (Uniwersytet Lódzki)

Fermat’s Principle in General Relativity via Herglotz Variational Formalism (preliminary title)

New form of Fermat's principle for light propagation in arbitrary (i.e. in general neither static nor stationary) gravitational field is proposed. It is based on Herglotz extension of canonical formalism and simple relation between the dynamics described by the Lagrangians homogeneous in velocities and the reduced dynamics on lower-dimensional configuration manifold. This approach is more flexible as it allows to extend immediately the Fermat principle to the case of massive particles and to eliminate any space-time coordinate, not only x_0.

Zapraszamy na spotkanie o godzinie 15:15

Mars Vermeeren (Loughborough University)

Contact variational integrators

Contact geometry is an odd-dimensional analogue to symplectic geometry. Hamiltonian systems based on contact geometry can describe systems with dissipation. The Lagrangian description of contact Hamiltonian systems is given by an variational principle known as Herglotz’ principle. In this seminar, we will discuss the continuous and discrete versions of Herglotz’ principle and how they can be used to construct variational integrators for contact systems. We will discuss abstract geometric considerations side by side with concrete examples in coordinates.

Zapraszamy na spotkanie o godzinie 15:15

A.M. Grundland (Centre de Recherches Mathematiques, CRM, of the University of Montreal)

On Riemann wave superpositions for the Euler system

This talk presents an analysis of the conditions for the existence of elastic versus non-elastic wave superpositions governed by the Euler system in (1+1)-dimensions. A review of recently obtained results is presented including the introduction of the notion of quasi-rectifiability of vector fields and its application to both elastic and non-elastic wave superpositions. It is shown that the smallest real Lie algebra containing vector fields associated with the waves admitted by the Euler system is isomorphic to an infinite-dimensional Lie algebra which is the semi-direct sum of an Abelian ideal and the three-dimensional real Lie algebra $K^+$ or $K^-$. The maximal Lie module corresponding to the Euler system can be transformed by an angle preserving transformation to a real algebra isomorphic to the Lie algebras $K^\pm$. The algebras $K^+$ and $K^-$ are quasi-rectifiable and describe the behavior of wave superpositions. Based on these facts, we are able to find a parametrization of the region of non-elastic wave superpositions which allows for the construction of the reduced form of the Euler system.

Zapraszamy na spotkanie o godzinie 15:15

Manuel de Leon (ICMAT)

to be announced

Zapraszamy na spotkanie o godzinie 15:15

Agustin Moreno (Heidelberg University)

A symplectic viewpoint on the restricted three body problem

In this talk, I will give an overview of some recent advances on the classical (circular, restricted) three-body problem, from the perspective of modern symplectic geometry.

Zapraszamy na spotkanie o godzinie 15:15

J. Ciesliński (Uniwersytet Białystoku)

Applications of Clifford algebras and Spin groups to integrable systems

A broad class of linear problems (Lax pairs) with values in Clifford algebras will be discussed. Such systems, through a suitable generalization of the Sym formula, give rise to interesting families of multidimensional submanifolds associated with integrable nonlinear partial differential equations. These equations can be interpreted as the Gauss–Codazzi–Ricci equations for the corresponding submanifolds.The development of this subject was stimulated by the discovery of a Lax pair associated with isothermic surfaces (Cieśliński, Goldstein, Sym 1995). Initially, this was formulated in terms of the groups SO(4,1) or Sp(2,2), and later expressed in a more elegant way using the Clifford algebra and the group Spin(4,1). The application of the Sym formula does not directly yield isothermic surfaces; instead, it produces flat submanifolds immersed in a six-dimensional space. Only after projection onto appropriate three-dimensional subspaces does one obtain a pair of isothermic surfaces, which are mutually dual (known as Christoffel transforms). In this way, the classical theory of isothermic surfaces, developed by Bianchi and Darboux, has been reformulated in terms of spectral problems within modern soliton theory.

Zapraszamy na spotkanie o godzinie 15:15

J. de Lucas (KMMF)

Lie systems: A 16 years restrospective about a 132-years-old theory

This talk offers a concise overview of Lie systems, commemorating the 16th anniversary of my PhD defense. I will trace the historical development of Lie systems, beginning with foundational work by Guldberg, Vessiot, and Lie, which had a relevant moment in 1893 with the celebrated Lie theorem. This will be followed by the analysis of the contributions from Winternitz and the CRM school in Canada. My presentation will then explore the geometric approach (Carinena, Marmo, Grabowski) developed at the begining of my PhD. Then, I will explore the developments during my PhD thesis culminating with the coalgebra method (developed with my coworkers), deformation theory, Lie systems related to geometric structures, and other developments that followed after that. I will finish the newer approaches to Lie systems and their generalisations: stochastic and super Lie systems, discrete Lie systems, and Lie groupoid methods in Lie systems. I will also pay special attention to the results by Marmo, Carinena, Grundland, Ballesteros, Herranz, Fernandez-Sainz, Sardon, Campoamor-Stursberg, Carballal, Odzijewicz, Ibragimov, etc.

Zapraszamy na spotkanie o godzinie 15:15

Robet Wolak (Jagiellonian University)

Sasakian geometry and foliations

A Sasakian manifolds can be considered as a Riemannian foliated manifold with a very particular transverse structure. In the lecture we present some basic facts about foliated manifolds and their transverse structures. Then we discuss how these notions apply to Sasakian manifolds and show some consequences of such an approach.

Zapraszamy na spotkanie o godzinie 15:15

Tomasz Sobczak (KMMF)

Introduction to PDE Lie systems

This talk aims to present the theory of systems of first-order partial differential equations that admit a (nonlinear) superposition rule: the so-called PDE Lie systems. These systems are characterized by the property that the partial derivatives of their solutions depend only on the independent and dependent variables, and their general solution can be expressed as a function of a family of particular solutions together with several constants. In particular, we will discuss the Lie–Scheffers theorem and the Lie group approach to PDE Lie systems. We will present the main features of these systems as well as a range of classical and novel results concerning Bäcklund transformations, Lax pairs, Floquet theory, geometric phases, conditional symmetries, and further potential applications in hydrodynamical systems.

Zapraszamy na spotkanie o godzinie 15:15

Bartołmiej Bąk (KMMF)

Classical model of particle with spin: geometric structure

The motion of point-like particles with mass and charge in the relativistic regime has been well understood for over a century. However, incorporating rotation proved to be far more challenging than initially expected. The first attempts to describe spinning objects were made independently by Jakov Frenkel [1], Myron Mathisson [2], Jan Weyssenhoff and Antoni Raabe [3], Achilles Papapetrou and Ernesto Corinaldesi [4], Barbara and W lodzimierz Tulczyjew [5], and William G. Dixon [6]. The classical spinning particle model on which I intend to rely is based on the approach proposed by Jerzy Kijowski [7]. He briefly outlines a procedure for deriving the equations of motion fromconservation laws and constructs the canonical (symplectic) structure together with the associated Lagrangian and Hamiltonian (cf. [8]). In my talk, I will focus on the symplectic structure of the theory, which exhibits several non-trivial features. In particular, it depends on acceleration (second derivatives of position with respect to time), which makes it a priori degenerate and leads to second-order Euler–Lagrange equations. To address this issue, the Lagrange multiplier method and a reduction procedure are introduced. If time permits, I will also discuss exact solutions and the Hamiltonian formulation of the theory. Most of the results to be presented are not yet published.References[1] J. Frenkel, Die Elektrodynamik des rotierenden Elektrons, Z. Phys. 37, 243 (1926);[2] M. Mathisson: Die Mechanik des Materieteilchens in der Allgemeinen Relativit¨atstheorie, Z. Phys. 67, 826 (1931); Neue Mechanik Materieller Systemes, Acta Phys. Pol. 6, 163 (1937); Das Zitternde Elektron und Seine Dynamik, Acta Phys. Pol. 6, 218 (1937);[3] J. Weyssenhoff, A. Raabe, Relativistic dynamics of spin-fluids and spin-particles, Acta Phys. Pol. 9, 7 (1947); J. Weyssenhoff: Relativistic dynamics of spin-particles moving with the velocity of light, Acta Phys. Pol. 9, 19 (1947); Further contributions to the dynamics of spin-particles moving witha velocity smaller than that of light, Acta Phys. Pol. 9, 26 (1947); Further contributions to the dynamics of spin-particles moving with the velocity of light, Acta Phys. Pol. 9, 34 (1947); On two relativistic models of Dirac’s electron, Acta Phys. Pol. 9, 46 (1947); [4] A. Papapetrou, Spinning test-particles in general relativity. I, Proc. R. Soc. Lond. A. 209, 248 (1951); E. Corinaldesi, A. Papapetrou, Spinning test-particles in general relativity. II, Proc. R. Soc. Lond. A. 209, 259 (1951);[5] W. Tulczyjew, Motion of Multipole Particles in General Relativity Theory, Acta Phys. Pol. 18, 393 (1959); B. Tulczyjew, W. Tulczyjew, On multipole formalism in general relativity, article in Recent Developments in General Relativity, Pergamon Press, New York, 465 (1962);[6] W.G. Dixon: A Covariant Multipole Formalism for Extended Test Bodies in General Relativity, Nuovo Cimento 34, 317 (1964); Mathisson’s new mechanics:its aims and realisation, Acta Phys. Pol. 1, 27 (2008);[7] J. Kijowski; Hamiltonian description of motion of charged particle with spin, Acta Phys. Pol. 1, 143 (2008);[8] J. Kijowski: Electrodynamics of moving bodies, Gen. Relat. Grav.Journal, 26, 167 (1994); On electrodynamical self-interaction, Acta Phys. Pol. 85, 771 (1994); D. Chruscinski, J. Kijowski, Equations of Motion from Field Equations and a Gauge- invariant Variational Principle for the Motion of Charged Particles, Journal Geom. Phys. 20, 393 (1996); H.P. Gittel, J. Kijowski, E. Zeidler, The Relativistic Dynamics of the Combined Particle-Field System in Nonlinear Renormalized Electrodynamics, Comm. Math. Phys. 198, 711(1998);