Schedule
March 13
11:00-12:00
Narciso Román Roy
Multisymplectic and Multicontact de Donder-Weyl Hamiltonian Formulations for Field Theories
The de Donder-Weyl formulation is a covariant description of Hamiltonian classical (first order) field theories.A geometric framework for this formulation is presented; first, for conservative Hamiltonian theories, using the multisymplectic approach, and, second, for nonconservative (or ``action-dependent´´) Hamiltonian field theories, by doing an extension of the above one which is based on a generalization of the contact structures,and is known as the multicontact formulationSpecial attention is devoted to the case of singular theories and, in particular, to the Hamiltonian formalism associated with regular and singular Lagrangian systems.
References:
[1] J.F. Cariñena, M. Crampin, and L.A. Ibort,
“On the multisymplectic formalism for first order field theories”,
Diff. Geom. Appl. 1(4) (1991) 345–374.
[2] M. de León, J. Marín-Solano, and J.C. Marrero,
“A Geometrical approach to Classical Field Theories: A constraint algorithm for singular theories”,
Proc. New Develops. Dif. Geom., L. Tamassi, J. Szenthe
eds., Kluwer Acad. Press, (1996) 291–312.
[3] M. de León, J. Marín-Solano, J.C. Marrero,
“The constraint algorithm in the jet formalism”,
Diff. Geom. App. 6(3) (1996) 275–300.
[4] J. Gaset, X. Gràcia, M. Muñoz-Lecanda, X. Rivas, and N. Román-Roy,
“A contact geometry framework for field theories with dissipation”,
Ann. Phys. 414 (2020) 168092.
12:10-12:40
Jagna Wisniewska
Lagrangian Rabinowitz Floer homology in the restricted three body problem
Throughout centuries the analysis of the three body problem has been one of the driving forces in the development of symplectic geometry. Can we fly a rocket between two arbitrary points in the gravitational field of the Earth and the Moon assuming that we accelerate and decelerate only at the start and the beginning of our journey? In my talk I will reformulate this problem in the language of symplectic geometry and show how to employ the Lagrangian Rabinowitz Floor homology to solve it. A project in progress with Kai Cieliebak, Urs Frauenfelder and Eva Miranda.
12:45-13:15
Pablo Nicolás
New perspectives in Poisson cohomology.
The approach to Poisson cohomology using spectral sequences is not new, but it has not been sufficiently explored.
In this talk, we will present new Poisson cohomology results for a wide class of Poisson manifolds using spectral sequences. Our technique allows us to prove a local Künneth formula for Poisson cohomology, which has straightforward application to computations for transversally linearizable Poisson structures, bm-Poisson structures, and Poisson structures with transverse simple singularities of corank 2. Our construction elucidates former computations for b-Poisson manifolds (by Guillemin, Miranda, Pires, and Marcut-Osorno) using the classical technique of unfolding applied to b^m-Poisson structures. This talk is joint work with Eva Miranda.
13:30-15.30
Lunch, Sala Restaurant EPSEB, 4th floor (Sala de PDI-PAS)
15:30 - 16:00
Søren Dyhr
Cosymplectic manifolds, compatible metrics, and fluid dynamics
Fluid mechanics can be described by PDEs, and one way to study them is by looking at the properties of their time-independent solutions. A
correspondence between Beltrami fields (a type of solutions to the
stationary Euler equations) and Reeb flows on contact manifolds is
known. This has enabled the construction of solutions to the stationary
Euler equations with specific properties.
In this talk I will briefly describe this construction and then discuss
our work in progress of studying similar correspondences with
cosymplectic manifolds. I will take a point of view of assigning
compatible metrics to cosymplectic manifolds, which then naturally leads to studying energy functionals on the space of these metrics. In the end I will discuss some possibilities for generalizations and further results.
This is joint work with Ángel Gonzalez-Prieto, Eva Miranda, and Daniel Peralta-Salas.
16:10 - 16:40
Xavier Rivas
A geometric framework for time-dependent contact systems
Contact geometry allows us to describe many dynamical systems which cannot be described by means of symplectic geometry. In this talk we introduce a new geometric structure in order to describe time-dependent contact systems: cocontact manifolds. Within this setting we develop the Hamiltonian and Lagrangian formalisms, both in the regular and singular cases. As a particular case we study contact systems with holonomic time-dependent constraints. We will also introduce several notions of symmetries for cocontact systems and stablish the relations between them. Some regular and singular examples are analyzed
16:50 - 17:20
Alfonso Garmendia
Two roads to E-Poisson
E-manifolds and almost regular foliated manifolds can be seen as ai (almost injective Lie) algebroids. These algebroids are always integrable to a Lie groupoid.
If we want to study Poisson geometry on these objects we get two options. On the one hand, some Poisson manifolds have symplectic foliations described by ai-algebroids. On the other hand, given a symplectic structure on an ai-algebroid there is a canonical Poisson structure. In this talk, we study the relations of these two structures and the implications on the groupoid level.
17:30-18:15
PhD thesis Rehearsal Floer Homology for b-symplectic manifolds
Joaquim Brugués