Young Researchers'
Virtual Multisymplectic Geometry Seminar

The purpose of this event is to provide a forum for early-career mathematicians working in and around multisymplectic geometry to share their research.

date: 15 July, 2020
time: 13:00 UTC

1st hour
Leonid Ryvkin (University Duisberg-Essen)
Homotopy Comoments on Spheres

We will briefly review the notion of homotopy comoments of multisymplectic manifolds and illustrate some of its characteristics by looking at the existence problem for spheres. The talk is based on joint work with Antonio Michele Miti.
Leyli Mammadova (KU Leuven)
Homotopy Comoments on Spheres

We compare the homotopy moment map introduced by M. Callies, Y. Fregier, C. L. Rogers, M. Zambon and the weak (homotopy) moment map introduced by J. Herman. In particular, we phrase the conditions for existence of the former in terms of existence of the latter. Joint work with L. Ryvkin.
Casey Blacker (East China Normal University)
Multisymplectic reduction and split Hamiltonian systems

We first review the Marsden–Weinstein–Meyer symplectic reduction theorem, and then show how this result extends to the setting of multisymplectic manifolds. We also exhibit a distinguished class of multisymplectic Hamiltonian systems that are particularly easy to understand.
2nd hour
Nicolás Martínez Alba (Universidad Nacional de Colombia)
Higher-Poisson structure: Definition and some remarks on its symmetries

Motivated by the well known relation between symplectic and Poisson geometries, we propose a definition of higher-order version of Poisson structures associate to multi-symplectic or poly-symplectic structures. Under such definition we study the reduction of the structure and give some application of this theory.
Roberto Rubio (Universitat Autònoma de Barcelona)
On higher complex Dirac structures

I will mention how recent work on higher Dirac structures (Bursztyn, Martnez-Alba, Rubio) can be combined with the latest developments in complex Dirac structures (Aguëro, Bursztyn, Rubio) in order to study higher complex Dirac structures, where premultisymplectic structures naturally fit.
Miquel Cueca Ten (Georg-August-Universitt Göttingen)
Multisymplectic and graded cotangent bundles

Since the 90's it is known that one can see symplectic geometry as Dirac structures in the standard Courant algebroid. For multisymplectic there are also higher Courant algebroids that play the same role. In this talk I will explain the relation of this higher Courant algebroids with graded cotangent bundles and how this viewpoint can be usefull for quantization.
3rd hour
Tom McClain (Washington and Lee University)
An introduction to a global version of Gunthers polysymplectic formalism using vertical projections

I construct a global version of the local polysymplectic approach to covariant Hamiltonian field theory pioneered by C. Gunther. After a brief overview of the geometric framework of the theory, I specialize to vertical vector fields to construct the (poly)symplectic structures, derive Hamiltons field equations, and construct a more-or-less natural Poisson bracket.
Marcin Zajac (University of Warsaw)
Hamilton-Jacobi theory for locally conformal symplectic manifolds

Locally conformal symplectic (lcs) manifolds have a very interesting geometry and provide a natural generalisation of a symplectic structure. It turns out that the generic example of a lcs manifold is given by a cotangent bundle of a manifold, equipped with certain closed one-form being a pull-back of a one-from on a base manifold. In my talk I will present main features of dynamics on locally conformal symplectic manfolds and present a geometrical version of Hamilton-Jacobi theory for this kind of structure.
Cristina Sardón (Instituto de Ciencias Matemáticas)
Hamilton-Jacobi theory on (conformal) multisymplectic manifolds
4th hour
Javier de Lucas (University of Warsaw)
Multisymplectic Forms and Lie Systems

A Lie system is a non-autonomous system of first-order ordinary differential equations whose general solution can be described as a function of a finite family of particular solutions and some constants: the so-called superposition rule. Lie systems cover as particular cases matrix Riccati equations, Smorodinsky-Winternitz oscillators, Schwarz equations, and control theory systems. The Lie–Scheffers theorem states that a Lie system amounts to a differential equation describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot-Guldberg Lie algebra.

In this talk, I will introduce and analyse a class of Lie systems admitting a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields relative to a multisymplectic form: the multisymplectic Lie systems. Geometric methods are developed to consider a Lie system as a multisymplectic one. By attaching a multisymplectic Lie system via its multisymplectic structure with a tensor coalgebra, we find geometric and coalgebra methods to derive superposition rules, constants of motion, and invariant tensor fields relative to its evolution. In particular, this extends the so-called coalgebra symmetry method for studying constants of motion of a class of Hamiltonian systems to a much more general realm. Our results are illustrated with examples occurring in physics such as Schwarz equations and other classical mechanical systems. Our results show that multisymplectic geometry, mainly aimed at partial differential equations and field theories, may also be applied to systems of ordinary differential equations by means of special new techniques.