Poster sesion

• Bogdan Balcerzak (Łodz University of Technology)
  Dirac operators on anchored vector bundles
Dirac type operators on anchored vector bundles with respect to different geometric structures will be defined and discussed.


• Andrew James Bruce (Polish Academy of Sciences)
  Higher order mechanics on graded bundles
Slides.
We discuss the applications of the recently discovered weighted Lie algebroids to the theory of higher order Lagrangian mechanics on graded bundles following the geometric ideas of Tulczyjew. As a particular example we will focus on higher order mechanics on Lie algebroids, which is motivated by reductions of higher order systems that posses symmetries.


• Ioan Bucataru (Alexandru Ioan Cuza University)
  Generalized Helmholtz conditions for non-conservative Lagrangian systems
(joint work with Oana A. Constantinescu)
In this paper we provide generalized Helmholtz conditions, in terms of a semi-basic 1-form, which characterize when a given system of second order ordinary differential equations is equivalent to the Lagrange equations, for some given arbitrary non-conservative forces. Our formulation allows, in some cases, to study the formal integrability of the proposed generalized Helmholtz conditions. These conditions, when expressed in terms of a multiplier matrix, reduce to those obtained previously by Mestdag, Sarlet and Crampin, for the particular case of dissipative or gyroscopic forces. We provide examples where the proposed generalized Helmholtz conditions, expressed in terms of a semi-basic 1-form, can be integrated and the corresponding Lagrangian and Lagrange equations can be found.


• Oğul Esen (Yeditepe University)
  Tulczyjew's triplet for Lie groups
(joint work with Hasan Gümral (Australian College of Kuwait))
All semidirect products and functorial trivializations of first order and iterated bundles over a Lie group are presented. Symplectic reductions of iterated bundles by right invariance result in Tulczyjew's triplet for reduced manifolds. The trivialized Euler-Lagrange and Hamilton's equations are obtained and presented as Lagrangian submanifolds of the trivialized Tulczyjew's symplectic space. Euler-Poincaré and Lie-Poisson equations are presented as Lagrangian submanifolds of the reduced Tulczyjew's symplectic space. Tulczyjew's generalized Legendre transformations for trivialized and reduced dynamics are constructed.


• Barbara Gołubowska, Wasyl Kowalczuk, Ewa Eliza Rożko
  (Institute of Fundamental Technological Research, Polish Academy of Sciences)
  On affine motion and nonholonomic constraints
In this work our goal is to carry out a thorough analysis of some geometric problems of the dynamics of affinely-rigid bodies. We present two ways to describe this case: the classical dynamical d'Alembert and variational (vakonomic) ones. So far, we can see that they give quite different results, but the vakonomic model from the mathematical point of view seems to be more elegant.


• Jacek Jezierski (University of Warsaw)
  Proof of positive energy theorem by spacetime foliations
(joined work with Piotr Waluk)
Around 1961 R. Arnowitt, S. Deser, and W. Misner proposed, in their collaborative work, a way of defining "total four-momentum" of a gravitating system. The idea consisted in calculating a surface integral, constructed of metric derivatives, at infinity of some spatial hypersurface ("a slice of constant time"). This integral turns out to be well-defined and quite independent of deformations of the chosen hypersurface, a long as the choice is asymptotically flat, i.e., gravitation field falls off quickly enough at infinity.
g_ab=\delta_ab+h_ab, with h_ab=o(r^-1)
The energy, or "mass", component of ADM four-momentum turned out to be especially useful in various applications. It is given by a following integral:
$$M_{ADM}=\lim_{r\to\infty}\frac{1}{16\pi}\oint_{S(r)}(h^j{}_{k,j}-h^j{}_{j,k})dS^k$$
In spite of importance of the concept, it took almost 20 years to settle such basic matter as the question of its positive definiteness.
It was only in 1979 that a complete proof was finally presented by Schoen and Yau, who succeeded by using variational arguments. Not much later, in 1981, another proof appeared (by Witten), based on the theory of spinors.
Here we present a yet alternative approach, requiring only basic tools of differential geometry.


• Igor Kanatchikov(University of St Andrews)
  From the polysymplectic structure to field quantization, YM mass gap and quantum gravity
I outline the algebraic structures which can be obtained from the polysymplectic structure in field theory as generalizations of the Poisson bracket in mechanics. One of them is the Poisson-Gerstenhaber bracket of differential forms. I show how a quantization of a Heisenberg subalgebra of Poisson-Gerstenhaber algebra of forms leads to a construction of quantum fields viewed as sections of the Clifford bundle over the finite dimensional covariant configuration bundle of fields. I also outline how this reformulation of quantum field theory based on the mathematical structures of the De Donder-Weyl covariant Hamiltonian theory can be applied to the mass gap problem in quantum Yang-Mills theory and quantum gravity.


• Antonio De Nicola (University of Coimbra)
  Geometry and topology of cosymplectic spheres
The notion of cosymplectic structure was introduced by P. Libermann in the late 50s as a pair ($\eta$, $\Omega$), where $\eta$ is a closed 1-form and $\Omega$ a closed 2-form on an $2n+1$-dimensional manifold $M$, such that $\eta\wedge\Omega^n$ is a volume form. Cosymplectic manifolds play an important role in the geometric description of time-dependent mechanics (see [B. Cappelletti Montano, A. De Nicola, I. Yudin Rev. Math. Phys. 25 (2013), 1343002] and references therein). Starting from 1967, when Blair defined an adapted Riemannian structure on a cosymplectic manifold, a study of the metric properties on these manifolds was also initiated.
We study the geometry and topology of cosymplectic circles and cosymplectic spheres, which are the analogues in the cosymplectic setting of contact circles and contact spheres, introduced by [H. Geiges, J. Gonzalo Invent. Math. 121, 147--209 (1995)], and then generalized by [M. Zessin Ann. Inst. Fourier (Grenoble) 55, 1167--1194 (2005)]. We provide a complete classification of 3-dimensional compact manifolds that admit a cosymplectic circle.
We introduce the notion of tautness and of roundness for a cosymplectic $p$-sphere. To any taut cosymplectic circle on a three-dimensional manifold $M$ we are able to associate canonically a complex structure and a conformal symplectic couple on $M\times \mathbb{R}$.
In dimension three a cosymplectic circle is proved to be round if and only if it is taut. In higher dimensions we provide examples of cosymplectic circles which are taut but not round and examples of cosymplectic circles which are round but not taut. Finally we show that the three cosymplectic structures of any 3-cosymplectic manifold generate a cosymplectic sphere which is both round and taut.


• Joana Nunes da Costa (University of Coimbra)
  Triples of non-degenerate 2-forms on a Lie algebroid
We show that starting with three non-degenerate 2-forms on a Lie algebroid that satisfy a simple condition, we may obtain several interesting structures.


• Yunhe Sheng (School of Mathematics Jilin University)
  Graded Poisson manifolds up to homotopy
In this paper, we introduce a notion of a graded Poisson manifold up to homotopy, namely a Poisson $[n,k]$-manifold, motivated by studying the dual of a Lie 2-algebra. We further study Maurer-Cartan elements on Poisson $[n,k]$-manifolds and symplectic $[n,n]$-manifolds. There are many interesting examples such as $n$-term $L_\infty$-algebras, twisted Poisson manifolds, quasi-Poisson $\g$-manifolds and twisted Courant algebroids. As a byproduct, we justify that the symplectic $[n,n]$-manifold is a homotopy version of the symplectic NQ-manifold and the Maurer-Cartan equation is a homotopy version of the master equation. The dual of an $n$-term $L_\infty$-algebra is a Poisson$[n,n]$-manifold. We prove that the cotangent bundle of a Poisson $[n,n]$-manifold is a symplectic NQ-manifold of degree $n+1$. In particular, we construct a Courant algebroid from a $2$-term $L_\infty$-algebra. By analyzing these structures, we obtain a Lie-quasi-Poisson groupoid from a Lie 2-algebra, which we propose to be the geometric structure on the dual of a Lie $2$-algebra. At last, we obtain an Ikeda-Uchino algebroid from a $3$-term $L_\infty$-algebra.


• Alfonso Giuseppe Tortorella (University of Florence)
  Deformations of coisotropic submanifolds in abstract Jacobi manifolds
(joint work with Y.-G. Oh (IBS Center for Geometry and Physics), H. V. Lê (Inst. of Math. at ASCR) and L. Vitagliano (University of Salerno))
In this work, using the Atiyah algebroid and first order multi-differential calculus on non-trivial line bundles, we attach an $L_\infty$-algebra to any coisotropic submanifold $S$ in an abstract (or Kirillov’s) Jacobi manifold. Our construction generalizes and unifies analogous constructions in the symplectic case (Oh and Park), the Poisson case (Cattaneo and Felder), locally conformal symplectic case (Lê and Oh). As a new special case, we attach an $L_\infty$-algebra to any coisotropic submanifold in a contact manifold, including Legendrian submanifolds. The $L_\infty$-algebra of a coisotropic submanifold $S$ governs the (formal) deformation problem of $S$.

(C) Michał Jóźwikowski, 2015