Plenary talks

• Jose-Fernando Carinena (University of Zaragoza)
  Revisiting Lie integrability by quadratures from a geometric perspective
Slides.
The classical result of Lie on integrability by quadratures will be reviewed and some generalizations will be proposed. After a short review of the classical Lie theorem, a finite dimensional Lie algebra of vector fields is considered and the most general conditions under which the integral curves of one of the fields can be obtained by quadratures in a prescribed way will be discussed, determining also the number of quadratures needed to integrate the system. The theory will be illustrated with examples and an extension of the theorem where the Lie algebras are replaced by some distributions will also be presented.


• Janusz Grabowski (Polish Academy of Sciences)
  New developments in geometric mechanics
Slides.
We introduce the concept of a graded bundle, generalizing that of a vector bundle, its linearization, and a double structure of this kind. Then, we present applications in classical field theories, including the Plateau problem, and mechanics with higher order Lagrangians.


• Partha Guha (S. N. Bose National Centre for Basic Sciences)
  Inverse problem of calculus of variations and the last Jacobi multiplier
Abstract: TBA


• Simone Gutt (Universite Libre de Bruxelles)
  Submanifolds in symplectic geometry and Radon transforms
We shall present in the symplectic framework an analogue to the classical space forms and study the spaces of their totally geodesic submanifolds.


• Alberto Ibort (Universidad Carlos III de Madrid)
  On the multisymplectic formalism of first-order Hamiltonian field theories on manifolds with boundary: an application to Palatini’s gravity
The multisymplectic formalism for first-order field theories on manifolds with boundary is discussed. A theory of boundary conditions and a canonical formalism near the boundary are obtained. Palatini’s gravity will be reviewed from this perspective.


• Madeleine Jotz Lean (University of Sheffield)
  Poisson Lie 2-algebroids, Courant algebroids and the geometrisation of positively graded manifolds of degree 2
Slides.
D. Roytenberg and P. Severa independently found an equivalence between symplectic Lie 2-algebroids and Courant algebroids, and D. Li-Bland established a correspondence between (Poisson) Lie 2-algebroids and (LA-) VB-Courant algebroids.
In this talk I will present a new manner to retrieve the Courant algebroid corresponding to a given symplectic Lie 2-algebroid. In order to do this, I will describe split Lie 2-algebroids via Dorfman 2-representations, split Poisson manifolds of degree 2 via self-dual representations up to homotopy, and the compatibility conditions for these structures to define a split Poisson Lie 2-algebroid.
In the second part of the talk, I will give a general overview of the equivalences between positively graded manifolds of degree 2 and metric double vector bundles, between Poisson manifolds of degree 2 and metric VB-algebroids, between Lie 2-algebroids and VB-Courant algebroids, and between Poisson Lie 2-algebroids and LA-Courant algebroids. If time permits, I will illustrate these equivalences with more examples.


• Jerzy Kijowski (Centre of Theoretical Physics Polish Academy of Sciences)
  Geometry of q-bit from geometric quantization
Slides.
The talk will contain basic notions and results of the Theory of Geometric Quantization. Its applicability and its limitations will be discussed. Finally, a miracle will be presented which I have discovered recently: a possibility to describe quantum mechanics of a spin (e.g. a q-bit) in this framework. This way we "discover" spinors (in a context which is entirely new) and propose a method to analyse problems of quantum informatics by means of a Wigner function for a q-bit.


• Yvette Kosmann-Schwarzbach (Ecole Polytechnique)
  Multiplicativity, from Lie groups to generalized geometry
Slides.
We survey the concept of multiplicativity, from its initial appearance in the definition of the Poisson Lie groups to the far-reaching generalizations for multivectors and differential forms in the geometry and the generalized geometry of Lie groupoids and Lie algebroids.


• Manuel de Leon (Instituto de Ciencias Matematicas)
  Hamilton-Jacobi theory in Cauchy data space
Slides.
We shall develop a Hamilton-Jacobi theory of Classical Field Theories using a multisymplectic setting, as well as the corresponding Hamilton-Jacobi theory in the Cauchy data space


• Charles-Michel Marle (Universite Pierre et Marie Curie)
  The works of William Rowan Hamilton in Geometric Optics and the Malus-Dupin theorem
Slides.
In this talk I will discuss the works of William Rowan Hamilton in Geometric Optics, with emphasis on the Malus-Dupin theorem. According to that theorem, a family of light rays depending on two parameters can be focused to a single point by an optical instrument made of reflexing or refracting surfaces if and only if, before entering the optical instrument, the family of rays is rectangular (i.e., admits orthogonal surfaces). Moreover, that theorem states that a rectangular system of rays remains rectangular after an arbitrary number of reflexions through, or refractions across, smooth surfaces of arbitrary shape. I will present the original proof of that theorem due to Hamilton, along with another proof founded in symplectic geometry. It was the proof of that theorem which led Hamilton to introduce his characteristic function in Optics, then in Dynamics under the name action integral.


• Giuseppe Marmo (Università di Napoli "Federico II")
  A quantum route to Hamilton-Jacobi theory
Slides.
There is a wide spread belief that the appropriate description of the physical world should be quantum, therefore one should require that the classical description should be an appropriate limit of the quantum one. From this point of view it is quite reasonable to ask about the fate of the complex structure and the linear structure available in the quantum setting but absent at the classical level. In this talk I shall discuss some of these questions, emerging when we go from the linear Schroedinger equation to the nonlinear Hamilton-Jacobi equation.


• Juan-Carlos Marrero (Universidad de La Laguna)
  Hamilton-Poincaré field equations
Slides.
In this talk, I will present several geometric descriptions of the Hamilton-Poincaré field equations for an equivariant hamiltonian section. First of all, I will discuss a local derivation of these equations. Next, I will present an intrinsic expression of the equations using the theory of prolongations of invariant vector fields to the reduced extended multimomentum bundle. A third description of the Hamilton-Poincaré field equations will be discussed using a principal connection (in the principal bundle with structural group, the symmetry group, and with total space, the configuration space of the theory). Finally, I will apply the previous results to some examples.


• Eduardo Martinez (University of Zaragoza)
  Jets and fields on Lie algebroids
Slides.
I will review on the extension of the concept of jet to the context of Lie algebroids and its application in Classical Field Theory.


• Guowu Meng (Hong Kong University of Science and Technology)
  Tulczyjew's approach for particles in gauge fields
Slides.
In this talk, we shall report that, via an idea due to Shlomo Sternberg, Tulczyjew's original approach to the dynamics of an "electrically-neutral" particle works equally well for the dynamics of an "electrically-charged" particle in non-abelian gauge fields.


• Norbert Poncin (University of Luxemburg)
  Multi-graded algebra and geometry
The aim of the talk is to present a generalization of superalgebra and supergeometry to Z_2^n-gradings, n>1. The corresponding sign rule is not given by the product of the parities, but by the scalar product of the involved Z_2^n-degrees. This Z_2^n - supergeometry exhibits interesting differences with classical supergeometry, provides a sharpened viewpoint, and has better categorical properties. Further, it is closely related to Clifford calculus: Clifford algebras have numerous applications in Physics, but the use of Z_2^n-gradings has never been investigated. If time permits, the Z_2^n-Berezinian determinant and the corresponding integration theory will be discussed.


• Olga Rossi (The University of Ostrava)
  Geometry of PDEs and Hamiltonian systems
Slides.
The talk will survey Hamiltonian field theory in jet bundles for first and some second order Lagrangians, and discuss geometric structures associated with Euler-Lagrange and Hamilton equations.


• Gennadi Sardanashvily (Moscow State University)
  Noether theorems in a general setting. Reducible graded Lagrangians
Slides.
Noether theorems are formulated in a general case of reducible degenerate Grassmann-graded Lagrangian theory of even and odd variables on graded bundles. A problem is that any Euler-Lagrange operator satisfies Noether identities, which therefore must be separated into the trivial and non-trivial ones. These Noether identities can obey first-stage Noether identities, which in turn are subject to the second-stage ones, and so on. Thus, there is a hierarchy of non-trivial Noether and higher-stage Noether identities. This hierarchy is described in homology terms. If a certain homology regularity conditions holds, one can associate to a reducible degenerate Lagrangian the exact Koszul-Tate chain complex possessing the boundary operator whose nilpotentness is equivalent to all complete non-trivial Noether and higher-stage Noether identities. Since this complex is necessarily Grassmann-graded, Lagrangian theory on graded bundles is considered from the beginning, and is formulated in terms of the Grassmann-graded variational bicomplex. Its cohomology defines a first variational formula whose straightforward corollary is the first Noether theorem. Second Noether theorems associate to the above mentioned Koszul-Tate complex a certain cochain sequence whose ascent operator consists of the gauge and higher-order gauge symmetries of a Lagrangian system. If gauge symmetries are algebraically closed, this ascent operator is prolonged to the nilpotent BRST operator which brings the gauge cochain sequence into a BRST complex, and thus provides a BRST extension of an original Lagrangian system. [G.Sardanashvily, arXiv: 1411.2910]


• Yunhe Sheng (School of Mathematics Jilin University)
  Lie 2-algebras, homotopy Poisson manifolds and Courant algebroids
Slides.
In this talk we study Maurer-Cartan elements on homotopy Poisson manifolds of degree n, which unify many twisted, or homotopy structures in Poisson geometry and mathematical physics, such as twisted Poisson manifolds, quasi-Poisson $\g$-manifolds and twisted Courant algebroids. We prove that the cotangent bundle of a homotopy Poisson manifold of degree n is a symplectic NQ-manifold of degree n+1. Using the fact that the dual of an n-term $L_\infty$-algebra is a homotopy Poisson manifold of degree n-1, we obtain a Courant algebroid from a 2-term $L_\infty$-algebra $\g$ via the degree 2 symplectic NQ-manifold $T^*[2]\g^*[1]$. Then, we derive a 2-term $L_\infty$-algebra from a given one. This construction could produce many interesting examples. By integrating the Lie quasi-bialgebroid associated to the Courant algebroid, we obtain a Lie-quasi-Poisson groupoid from a 2-term $L_\infty$-algebra, which is proposed 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.


• Alexandre Vinogradov (Levi-Civita Institute)
  DCinCA, NPDEs and (Q)FT : some problems and perspectives
Slides.
I shall try to present an unifying view on the current "differential mathematics" including classical differential geometry (CDE) and its natural generalization known as modern geometry of NPDEs.
In the first part of my talk I'll sketch how all structures of modern CDE and many others are almost algorithmically deduced from mathematically formalized observability mechanism in classical physics. In particular, it will be shown that the "observability considerations" leads to reveal a purely algebraic nature of differential calculus and hence its logical structure in the form of functors of differential calculus over commutative (in any sense) algebras (DICOCA). Some immediate applications to physics and mechanics will be outlined.
In the second part of the talk I will informally explain how to develop differential calculus and, in particular, differential geometry, on the "manifold of all solutions of a given NPDE". Essentially, this will be a simplified presentation of Secondary Calculus. Also my intention is to formulate some problems and to discuss some perspectives.


• Elizaveta Vishnyakova (MPI Bonn)
  Flag supermanifolds: definition, properties and applications
Yu.I. Manin introduced four series of compact complex homogeneous supermanifolds corresponding to four series of classical linear complex Lie superalgebras: the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb{C})$; the orthosymplectic Lie superalgebra $\mathfrak{osp}_{m|2n}(\mathbb{C})$ that annihilates a non-degenerate even symmetric bilinear form; the linear Lie superalgebra $\pi\mathfrak{sp}_{n|n}(\mathbb{C})$ that annihilates a non-degenerate odd skew-symmetric bilinear form; the linear Lie superalgebra $\mathfrak{q}_{n|n}(\mathbb{C})$ that commutes with an odd involution. These supermanifolds are called supermanifolds of flags in Case 1, supermanifolds of isotropic flags in Cases 2 and 3, and supermanifolds of $\pi$-symmetric flags in Case 4. As in the classical case flag supermanifolds are $\mathcal{G}$-homogeneous, where $\mathcal{G}$ is one of the following supergroups: $\GL_{m|n}(\mathbb{C})$, $\OSp_{m|2n}(\mathbb C)$, $\Pi\Sp_n(\mathbb C)$ or $\Q_n(\mathbb C)$, respectively.
We will give the notion of a flag supermanifold via functor of points and as a factor of a Lie supergroup modulo a parabolic subsupergroup. Further, we will discuss several unexpected properties of flag supermanifolds. For example, almost all flag supermanifolds fail to posses an embedding in a projective superspace. Another interesting question here is whether a flag supermanifold has a non-trivial local deformation. If we have time we will give some applications of flag supermanifolds in representation theory of classical Lie superalgebras and in the theory of super Riemann surfaces.


• Luca Vitagliano (University of Salerno)
  Vector bundle valued differential forms on NQ-manifolds
Slides.
Geometric structures on $NQ$-manifolds, i.e. non-negatively graded manifolds with an homological vector field, encode non-graded geometric data on Lie algebroids and their higher analogues. A particularly relevant class of structures consists of vector bundle valued differential forms. (Pre-)symplectic forms, (pre-)contact structures and, more generally, distributions are in this class. I describe vector bundle valued differential forms on non-negatively graded manifolds in terms of non-graded geometric data. Moreover, I use this description to present, in a unified way, novel proofs of known results, and completely new results about degree one $NQ$-manifolds equipped with certain geometric structures, namely symplectic structures, contact structures, (already present in literature) and, more generally, locally conformal symplectic, presymplectic and precontact structures (not yet present in literature). I also discuss the case of generic vector bundle valued higher order forms, in particular multisymplectic structures.


• Cornelia Vizman (West University of Timisoara)
  Central extensions of the Lie algebra of Hamiltonian vector fields
For a connected symplectic manifold, we classify the continuous central extensions of the Poisson Lie algebra, the Lie algebra of Hamiltonian vector fields, and the Lie algebra of symplectic vector fields.
This is joint work with Bas Janssens from Utrecht University.


• Aissa Wade (Penn State University)
  On cosymplectic groupoids
A cosymplectic groupoid is a Lie groupoid $G\rightrightarrows M$ endowed with a multiplicative cosymplectic structure. It's clear that the base manifold $M$ of any cosymplectic groupoid $G$ is equipped with a Poisson tensor together with a Poisson vector field: these are the push-forward (under the source map $s$) of the Poisson tensor on $G$ and its Reeb vector field, respectively. Consequently, the 1-jet of bundle $J^1M$ of the base manifold $M$ has a canonical Lie algebroid structure. We will discuss how certain Lie algebroid structures on the 1-jet of bundle of some Poisson manifolds can be integrated into cosymplectic Lie groupoids.


• Ping Xu (Penn State University)
  Infinite jets of exponential maps and $L_\infty$-algebras
Exponential maps arise naturally in the contexts of Lie theory and connections on smooth manifolds. The infinite jets of these classical exponential maps are related to the Poincaré-Birkhoff-Witt isomorphism and the complete symbol of differential operators. We will explain how to extend them to a variety of diverse situations including foliations and complex manifolds. In particular, we will show how such maps induce an interesting class of $L_\infty$-algebras.

(C) Michał Jóźwikowski, 2015