Workshop on
Contact and Poisson Geometry
Timișoara, October 31 - November 02, 2019
Program
Lectures will be held in the amphitheater A11 (first floor), West University of Timișoara, main building (Address: Blvd Vasile Pârvan No. 4, Timișoara). To reach the lecture room, enter the main building, take the right stairs in front of you, make an U turn at the end of the stairs: the amphitheater is on the left.
For the Speakers
In the amphitheater there will be a beamer, and two small whiteboards. There will be also a laptop and an electronic pen that you can use to write directly on the screen.
Schedule (Tentative)
Thursday Oct 31
09:00 - 09:30 welcome coffe
09:30 - 09:45 official opening
09:45 - 10:30 Marrero
10:30 - 11:00 Oms
11:00 - 11:30 coffe break
11:30 - 12:00 Jovanovic
12:00 - 12:45 Crainic
Lunch break
14:45 - 15:30 Nunes da Costa
15:30 - 16:15 Gay-Balmaz
16:15 - 16:45 coffe break
16:45 - 17:15 Tortorella
17:15 - 18:00 del Pino Gómez
Talks
Henrique Bursztyn
Title: Integration of Poisson Homogeneous Spaces (Slides)
Abstract: An important issue in Poisson geometry is identifying integrable Poisson structures and describing their symplectic groupoids. We will use tools from Dirac geometry to prove that any Poisson homogeneous space of any Poisson Lie group is integrable, providing explicit constructions of their symplectic groupoids. Based on joint work with D. Iglesias and J.-H. Lu. Eugen-Mihăiţă Cioroianu
Title: On the Linear Setting for Dual Pairs (Slides)
Abstract: We consider the problem of dual pairs in the light of linear Jacobi structures. In view of this, we initially introduce the category of Jacobi vector spaces with emphasis on the 'transitive' objects. Then, we construct linear dual pairs with transitive Jacobi vector spaces as 'bases' and prove the characteristic subspace correspondence. Authors: E.M. Cioroianu and C. Vizman. Marius Crainic
Title: Almost Γ-Structures (Slides)
Abstract: TBA Álvaro del Pino Gómez
Title: Convex Integration without Ampleness? (Slides)
Abstract: Following the ideas of J. Nash for the C1-isometric embedding problem, M. Gromov developed in 1973 a method, called convex integration, for finding solutions of partial differential relations (PDRs) that are open and ample. Ampleness can be loosely understood as saying that the PDR is the complement of a set of codimension at least 2. There are several ways in which convex integration may be pushed further. For instance, openness can sometimes be dropped: the isometry PDR in Nash's original application is, in fact, closed. But what about ampleness? In dimension 1, both openness and ampleness are not essential. In this talk I will explain first how the Chow-Rashevskii theorem in Control Theory can be understood as a particular case of this. I will then discuss joint work with F.J. Martínez-Aguinaga in which we produce a scheme, similar to convex integration, which applies to certain non-ample PDRs. Tobias Diez
Title: Smooth Path Groupoids and the Smoothness of the Holonomy Map (Slides)
Abstract: I will discuss smoothness properties of the holonomy map of a smooth principal G-bundle with connection. For this, the space PM of piecewise smooth paths is endowed with a natural infinite-dimensional manifold structure, which turns PM into a Lie groupoid (up to reparameterization and retrace). With respect to the derived manifold structure on the space LM of piecewise smooth loops, the holonomy of a connection turns out to be smooth as a map Hol: LM → G. An explicit formula for the derivative of the holonomy map at a loop γ in terms of the curvature and the horizontal lift of γ will be given. Conversely, every smooth map H: LM → G satisfying certain natural morphism properties gives rise to a smooth principal G-bundle with a connection having H as its holonomy. This reconstruction theorem and Maurer-Cartan theory in the infinite-dimensional setting shines a new light on well-known facts about the topological classification of principal bundles and Chern-Weil theory. Rui Loja Fernandes
Title: Prequantization, Differential Characters and the Genus Integration (Slides)
Abstract: The genus integration of a Lie algebroid was introduced recently by Ivan Contreras and myself as the space of A-paths modulo A-homologies. We have shown that the obstructions to smoothness of the genus integration are related with extended monodromy groups. In this talk I will briefly survey these results, and look at the special case of the prequantization Lie algebroid associated with a closed 2-form, showing how our results recover the well-known prequantization condition as well as the usual description of principal S1-bundles with connection via differential characters. Based on joint work with Alejandro Cabrera. François Gay-Balmaz
Title: Dirac Structures in Nonequilibrium Thermodynamics (Slides)
Abstract: Dirac structures are geometric objects that generalize both Poisson structures and presymplectic structures on manifolds. They naturally appear in the formulation of constrained mechanical systems. In this talk, we show that the evolution equations for nonequilibrium thermodynamics admit an intrinsic formulation in terms of Dirac structures, both on the Lagrangian and the Hamiltonian settings. In absence of irreversible processes these Dirac structures reduce to canonical Dirac structures associated to canonical symplectic forms on phase spaces. This geometric formulation of nonequilibrium thermodynamic thus consistently extends the geometric formulation of mechanics, to which it reduces in absence of irreversible processes. The Dirac structures are associated to a variational formulation of nonequilibrium thermodynamics that we developed earlier and are induced from a nonlinear nonholonomic constraint given by the expression of the entropy production of the system. Several examples will be presented. Janusz Grabowski
Title: Linear Contact Structures (Slides)
Abstract: It is well known that the cotangent bundles with their canonical symplectic structures are, up to isomorphisms, the only linear symplectic manifolds. We prove an analogous result for contact structures. Stefan Haller
Title: A Dual Pair for the Contact Group (Slides)
Abstract: We consider an infinite dimensional non-linear Stiefel manifold of weighted embeddings into a contact manifold. This space carries a symplectic structure such that the contact group and the group of reparametrizations act in a Hamiltonian fashion with equivariant moment maps, respectively, giving rise to a dual pair, called the EPContact dual pair. Via symplectic reduction, this dual pair permits to identify certain coadjoint orbits of the contact group. Moreover, the EPContact dual pair gives rise to singular solutions for the geodesic equation on the group of contact diffeomorphisms. For the projectivized cotangent bundle, the EPContact dual pair is closely related to the EPDiff dual pair due to Holm and Marsden. This talk is based on joint work with Cornelia Vizman. Bozidar Jovanovic
Title: Bi-Hamiltonian, Algebraic and Geometric Aspects of Integrability of Natural Mechanical Systems on Stiefel Varieties (Slides)
Abstract: We study integrable geodesic flows and Neumann systems on Stiefel varieties. We give geometrical description of the flows and construct integrable discrete models. The results are obtained in collaboration with Yuri Fedorov (UPC, Barcelona). Honglei Lang
Title: Characteristic Pairs of Multiplicative Structures on Lie groupoids (Slides)
Abstract: Various multiplicative structures on Lie groupoids have been studied by Bursztyn-Cabrera-Ortiz, Bursztyn-Cabrera, Bursztyn-Drummond, Cabrera-Marcut-Salazar, Crainic-Salazar-Struchiner, Iglesias-Ponte-Laurent-Gengoux-Xu, Jotz-Lean-Stienon-Xu and many others. Based on these works, we describe characteristic pairs of multiplicative polyvector fields, differential forms and (1,1)-tensors on Lie groupoids. Some particular multiplicative structures on Lie groupoids are discussed. This is a joint work with Zhuo Chen. Vân Lê Hông
Title: Higher Dimensional Knot Space over G2 and Spin(7)-manifolds
Abstract: Let S be a compact oriented finite dimensional manifold and M a Riemannian manifold. We denote by Imm0(S ) the space of all immersions φ : S → M such that there is a closed subset S 0 ⊂ S of codimension at least one where φ|S \S 0 is injective. In my talk I shall show that, if M admits a parallel r-fold vector cross product φ ∈ Ωr(M,TM), and dimS = r − 1, then the higher dimensional knot space K(S, M) := Imm0(S )/Diff+(S ) is a formally Kähler manifold. This is a joint work in progress with Domenico Fiorenza. Our theorem generalizes Brylinski’s and Verbitsky’s results for the case S = S1, M = M3 and S = S1, M is a G2- manifold respectively. Our theorem also implies that the double loop space K(T2, M8) over a Spin(7)-manifold M8 is a formally Kähler manifold. Ioan Marcut
Title: Poisson Cohomology of sl2(R) (Slides)
Abstract: Together with Florian Zeiser, we have computed the Poisson cohomology of the linear Poisson structure corresponding to sl2(R). I will explain the methods we used, and what the result conjecturally says about outer Poisson automorphisms and deformations of sl2(R). Juan Carlos Marrero
Title: Contact Hamiltonian Systems, Reeb-Liouville Dynamics and Invariant Measures (Slides)
Abstract: In this talk, I will present some results on contact Hamiltonian dynamics. In fact, we will see that a Hamiltonian function H on a contact manifold C produces Reeb dynamics on the open subset U = {x ∈ C : H(x) is not 0}. Moreover, if the Reeb vector field of C is transverse to the submanifold S = {x ∈ C : H(x) = 0} then S is an invariant exact symplectic submanifold and the dynamics on S is, up to reparametrization, Liouville dynamics. The previous results will be used to discuss the existence of invariant measures for the dynamics. Joana Nunes da Costa
Title: Courant Algebroids as Curved Lie-infinity Algebras
Abstract: We extend a result of Roytenberg and establish a correspondence between Courant algebroids, which are doubles of proto-bialgebroids, and curved Lie-infinity algebras. We discuss applications to Nijenhuis tensors and recover some known structures. Cédric Oms
Title: Geometry, Topology and Dynamics of Singular Contact Manifolds (Slides)
Abstract: We will extend contact geometry to a more general setting by including singularities in the contact form. We interpret the associated contact structure as a non-integrable distribution on a manifold with boundary, where the distribution is tangent to the boundary. We analyze the geometry and topology of those manifolds and discuss the dynamical results of the associated Reeb vector field. This is joint work with Eva Miranda. Cristian Ortiz
Title: Reduction of IM-2-Forms and Their Lie Theory (Slides)
Abstract: IM-2-forms are the infinitesimal counterparts of multiplicative 2-forms on Lie groupoids. In this talk I will explain how IM-2-forms can be reduced along a given surjective submersion. This unifies several known reduction schemes but also provides new reduction procedures with Poisson outcome. The integration of reduced IM-2-forms will be discussed. This is based on an ongoing project with Alejandro Cabrera (Rio de Janeiro). Jonas Schnitzer
Title: Semi-local Structure of Jacobi-related Geometries (Slides)
Abstract: After a short introduction to Jacobi-related geometries and their connection to Poisson and contact geometry, I want to present some results on their (semi-)local structure around transversal submanifolds. As applications of these local structure theorems, I discuss the Dazord-Marle-Lichnerowicz splitting theorem for Jacobi pairs and a splitting theorem for homogeneous Poisson structures. Miron Stanciu
Title: Locally Conformally Symplectic Cotangent Bundle Reduction (Slides)
Abstract: I will show the construction of a generalization of symplectic reduction for locally conformally symplectic manifolds with respect to any regular value of the momentum mapping. This procedure is, under certain conditions, compatible with the existence of additional structures on the manifold (e.g. a complex structure or one related to contact geometry). I then present a theorem characterizing the reduced spaces of cotangent bundles. Alfonso Tortorella
Title: Homogeneous G-Structures (Slides)
Abstract: The theory of G-structures provides us with a unified framework for a large class of geometric structures, including symplectic, complex and Riemannian structures, as well as foliations and many others. Surprisingly, contact geometry - the "odd-dimensional counterpart" of symplectic geometry - does not fit naturally into this picture. In this paper, we introduce the notion of a homogeneous G-structure, which encompasses contact structures, as well as some other interesting examples that appear in the literature. This is joint work with L. Vitagliano and O. Yudilevich. Aïssa Wade
Title: Stability of Singular Leaves of Jacobi Manifolds
Abstract: In this talk, we will discuss conditions under which singular leaves of Jacobi manifolds are stable. Marco Zambon
Title: Coisotropic Submanifolds in b-Symplectic Geometry (Slides)
Abstract: I will report on joint work with Stephane Geudens. A b-symplectic structure is a certain kind of Poisson structure which is symplectic outside of a hypersurface. We single out two classes of coisotropic submanifolds: those transverse to the hypersurface turn out to admit a normal form theorem, which extends Gotay's theorem in symplectic geometry. Those that satisfy a stronger transversality property admit a coisotropic quotient which locally is always smooth, and which inherits a reduced b-symplectic structure.Posters
Marvin Dippel
Title: Generalized Coisotropic Reduction
Jordi Gaset
Title: k-Contact Formalism: a Framework for Field Theories with Dissipation.
Cristina-Elena Hreţcanu
Title: Warped Product Pointwise Submanifolds in Golden Riemannian Manifolds
Varun Jain
Title: On Induced Ricci Type Tensor of GCR-Lightlike Submanifolds of Indefinite Sasakian Manifolds
Andreas Kraft
Title: Reduction of Equivariant Polyvector Fields and Ideas for Polydifferential Operators
Rakesh Kumar
Title: On the Geometry of Hemi-Slant Lightlike Submanifolds of Indefinite Cosymplectic Manifolds
Pier Paolo La Pastina
Title: Deformations of VB-groupoids
João Nuno Mestre
Title: Deformations of Symplectic Groupoids
Aldo Witte
Title: Normal Forms of Mildly Degenerate Poisson Structures
Federico Zadra
Title: Numerical Methods for Contact Geometry and their Applications
Carlos Zapata-Carratalá
Title: Dimensioned Mechanics via Contact Geometry