Friday Fish seminar

I will survey some old ideas and some recent developments in the study of spaces of smooth embeddings, starting from the early work of Haefliger, and leading on to the embedding calculus of Goodwillie-Weiss.

In the second part, I will focus on the case of classical knots and discuss some questions and recent progress on relating the homotopical approach to finite type theory.

In this talk, we will discuss the representation theory of higher Lie algebroids and higher VB-algebroid structures on double vector bundles. We will give the basic definitions about representations (up to homotopy) of Lie n-algebroids and construct some basic examples, such as the adjoint and coadjoint representations. Next, we will define the notion of a VB-Lie n-algebroid and explain how they correspond to (n+1)-representations of their side Lie n-algebroid, generalizing thus a well-known result from representation theory of ordinary Lie algebroids.

The talk is based on a joint work with Madeleine Jotz Lean and Rajan Mehta.

On the one hand, loose Legendrian submanifolds of contact manifolds are topologically flexible according to Murphy's h-principle. On the other hand, Legendrians are often times rigid with respect to the energy of contactomorphisms. For example, Dimitroglou Rizell and Sullivan showed that the displacement energy of closed Legendrians in certain contact manifolds is bounded away from zero. In this talk, I will explain that loose Legendrians with nice loose charts are as flexible as they can be according to Dimitroglou Rizell and Sullivan's rigidity result. The result can be interpreted as a quantitative version of Murphy's h-principle.

Manifold calculus is a useful theory which imports homotopy theoretical tools to analyse spaces of maps between manifolds. It provides us with a tower of fibrations, approximating the space of morphisms. In this talk, I’ll give an introduction of manifold calculus, via the example of embedding spaces of manifolds.

The concept of modular class is best known for Poisson structures, but is naturally defined for any Lie algebroid: It is a class in the first Lie algebroid cohomology. Poisson structures as Lie algebroids have the special feature that their dual is isomorphic to the tangent bundle and thus representatives are vector fields, which allows for the definition of the so-called modular foliation, locally spanned by Hamiltonian vector fields and the modular vector field. This modular foliation can in turn be viewed as the foliation of a Poisson structure on the total space of the real line bundle det(T^*M) (Gualtieri-Pym).

In this talk, I will show how to extend these concepts to general real or complex Dirac structures in exact Courant algebroids and discuss the information contained in the modular class of a Dirac structure in some non-Poisson examples.

This is joint work in progress with Ralph Klaasse.

Differential graded Lie algebras play an important role in deformation theory as algebraic objects classifying the infinitesimal neighbourhoods of moduli spaces around a basepoint. An informal principle asserts that geometric objects without a fixed basepoint should admit a similar description in terms of curved Lie algebras, which have a `differential' whose square is controlled by a curvature element. In this talk, I will discuss the relation between two algebraic models for the formal neighbourhood of a moduli space around a manifold, rather than around a single point: in terms of dg-Lie algebroids and in terms of curved Lie algebras over the de Rham complex. In particular, I will describe an embedding of the homotopy category of dg-Lie algebroids into the homotopy category of such curved Lie algebras. Joint work with Damien Calaque and Ricardo Campos.

An integrable Poisson manifold is said to be of strong compact type if the source 1-connected groupoid integrating it is compact. One class of such manifolds is that of compact symplectic manifolds with finite fundamental group, but beyond that finding examples is difficult. The first non-symplectic example was given by D. Martínez Torres in 2014. The construction there is inspired by D. Kotschick’s construction of a free symplectic circle action with contractible orbits. In this talk I will go over the original construction, recalling the relevant results on Poisson manifolds of compact types as well as the geometry of the moduli spaces of K3 surfaces, and then modify the construction to obtain more examples. In the end, we will have for every strongly integral affine 2-torus (i.e. integral affine 2-torus with integral translational part) a Poisson manifold of strong compact type having that torus as its leaf space.

The globalization procedure of the Kontsevich formality by Dolgushev depends on the choice of a torsion-free covariant derivative. We show that the globalized formalities with respect to two different covariant derivatives are homotopic. More explicitly, we derive the statement by proving a general homotopy equivalence between $L_infty$-morphisms that are twisted with gauge equivalent Maurer-Cartan elements. This talk is based on joint work with Jonas Schnitzer.

In this talk, we will explore relationships between Riemannian geometry and Lie groupoids. We will begin by reviewing the correspondence between effective proper isometric actions and closed subgroups of the isometry group, then we will see how to translate it to Lie groupoids. Next, we discuss the linear holonomy groupoid for singular Riemannian foliations and show that its closure is a proper Lie groupoid. We finish by giving results about how far a Riemannian groupoid is from being proper. This talk is based on joint work with Marcos Alexandrino, Marcelo Inagaki, and Ivan Struchiner.

Start with a two-variable complex polynomial f with an isolated critical point at the origin. We will survey a range of classical structures associated to f, and explain how these can be revisited and enhanced using insights from symplectic topology. No prior knowledge of singularity theory will be assumed.

Let $H$ be a Cartan subgroup of a semisimple algebraic group $G$ over the complex numbers. The wonderful compactification $bar H$ of $H$ was introduced and studied by De Concini and Procesi. For the Lie algebra $fh$ of $H$, we define an analogous compactification $bar fh$ of $fh$, to be referred to as the wonderful compactification of $fh$. We establish a bijection between the set of irreducible components of the boundary $bar fh - fh$ of $fh$ and the set of maximal closed root subsystems of the root system for $(G, H)$ of rank $r - 1$, where $r$ is the dimension of $fh$. An algorithm based on Borel-de Siebenthal theory that constructs all such root subsystems is given. We prove that each irreducible component of $bar fh - fh$ is isomorphic to the wonderful compactification of a Lie subalgebra of $fh$ and is of dimension $r - 1$. In particular, the boundary $bar fh - fh$ is equidimensional. We
describe a large subset of the regular locus of $bar fh$. As a consequence, we prove that $bar fh$ is a normal variety.

I will begin with a brief account on the theory of generalized connections and Dirac generating operators on transitive Courant algebroids. I will develop a T-duality for transitive Courant algebroids, which is a generalization of the T-duality of Cavalcanti and Gualtieri in the exact case. I will show that T-duality between transitive Courant agebroids E ra M and tilde{E} ra tilde{M} induces a map between the spaces of sections of the corresponding canonical weighted spinor bundles of E and tilde{E} which intertwines the canonical Dirac generating operators. I will present a general existence result for T-duals under assumptions generalizing the cohomology integrality conditions for the T-duality in the exact case.

If time allows, I will explain that the T-dual of a heterotic Courant algebroid is again heterotic. This is joint work with Vicente Cortes.

Meromorphic connections on Riemann surfaces (i.e., an invariant point of view on complex singular ODEs) can be understood as representations of relevant holomorphic Lie algebroids or equivalently (by integration, called the Riemann-Hilbert correspondence) as Lie groupoid representations. I will describe an approach to studying these objects called abelianisation. It uses simple combinatorial data attached to the Riemann surface (certain topological graphs) to recast the representations into the nonabelian group GL(n,C) instead as the rather more amicable representations into the abelian group C* but on a suitable n-fold covering Riemann surface. This technique has its origins in the works of Fock-Goncharov (2006) and Gaiotto-Moore-Neitzke (2013), as well as the WKB analysis. I will explain this correspondence in the case n=2 as an equivalence of categories of representations.

This presentation is based on arXiv:1902.03384 and the more recent work in progress extending those results.

In this talk I will talk about a joint work with Alfonso Garmendia. Singular foliations are a class of subsheaves of the sheaf of vector fields. Although they are not Lie algebroids, singular foliations are sheaves of Lie-Rinehart algebras and therefore have many features in common with Lie algebroids. In this talk I will discuss how to generalize the path model for integrating Lie algebroids to the setting of singular foliations.

A folklore theorem of Eliashberg and Gromov, which states that the group of symplectic diffeomorphisms is closed in the group of all diffeomorphisms w.r.t. C^0-topology, is considered the beginning of $C^0$-symplectic geometry which roughly investigates which symplectic phenomena persist under $C^0$-limits. It also allows us to define symplectic homeomorphisms (as $C^0-limits of symplectomorphisms) and $C^0$-symplectic manifolds. So far it is not known whether there are manifolds that admit a $C^0$-symplectic, but not smooth symplectic structures. One of the central question in the field, due to Hofer, is whether the spheres admit such structure in dimension greater than 2.

In this (mostly overview) talk we will state and prove Elisahberg-Gromov's theorem using another result from $C^0$-symplectic geometry, namely, the $C^0$-rigidity of Poisson brackets due to Cardin-Viterbo, Entov-Polterovich, and Buhovski. If time allows, we will discuss Hofer's question as well.

Symplectic groupoids are Lie groupoids equipped with multiplicative symplectic forms, and serve as global counterparts to (integrable) Poisson manifolds, in the same way that Lie groups are global counterparts to Lie algebras.

In this talk I will present the construction of the deformation cohomology controlling deformations of symplectic groupoids, built out of the deformation complexes of a Lie groupoid and of a multiplicative form. We will see that it can be identified with the total complex of a sub double complex of the Bott-Shulman-Stasheff double complex.

I will then compute this cohomology in some particular cases (linear Poisson structures and proper groupoids), explain how to use it in a Moser path argument, and describe some relations to the deformation theory of the corresponding Poisson manifolds.

This is joint work with Cristian Cárdenas (UFF) and Ivan Struchiner (USP).

One of the building blocks in the study of the homotopy type of the diffeomorphism group of a 3-manifold is the positive answer of A. Hatcher to the Smale conjecture. This result has its contact counterpart: Eliashberg's theorem about the contractibility of the contactomorphism group of the tight contact 3–ball.

In order to adapt the smooth techniques to the contact world, it is necessary to develop an understanding of the space of embeddings of surfaces into a given tight contact 3-manifold. In this talk we will see how to do this for simple surfaces such as disks and spheres. As a consequence, we will conclude that, with the exception of connected components, all the remaining homotopy groups of the group of contactomorphisms of a tight contact 3-manifold are controlled by topological invariants.

This talk is a continuation of the previous talk "Flexibility in contact 3-manifolds: from contactomorphisms to legendrian knots" by Javier Martínez-Aguinaga. Joint work with Javier Martínez-Aguinaga and Fran Presas.

The study of the homotopy type of the space of legendrian submanifolds in a contact manifold is a central problem in Contact Topology. In this talk, mimicking the techniques developed in Smooth Knot Theory (work of A. Hatcher and R. Budney), we will relate the homotopy type of such spaces with the homotopy type of the contactomorphism group of the complement.

In particular, we show that the inclusion of each connected component of the space of long legendrian embeddings into the space of smooth long embeddings is a homotopy equivalence. This result shows that, except for the number of connected components, the homotopy of the space of Legendrians is governed by the topology of the space of smooth knots.

This talk will be the first part of a 2-talk session, followed by "On the homotopy type of the contactomorphism group of a tight contact 3-manifold" by Eduardo Fernández. Joint work with Eduardo Fernández and Francisco Presas.

By complexifying a Hamiltonian system one obtains dynamics on a holomorphic symplectic manifold. To invert this construction we present a theory of real forms which not only recovers the original system but also yields different real Hamiltonian systems which share the same complexification. In this talk I will explain the notion of real form for a symplectic manifold, more generally, for a Poisson manifold and will discuss their applications to integrable Hamiltonian systems. This is based on joint work with P. Arathoon.

A strict Lie 2-algebra is a groupoid object in the category of Lie algebras. These can naturally be seen as an infinitesimal version of strict Lie 2-groups which are groupoids in the category of Lie groups. Lie 2-algebras were proven to be integrable in this sense using the path strategy to integration, we present an alternative proof based on the van Est strategy.

Many important geometric structures are geometrically rigid in the sense that their Lie algebra of infinitesimal automorphisms is finite-dimensional. Prominent examples of such structures are Riemannian and conformal manifolds, and in general all geometric structures admitting equivalent descriptions as so-called Cartan geometries. Generically these geometric structures have trivial automorphism groups and so the ones among them with large automorphism groups or special types of automorphisms are typically geometrically and topologically very constrained and hence can often be classified.

In this talk I will discuss several such classification results and explain how Cartan connections can be used to study local and global questions of geometric rigidity. The talk will also provide an introduction to Cartan geometries with a focus on the subclass of parabolic geometries.

I will discuss the existence of hyperkähler structures on symplectic realizations of holomorphic Poisson manifolds, and show that they always exist when the Poisson manifold has complex dimension two. We obtain this structure by constructing a twistor space by lifting special deformations of the Poisson surface adapted from Hitchin's unobstructedness theorem. In the case of the zero Poisson structure, we recover the Feix-Kaledin hyperkähler structure on the cotangent bundle of a Kähler manifold. This talk is based on arXiv:2011.09282.

Given a Hamiltonian symmetry on a Poisson manifold one can construct a Poisson structure on a reduced manifold. This can be achieved with the Poisson version of the Marsden-Weinstein reduction or equivalently with the BRST-method.

Fixing a Lie group action on a manifold, one can define a curved Lie algebra whose Maurer-Cartan elements are Poisson structures together
with momentum maps. Poisson structures on the reduced manifold are Maurer-Cartan elements of the usual DGLA of polyvector fields. Thus, reduction is just a map between these two sets of Maurer-Cartan elements.

In my talk I want to show that this map is actually the map on Maurer-Cartan elements induced by an $L_infty$-morphism.

The group of symplectomorphisms of a symplectic manifold is a well-studied object. However, little is known about the Poisson diffeomorphism group of other Poisson structures.

In this talk, we present a first step in this direction. First, we discuss the intricate algebraic structure of the Poisson diffeomorphism group and its connection with coisotropic bisections of a Poisson groupoid. Then, we reformulate the problem of finding smooth structures on these objects to a linearization problem of Poisson structures around Lagrangian submanifolds.

In the second part, we dive deeper into linear Poisson structures and present a linearization result, generalizing the Lagrangian neighbourhood theorem to the setting of Lie algebroids and cosymplectic structures. This is applied to obtain smooth structures on Poisson diffeomorphism groups of several classes of Poisson manifolds.

This is part of my master thesis, supervised by Ioan Marcut.

Motivated by semilocal normal forms and invariants for (non)-commutative integrable systems on manifolds with boundary, we prove a singular symplectic slice theorem for $b^m$-symplectic manifolds and discuss some generalizations. This is joint work with Eva Miranda.

We describe the coadjoint orbits of the group of volume preserving diffeomorphisms of R3 associated to the motion of closed vortex sheets in ideal 3D fluids. We show that these coadjoint orbits can be identified with nonlinear Grassmannians of compact surfaces enclosing a given volume and endowed with a closed 1-form describing the vorticity density. Goldin, Menikoff, and Sharp present in [1] the case of open vortex sheets (tubes/ribbons).

If the vorticity density has a discrete period group and is nonvanishing, the vortex sheet is given by a surface of genus one fibered by its vortex lines over a circle. We determine the Hamilton equations for such vortex sheets relative to the Hamiltonian function suggested by Khesin in [2] and we prove that there are no stationary solutions having rotational symmetries. These coadjoint orbits are shown to be prequantizable if the period group of the 1-form and the volume enclosed by the surface satisfy an Onsager-Feynman relation.

Joint work with Francois Gay-Balmaz, ENS Paris.

[1] Goldin, G.A., Menikoff, R., Sharp, D.H., Quantum vortex configurations in three dimensions, Phys. Rev. Lett., 67 (1991), 3499-3502.
[2] Khesin, B., Symplectic structures and dynamics on vortex membranes, Moscow Math. Journal 12 (2012), 413-434.

In this talk, the idea is to describe with some detail certain singular cotangent (algebroid) homotopies, associated to a Poisson manifold M, which appear as a semiclassical limit in the Poisson Sigma Model (PSM). To that end, first, we will review the heuristic motivation for the PDEs that define these homotopies and their role in quantization. We then proceed to describe the solutions and relate them to certain triangles in an integrating groupoid. The final aim is to show how these homotopies, when fed into a PSM action functional, yield a generating function for the structure of a (local) symplectic groupoid integrating the underlying Poisson M.

Log-symplectic structures are a type of Poisson structures that are symplectic outside of a hypersurface. The aim of this talk is to discuss whether the deformation theory of Lagrangian submanifolds in this setting is as well-behaved as in symplectic geometry.

We will focus on deformations of a Lagrangian submanifold contained in the singular locus of a log-symplectic manifold. Using a normal form around the Lagrangian, we show that the deformation problem is governed by a DGLA. We discuss whether the Lagrangian admits deformations not contained in the singular locus, and we give criteria for unobstructedness. If time permits, we also address equivalences of Lagrangian deformations under Hamiltonian and Poisson isotopies.

This is joint work with Marco Zambon.

Let G be a Lie groupoid. The category BG of principal G-bundles defines a differentiable stack. On the other hand, given a differentiable stack D, there exists a Lie groupoid H such that BH is isomorphic to D. We define a gerbe over a stack as a morphism of stacks F : D -> C, such that F and the diagonal map Delta_F : D -> D times_C D are epimorphisms.

In this talk we explore the relationship between a gerbe, as defined above, and a Morita equivalence class of a Lie groupoid extension. This talk is based on our paper (j/w Saikat Chatterjee) titled "On two notions of a gerbe over a stack" (https://doi.org/10.1016/j.bulsci.2020.102886).

Differentiable stacks are useful models for singular spaces. Fifteen years ago, Tseng and Zhu defined stacky Lie groupoids and used them to study the integrability of Lie algebroids. In this talk we will introduce their infinitesimal counterpart, stacky Lie algebroids, and we shall give their basic properties.

If time permits, we will comment on the integrability of Courant algebroids. This is a joint work with Daniel Alvarez.

This talk will give an introduction to multisymplectic geometry and homotopy moment maps. I will start from the basics, defining notions like n-plectic manifolds, Hamiltonian vector fields, and Hamiltonian (n-1)-forms.

Then, before defining the Lie n-algebra of observables corresponding to an n-plectic manifold (definition due to C. Rogers), I will give a brief introduction to $L_{infty}$-algebras.

Finally, I will introduce two notions of moment maps, the first one due to M. Callies, Y. Fregier, C. L. Rogers, and M. Zambon, and the second one due to J. Herman. If time permits, we will also compare the two notions (this last part would be based on joint work with L. Ryvkin).

I will describe a variant of the normal bundle to a submanifold that is appropriate when one wants to treat normal directions to the submanifold as having different `weights'. The construction has applications to normal form theorems. This is joint work with Eckhard Meinrenken.

Let G be a semisimple complex algebraic group of adjoint type. We consider a multiplicative analogue of the universal centralizer of G --- a family of centralizers parametrized by the regular conjugacy classes of the simply-connected cover of G. This multiplicative analogue has a natural symplectic structure and sits as a transversal in a variation of the quasi-Poisson double D(G). We show that D(G) extends to a smooth groupoid over the wonderful compactification of G, and we use this to construct a partial compactification of the multiplicative universal centralizer.

The differentiable cohomology of a pseudogroup on a manifold M was defined by André Haefliger in the seventies, in the development of characteristic classes of foliations. In Haefliger's approach to the subject, foliations are seen as cocycles valued in the étale groupoid Gamma^q of germs of diffeomorphisms of R^q. From this point of view, the theory of characteristic classes for foliations is reminiscent of the theory of characteristic classes for flat principal g-bundles, which can be represented as cocycles valued in the discrete group G^delta.

For such bundles, a "geometric" characteristic map is defined. It is a map from the relative cohomology H*(g, K) of the Lie algebra g of G to the cohomology of M. The classical Van-Est isomorphism for Lie groups allows one to interpret it as a map from the differentiable cohomology H_d*(G) of G to H*(M). A similar "geometric" map is available for foliations and is defined from the cohomology GF^q of the Gelfand-Fuchs Lie algebra of formal vector fields on R^q relative to O(q). Haefliger defined a differentiable complex for the groupoid Gamma^q and proved a Van Est isomorphism between the differentiable cohomology H*_{diff}(Gamma^q) and the cohomology of the relative Gelfand-Fuchs Lie algebra, making the analogy with flat principal bundles complete.

His construction focuses on the groupoid Gamma^q and, probably also because of the seemingly ad-hoc approach, did not receive much attention. Furthermore, it is not clear which is the structure that makes the definition work. Our main driving question is to find such a structure. The outcome is the structure of "flat Cartan groupoid" on the infinite jet groupoid J^inftyGamma^q, which is well known for the role that plays in the geometric theory of PDE. After reviewing Haefliger's approach and introducing flat Cartan groupoids, we will define their "Haefliger cohomology", investigate its infinitesimal counterpart and relate the two by a Van Est map. Finally, we will construct a characteristic map for cocycles valued in a flat Cartan groupoid, which reduces to the "geometric" map for foliations when the groupoid is J^inftyGamma^q.

This is joint work with Marius Crainic.

We study Lie-(Hamilton) systems on the plane, i.e. systems of first-order (nonlinear) ODEs describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar (Hamiltonian) vector fields with respect to a geometric structure.

Lie-(Hamilton) systems enjoy a plethora of properties, e.g. they admit their general solution expressed as a (nonlinear) function through the so-called superposition rule of a finite set of particular solutions and some constants.

Lie-Hamilton systems are important because of their appearance in the physics, mathematics and biology literature. For example, they can be used to study Milne–Pinney, second-order Kummer–Schwarz, complex Riccati and Buchdahl equations, which occur in cosmology, relativity and classical mechanics. They also appear in the investigation of Lotka–Volterra, predator-prey or growth of a viral infection models. We are particulary interested in the latter.

In this talk, I present the geometrical properties of Lie-(Hamilton) systems and their application to SIS-pandemic models. We derive complete solutions for a SIS-pandemic model with fluctuations through the intrinsic properties of Lie systems and through the coalgebra method. We will present graphic representations of the solutions, and we will see how the number of infected individuals grows accordingly with a sigmoid-like function. This is precisely the expected behavior, and it is retrieved in two different geometric ways, from a symplectic point of view, and from a Poisson framework.

We finish this talk discussing whether this model is applicable to the current Covid pandemic.

In this talk I'll describe a Lie groupoid based approach to the study of flat connections with logarithmic singularities on a hypersurface.
Flat connections on the affine line with logarithmic singularity at the origin are equivalent to representations of a groupoid associated to the exponentiated action of C. I'll describe a canonical Jordan-Chevalley decomposition for these representations, and show how this leads to a functorial classification.
Flat connections on a general complex manifold with logarithmic singularities along a hypersurface are equivalent to representations of a twisted fundamental groupoid. By using a Morita equivalence, the category of logarithmic flat connections can be localized to the normal bundle of the hypersurface. I'll explain how this can be used to prove a functorial Riemann-Hilbert correspondence for logarithmic flat connections.

This talk describes the Atiyah class of a holomorphic vector bundle and explains how it can be understood as the Atiyah class of an infinitesimal ideal system. Then it explains how Molino's Atiyah class also fits in this context.

The Atiyah class of a Lie pair is an obstruction to the existence of an ideal system (à la Mackenzie and Higgins) on the Lie pair.

In this talk we take a look at the linearization question in Poisson geometry as first asked by Weinstein. We briefly recall what is
currently known for semisimple Lie algebras. Moreover, we outline a proof for the Poisson non-degeneracy of the Lie algebra so(3,1). Therefore so(3,1) is the first example of a non-compact semisimple Lie algebra which is Poisson non-degenerate. This is based on joint work with Ioan Marcut.

In this talk, we study contact structures that admit certain types of singularities, called $b^m$-contact structures. Those structures can be viewed as a particular case of Jacobi manifolds satisfying some transversality conditions. The motivation to study this generalization of contact structures arises from classical examples in celestial mechanics, as for example the restricted planar circular three body problem (RPC3BP), but also appears in the study of fluid dynamics on manifolds with cylindrical ends.

We will focus on understanding the dynamics of the associated Reeb vector field of $b^m$-contact forms. Due to singularities, the dynamics are fundamentally different to smooth Reeb dynamics and we will discuss a singular version of Weinstein conjecture on the existence of periodic orbits on those manifolds. Time permitting, we talk about generic existence of so called singular periodic orbits.

This is joint work with Eva Miranda and work in progress with Eva Miranda and Daniel Peralta-Salas.

Let G be a simply connected Lie group. We denote by C(G) the differential graded Hopf algebra of smooth singular chains on G. We will
explain how the category of modules over C(G) can be described infinitesimally in terms of representations of the differential graded Lie
algebra Tg, which is universal for the Cartan relations. In case G is compact, this correspondence can be promoted to an A-infinity equivalence of differential graded categories. We will also explain how this equivalence is related to Chern-Weil theory and higher local systems on classifying spaces.

This talk is based on joint work with Alexander Quintero Vélez.

Characteristic classes are an important tool in algebra, geometry and topology as they encode invariants that can often be calculated explicitly. In this talk I will present some recent ongoing work about the definition of characteristic classes associated to representations of Lie groupoids. I will concentrate on the construction of such classes as this will shine some light on the underlying geometry and the relation with the classifying space.

In this talk we describe an ongoing project to construct the adjoint (symplectic) groupoids associated to elliptic Poisson structures. These are a type of Poisson structure that is nondegenerate outside of a submanifold of codimension two. We will describe the geometry of these structures and see that their behavior depends on their so-called elliptic residue, and consequently so do the groupoids. Tools that are used in our constructions are the blow-up procedure for groupoids of Gualtieri-Li, a normal form result due to Witte, and the fact that elliptic Poisson manifolds can be blown-up to be log-Poisson, as noticed by Cavalcanti-Gualtieri.

While contact geometry is often described as the odd-dimensional analogue of symplectic geometry, Arnol'd pointed out that the relation between these two geometries is analogous to that between affine and projective geometries. A natural (vague!) question is to investigate the extent to which this analogy extends to the degenerate versions of symplectic and contact structures, namely Poisson and Jacobi geometries.

The aim of this talk is to attempt to formalise the above question and to illustrate the intimate relation between these six geometries through a few specific problems, including isotropic realisations and Poisson/Jacobi manifolds of compact types. The talk is based on joint work with María Amelia Salazar, and on ongoing joint work with Camilo Angulo and María Amelia Salazar.