Schedule for: 17w5023 - Geometric Structures on Lie Groupoids

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We show that the category of vector fields on a geometric stack is a Lie 2-algebra. I will start by sketching out the definitions of a stack, a geometric stack, vector field on a stack and of a (Baez-Crans) Lie 2-algebra, which is a categorified version of a Lie algebra. (joint work with Daniel Berwick-Evans)

Heuristically, it is known that Courant algebroids should "integrate" to symplectic 2-groupoids, but very little of this correspondence has been developed in a precise way. I will describe in detail the case of a linear 2-groupoid equipped with a constant symplectic form, and I will explain how these "constant symplectic 2-groupoids" correspond to a certain class of Courant algebroids. The study of constant symplectic 2-groupoids is intended to be a first step toward a more general study of symplectic 2-groupoids, in analogy to how a student usually first learns about symplectic vector spaces before moving on to symplectic manifolds. Symplectic 2-groupoids are closely related to the shifted symplectic structures studied by Pantev, et al, although the definition is more "strict" in certain ways. As part of the talk, I will give some context to explain why the additional strictness is appropriate for the problem of integrating Courant algebroids.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

When working with Lie groupoids, representations up to homotopy arise naturally, and they are useful, for instance, to make sense of the adjoint representation. The idea behind them is to use graded vector bundles and allow non-associativity. We discuss the symmetries of a graded vector bundle and show that, in the 2-term case, they can be regarded as a Lie 2-groupoid. We show that the nerve of a Lie 2-groupoid is a simplicial manifold, and use this construction to realize 2-term representations up to homotopy as pseudo-functors. Based in a joint work with D. Stefani.

The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra (dgLa) which depends on the choice of an auxillary transverse Dirac structure. In this talk, we show that different choices of transverse Dirac structure may lead to dgLas which are not isomorphic (as dgLas), but which are isomorphic as $L_{\infty}$-algebras. We apply our results to study the Kodaira-Spencer deformation complex of a complex manifold.

We explain how extending Haefliger's approach to transverse measures for foliations to general Lie groupoids allows us to define and study measures and geometric measures (densities) on differentiable stacks. The abstract theory works for any differentiable stack, but it becomes very concrete for those presented by proper Lie groupoids - for example, when computing the volume associated with a density, we recover the explicit formulas that are taken as definition by Weinstein. This talk is based on joint work with Marius Crainic.

In this talk, I will present some recent results on the geometric construction of the exact discrete Lagrangian function associated with a continuous regular Lagrangian function. This Lagrangian function is defined on the Lie algebroid of a Lie groupoid. In the last part of the talk, I will discuss two applications of the previous construction: i) Analysis of the error between the exact solutions of the Euler-Lagrange equations for the continuous Lagrangian function and the discrete trajectories derived by a variational integrator and ii) A relation with the Hamilton-Jacobi theory for the Hamiltonian function associated with the regular Lagrangian function.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We present a survey of results regarding multiplicative structures on Lie groupoids (e.g. tensor fields, differential forms with values in representations, foliations). The Lie theory of such structures establishes an infinitesimal vs. global correspondence with relevant data on the Lie algebroid. Our aim is to show how the simple idea of treating tensor fields as functions on Whitney sums of vector bundles allows us to unify various results on the literature as well as extend some of them to more general contexts (e.g. differential forms with values in representations up to homotopy).

We define a class of Poisson manifolds that is well-behaved from the point of view of singular foliations, understood as as submodules of vector fields rather than partitions into leaves: the class of almost regular Poisson manifolds. The latter admit a geometric characterization in terms of the symplectic leaves alone, and contain the class of log-symplectic manifolds. We study the holonomy groupoid integrating the singular foliation of an almost regular Poisson structure. We shot that it is a Poisson groupoid, integrating a naturally associated Lie bialgebroid. The Poisson structure on the holonomy groupoid is regular, and as such it provides a desingularization.

We use methods from Dirac geometry to prove that any Poisson homogeneous space admits an integration to a symplectic groupoid. This is joint work with Henrique Bursztyn and Jiang Hua Lu.

We give a direct, explicit and self-contained construction of a local Lie groupoid integrating a given Lie algebroid which only depends on the choice of a connection. We also give a complete account of local Lie theory based on these explicit constructions. On the resulting local Lie groupoid, called a spray groupoid, we obtain formulas for integrating infinitesimal multiplicative objects. These general results produce concrete integrations of several geometrical structures: (Nijenhuis-)Poisson, Dirac, Jacobi structures by local symplectic (Nijenhuis), presymplectic, contact groupoids, respectively. Joint work with A. Cabrera and I. Marcut.

The infinitesimal data attached to a (“finite type” class of) G-structures with connections are its structure equations. Such structure equations give rise to Lie algebroids endowed with extra geometric information following from the fact that they come from G-structures. For example, the Lie algebroid is trivial as a vector bundle, its bracket encodes the Lie bracket and the natural representation on $R^n$ of the Lie algebra of the structure group G of the G-structure, the Lie algebroid comes equipped an action of G by inner Lie algebroid automorphisms, etc… Conversely, given a Lie algebroid as above (called a G-algebroid), a natural question is that of finding G-structures which correspond via differentiation to the Lie algebroid. This integration problem is known as “Cartan’s Realization Problem for G-Structures”. In this talk I will show that if a G-algebroid is integrable by a Lie groupoid endowed with an action of G, then each s-fiber of the groupoid can be identified with the total space of a G-structure with connection solving the realization problem. If time permits I will also explain the obstructions for finding such “G-integrations” of G-algebroids. The talk will be based on joint work with prof. Rui Loja Fernandes

We consider topological orbifolds as proper \'etale groupoids, i.e., topological groupoids with a proper diagonal and \'etale structure maps. We call these orbigroupoids. To describe maps between these groupoids and 2-cells between them, we will use the bicategory of fractions of the 2-category of orbigroupoids and continuous functors with respect to a subclass of the Morita equivalences which is gives a bicategory of fractions that is equivalent to the usual one and renders mapping groupoids that are small. We will present several nice results about the equivalence relation on the 2-cell diagrams in this bicategory that then enable us to obtain a very explicit description of the topological groupoid $\mbox{Map}\,(G,H)$ encoding the new generalized maps from $G$ to $H$ and equivalence classes of 2-cell diagrams between them, for any orbigroupoids $G$ and $H$. In particular, we obtain the arrow space as a retract of the space of all 2-cell diagrams. When $G$ has a compact orbit space we show that the mapping groupoid is an orbigroupoid and has the appropriate universal properties to be the mapping object. In particular, sheaves on this groupoid for the mapping topos for geometric morphisms between the toposes of sheaves on $G$ and $H$. This groupoid can also be viewed as a pseudo colimit of mapping groupoids in the original 2-category of topological groupoids and continuous functors.

Given a proper, cocompact action of a Lie groupoid, we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We will explain a cohomological index formula computing this pairing. This is joint work with Markus Pflaum, and Hessel Posthuma.

I will give an overview on the recent (and ongoing) joint work with R.L. Fernandes and D. Martinez on Poisson manifolds of compact types. These are the analogues in Poisson Geometry of the compact Lie groups from Lie theory. However, within this Poisson geometric context, one finds various parts of geometry coming in and interacting with each together (symplectic, foliations, integrable systems, gerbes). My plan is to exhibit these by concentrating on Duistermaat-Heckmann-types of formulas.

In this talk I will present a new approach to Generalized Kahler geometry in which a GK structure of symplectic type can be described in terms of a holomorphic symplectic Morita equivalence along with a brane bisection. I will then explain how this new approach can be applied to the problem of describing a GK structure in terms of holomorphic data and a single real-valued function (the generalized Kahler potential). This is joint work with Marco Gualtieri and Maxim Zabzine.

Lie groupoids with Morita equivalence are a convenient model for the study of singular manifolds. This talk will explain how one can interpret a symplectic groupoid and symplectic Morita equivalence as a model for a singular Dirac manifold. To that end, we will define a site (a category with a topology) whose objects are Dirac manifolds, DMan, and explain how to associate a stack over DMan to any symplectic groupoid. Furthermore, we will relate isomorphisms of such stacks with symplectic Morita equivalences.

The aim of this talk is to dicuss index theory of elliptic pseudodifferential operators on groupoids. It will be very much audience dependent... We will introduce some convolution algebras - including the C*-algebras- of a Lie groupoid, their K-theory, the construction of the index, the link with the tangent (or adiabatic) groupoid, and maybe the Baum-Connes conjecture.

Poisson brackets of special type on n-tuples of N by N matrices may be encoded by double brackets in the sense of van den Bergh. Interesting examples include constant and linear (KKS) Poisson brackets. In particular, these brackets admit moment maps for the GL(N) action by simultaneous conjugation of matrices in the n-tuple. Surprisingly, there are instances where the theory of double brackets deviates from the standard wisdoms of Poisson geometry. For instance, KKS brackets turn out to be non-degenerate, and under some assumptions a moment map uniquely determines the double bracket. These observations give rise to a new proof of the theorem by L. Jeffrey on symplectomorphisms between moduli of flat connections and reduced spaces of products of coadjoint orbits. The talk is mostly based on the work by F. Naef. If time permits, I’ll sketch how these results are related to the Kashiwara-Vergne theory and to the Goldman-Turaev Lie bialgebra.

In this talk we will present natural constructions of Lie groupoids coming from deformation and blowup procedure. We will see that these constructions enable one to recover many known constructions of Lie groupoids involved in index theory and that they lead to new index problems. This is a joint work with G. Skandalis.

The inertia space of a compact Lie group action or more generally of a proper Lie groupoid has an interesting singularity structure. Unlike the quotient space of the group action, respectively the groupoid, the inertia space can not be stratified by orbit types, in general. In the talk we explain this phenomenon and provide a stratification and local description of the inertia space. Moreover, we show that that leads naturally to the concept of a stratified groupoid which lies in between the one of a Lie groupoid and the one of a topological groupoid. Finally we show that a de Rham theorem holds for inertia spaces and explain the connection of the inertia space with the non-commutative geometry of the underlying groupoid.

The tangent groupoid is a geometric device for glueing a pseudodifferential operator to its principal symbol, via a deformation family. We will discuss a converse to this: briefly, pseudodifferential kernels are precisely those distributions that extend to distributions on the tangent groupoid that are essentially homogeneous with respect to the natural $R^+$-action. One could see this as a simple new definition of a classical pseudodifferential operator. Moreover, we will show that, armed with an appropriate generalization of the tangent groupoid, this approach allows us to easily construct more exotic pseudodifferential calculi, such as the Heisenberg calculus. (Joint work with Erik van Erp.)

In the 1980s, Alain Connes gave a conceptually appealing proof of the Atiyah-Singer index theorem by means of the so-called "tangent groupoid". Over time, the tangent groupoid was generalized to more complicated analytic settings. I will discuss the role played by groupoids in work on the index problem for hypoelliptic differential operators on contact manifolds. The groupoid perspective proved to be very fruitful in pointing the way to a solution of this problem. A non-trivial hurdle in this case is to construct the correct groupoid. The (by now) standard construction of "adiabatic groupoids" does not give the desired object. The integration theorem for Lie algebroids of Crainic-Fernandes provides the necessary technical tool for our construction.

Morse and Morse-Bott singular foliations, which have been studied my many authors, are of codimension 1. A natural question arises: what are higher-dimension analogs of these foliations? The aim of this talk is to give an answer to this question, in terms of what I will call "Dufour foliations", in honour of Jean Paul Dufour. I'll show some results on deformations and structural stability of Dufour foliations.

The constraint manifold for the initial value problem of general relativity is a coistropic subset in the symplectic manifold $P$ of 1-jets of lorentzian metrics along a space-like hypersurface. Coistropic submanifolds often arise as the zero sets of momentum maps for Lie algebra actions, but there is no evident Lie algebra action on $P$ to produce all of the constraints as momenta. Perhaps the Lie algebra should be replaced by a Lie algebroid. In an attempt to find the appropriate symmetry structure, Christian Blohmann, Marco Cezar Fernandes, and the speaker constructed a Lie algebroid over a space of \underline{infinite} jets for which the brackets relations among constant sections exactly matched the bracket relations among constraints, but this was not enough to explain the coisotropic nature of the constraint set. Two unanswered questions remain. (1) What are the appropriate notions of ``hamiltonian Lie algebroid" over a symplectic (or Poisson) manifold and associated ``momentum map" which make the zero sets of momentum maps coisotropic? (2) Is there a hamiltonian Lie algebroid over some ``extended phase space" $P'$ closely related to $P$ in which the constraint functions can be understood as the components of a momentum map? This talk will be a report on ongoing work with Blohmann and Michele Schiavina, the aim of which is to resolve these problems. Our working notion of hamiltonian structure on a Lie algebroid involves a connection with conditions relating its torsion to the symplectic structure on the base. A relevant extended phase space $P'$ appears to require a version of the BV-BFV construction currently under investigation by Cattaneo, Mnev, Reshetikhin, and Schiavina. Most of the talk will be devoted to question (1), with some remarks on question (2).

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.