Study index theory for foliated manifolds with boundary using blup groupoids.
problem Index theory for foliated manifolds with boundary.
method Use blup groupoids and pseudodifferential calculus to study index problems.
result Construct and prove new index theorems for fully elliptic operators.
Develops a multilevel method for solving BLUP and GLS models with high dimensional data.
problem Numerical instability and ill-conditioned covariance matrices in high-dimensional data.
method Multilevel basis construction and transformation of covariance matrices using kD-tree partitioning.
result The multilevel method solves BLUP and GLS models accurately and efficiently, scaling well with dimensions and observations.
Study on cosymplectic groupoids with structural results.
problem Understanding cosymplectic groupoids and their properties.
method Structural analysis of cosymplectic groupoids, comparison with other groupoids.
result Several structural results for cosymplectic groupoids.
Paper defines PB-groupoids and their relation to VB-groupoids.
problem Establishing PB-groupoids as a Lie 2-groupoid counterpart.
method Defining PB-groupoids as Lie 2-groupoid actions on diagrams of Lie groupoids and principal bundles.
result PB-groupoids generalize VB-groupoids and provide a new correspondence between them and principal bundles.
The paper studies groupoid structures from different viewpoints.
problem Understanding groupoid morphisms and their structures.
method Constructing two groupoids from morphisms of groupoids, one from a categorical viewpoint and the other from a geometric viewpoint. Showing equivalence of the two kinds of groupoids of morphisms.
result Equivalence of two kinds of groupoids of morphisms for each pair of groupoids.
The paper generalizes bundle gerbes over groupoids and their correspondence with PB groupoids.
problem Generalizing bundle gerbes over groupoids and their properties.
method Developed a functorial correspondence between PB groupoids and bundle gerbes over groupoids.
result Built a correspondence between PB groupoids and bundle gerbes over groupoids, including partial quotients.
We apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. As an application, we recover Haefliger's fundamental groupoid from the fundamental double groupoid of a Lie groupoid. In the case of a symplectic double groupoid, we study the induced closed 2-form on the associated l…
Differentiable groupoids and their inertia spaces are studied with a de Rham theorem.
problem Understanding the de Rham cohomology of inertia spaces.
method Introducing differentiable stratified groupoids and proving a de Rham theorem.
result Inertia groupoids of proper Lie groupoids are locally contractible differentiable stratified groupoids.
A groupoid is a small category in which each morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all groupoid structure maps are continuous. The notion of monodromy groupoid of a topological groupoid generalises those of fundamental groupoi…
Lie Calculus connects differential and Lie theory using groupoids.
problem Understanding the relationship between differential and Lie theories.
method Using groupoids to link differential and Lie theories.
result Higher order theory involves higher algebra (n-fold groupoids).
We outline the construction of the holonomy groupoid of a locally Lie groupoid and the monodromy groupoid of a Lie groupoid. These specialise to the well known holonomy and monodromy groupoids of a foliation, when the groupoid is just an equivalence relation.
We complete the construction of the double Lie algebroid of a double Lie groupoid begun in the first paper of this title. We show that the Lie algebroid structure of an LA--groupoid may be prolonged to the Lie algebroid of its Lie groupoid structure; in the case of a double groupoid this prolonged structure for either …
We discuss two sorts of generalization of Lie groupoids. One is Lie n-groupoids defined as simplicial manifolds with trivial πk≥n+1. The other is the stacky Lie groupoid $\cG\rra M$ with $\cG$ a differentiable stack. We build 1-1 correspondence between Lie 2-groupoids and stacky Lie groupoids up to a certain…
In this work we deal with coverings and actions of Lie group- groupoids being a sort of the structured Lie groupoids. Firstly, we define an action of a Lie group-groupoid on some Lie group and the smooth coverings of Lie group-groupoids. Later, we show the equivalence of the category of smooth actions of Lie group-grou…
We introduce the notion of Glanon groupoids, which are Lie groupoids equipped with multiplicative generalized complex structures. It combines symplectic groupoids, holomorphic Lie groupoids and holomorphic Poisson groupoids into a unified framework. Their infinitesimal, Glanon Lie algebroids are studied. We prove that …
Paper solves Hilbert's fifth problem for specific groupoids.
problem When are transitive groupoids continuously isomorphic to Lie groupoids?
method Investigates proper and transitive groupoids with compact source fibers.
result Solves Hilbert's fifth problem for specified groupoids.
Introduces multiplicative differential forms on Lie groupoids with VB-groupoids values.
problem Describing multiplicative differential forms on Lie groupoids with VB-groupoids values.
method Introduces multiplicative differential forms on Lie groupoids with values in VB-groupoids, presents a Lie theory for differential forms on Lie groupoids with values in 2-term representations up to homotopy, defines a differential complex whose 1-cocycles are multiplicative forms with values in VB-groupoids.
result Complete description of multiplicative differential forms on Lie groupoids with values in VB-groupoids.
Solves open problem on Lie groupoids equivalence.
problem Whether Lie groupoids Morita equivalent are diffeologically Morita equivalent.
method Localisation of 2-categories, anafunctors, Lie groupoids, diffeological groupoids.
result Two Lie groupoids diffeologically Morita equivalent are Morita equivalent in the Lie sense.
The paper extends Hilbert's fifth problem to transitive groupoids.
problem When is a transitive topological groupoid continuously isomorphic to a Lie groupoid?
method Investigation of transitive topological groupoids, focusing on proper and transitive groupoids with compact source fibers.
result Presentation of results generalizing Hilbert's fifth problem to transitive groupoids, including solutions for proper and transitive groupoids with compact source fibers.
Characterizes actions of stacky Lie groupoids on differentiable stacks.
problem Understanding actions of stacky Lie groupoids on differentiable stacks.
method Characterization of principal actions of stacky Lie groupoids.
result Principal actions of stacky Lie groupoids yield differentiable stack quotients as principal bundles.
In this note a functorial approach to the integration problem of an LA-groupoid to a double Lie groupoid is discussed. To do that, we study the notions of fibred products in the categories of Lie groupoids and Lie algebroids, giving necessary and sufficient conditions for the existence of such. In particular, it turns …
Analyzes how suborbifolds relate to groupoid embeddings.
problem Understanding the relationship between suborbifolds and groupoid embeddings.
method Examines the correspondence between suborbifolds and groupoid embeddings via atlases and effective orbifold groupoids.
result Identifies classes of suborbifolds that naturally lead to groupoid embeddings.
We approach Mackenzie's LA-groupoids from a supergeometric point of view by introducing Q-groupoids, which are groupoid objects in the category of Q-manifolds. There is a faithful functor from the category of LA-groupoids to the category of Q-groupoids. We associate to every Q-groupoid a double complex that provides a …
In this paper we study the Lie groupoids which appear in foliation theory. A foliation groupoid is a Lie groupoid which integrates a foliation, or, equivalently, whose anchor map is injective. The first theorem shows that, for a Lie groupoid G, the following are equivalent: - G is a foliation groupoid, - G has discrete…
The article proves Lie's third theorem for Lie algebroids via singular Lie groupoids.
problem Lie's third theorem does not hold for Lie groupoids and Lie algebroids.
method Introducing a subcategory of diffeological spaces called quasi-etale, constructing a functor mapping singular Lie groupoids to Lie algebroids.
result Lie's third theorem is valid for Lie algebroids within the context of singular Lie groupoids.
Locally conformal symplectic (l.c.s.) groupoids are introduced as a generalization of symplectic groupoids. We obtain some examples and we prove that l.c.s. groupoids are examples of Jacobi groupoids in the sense of \cite{IM}. Finally, we describe the Lie algebroid of a l.c.s. groupoid.
The study examines fibrations of Lie groupoids and their classification.
problem Understanding and classifying fibrations of Lie groupoids.
method Introduced locally topological product Lie groupoid fibrations and used groupoid cohomology to obstruct and classify these fibrations.
result The existence of locally topological product Lie groupoid fibrations is obstructed by a specific groupoid cohomology.
Paper defines weak representations and their equivalence to VB-groupoids.
problem Understanding representations of Lie groupoids and VB-groupoids.
method Introduces weak representations and shows equivalences between categories.
result Equivalence between 2-term representations up to homotopy and weak representations of G.
The paper studies deformations of symplectic groupoids using cohomology and spectral sequences.
problem Deformations of symplectic groupoids and their cohomology.
method Deformation cohomology, Moser path methods, Lie groupoids, multiplicative forms, de Rham models, spectral sequences.
result Computations and constructions of deformation cohomology for various types of symplectic groupoids.
Integrates singular subalgebroids using diffeological groupoids.
problem Integration of singular subalgebroids.
method Definition of integration via diffeological groupoids with specific properties.
result Holonomy groupoids correspond to singular subalgebroids with submersive property.
We discuss two generalizations of Lie groupoids. One consists of Lie n-groupoids defined as simplicial manifolds with trivial πk≥n+1. The other consists of stacky Lie groupoids $\cG\rra M$ with $\cG$ a differentiable stack. We build a 1-1 correspondence between Lie 2-groupoids and stacky Lie groupoids up to …
New approach to principal groupoid bundles with connections using dg-Lie groupoids.
problem Developing a new perspective on principal bundles with connections.
method Using dg-Lie groupoids and additional adjustment data for Lie groupoids.
result Adjusted connections provide a global formulation of curved Yang-Mills-Higgs theories.
Monodromy groups help test Lie groupoids for faithful representations.
problem Testing Lie groupoids for faithful representations.
method Constructing monodromy groups and using them to test Lie groupoids.
result Monodromy groups can be used to determine if Lie groupoids lack faithful representations.
Lie groupoids and their orbit spaces are linked through equivalence classes.
problem Understanding the relationship between Lie groupoids and their orbit spaces.
method Introducing lift-complete Lie groupoids and showing equivalence between categories.
result Morita equivalence class of a lift-complete Lie groupoid is determined by its orbit space.
Study infinitesimal automorphisms of VB-groupoids and algebroids.
problem Understand transformations preserving the structure of VB-groupoids and algebroids.
method Examine vector fields generating flows that preserve both linear and groupoid/algebroid structures.
result Infinitesimal automorphisms of a special class are multiplicative sections of a derivation groupoid/algebroid.
We address the question of duality for the dynamical Poisson groupoids of Etingof and Varchenko over a contractible base. We also give an explicit description for the coboundary case associated with the solutions of the classical dynamical Yang-Baxter equation on simple Lie algebras as classified by the same authors. O…
We study precontact groupoids whose infinitesimal counterparts are Dirac-Jacobi structures. These geometric objects generalize contact groupoids. We also explain the relationship between precontact groupoids and homogeneous presymplectic groupoids. Finally, we present some examples of precontact groupoids.
Post-groupoids help solve Yang-Baxter equation using quivers.
problem Solving the Yang-Baxter equation using algebraic structures.
method Introducing post-groupoids and showing their connection to quivers.
result Post-groupoids provide solutions to the Yang-Baxter equation.
Leaves of Lie algebroids are Lie groupoids under certain conditions.
problem Understanding the structure of leaves in Lie algebroids.
method Analyzing cotangent Lie algebroids and Lie 2-group coadjoint orbits.
result Coadjoint orbits of Lie 2-groups are symplectic groupoids.
Extends connections on Lie groupoids, proving completeness conditions.
problem Existence and completeness of multiplicative connections on Lie groupoid fibrations.
method Introduces and investigates multiplicative Ehresmann connections on Lie groupoid fibrations.
result Conditions for completeness of multiplicative connections on Lie groupoid fibrations.
Defines symplectic 2-groupoids and their relation to Courant algebroids.
problem Integrating Courant algebroids and understanding symplectic 2-groupoids.
method Proposes and studies constant symplectic 2-groupoids, finding correspondences with Courant algebroids.
result Constant symplectic 2-groupoids are equivalent to constant Courant algebroids.
We prove that the cotangent of a double Lie groupoid S has itself a double groupoid structure with sides the duals of associated Lie algebroids, and double base the dual of the Lie algebroid of the core of S. Using this, we prove a result outlined by Weinstein in 1988, that the side groupoids of a general symplectic do…
A VB-groupoid is a Lie groupoid equipped with a compatible linear structure. In this paper, we describe a correspondence, up to isomorphism, between VB-groupoids and 2-term representations up to homotopy of Lie groupoids. Under this correspondence, the tangent bundle of a Lie groupoid G corresponds to the "adjoint repr…
The paper defines Haar system preserving morphisms and applies them to groupoid C∗-algebras.
problem Understanding and constructing inverse systems of groupoids.
method Defining Haar system preserving morphisms and using them to induce *-morphisms between convolution algebras.
result Inverse systems of groupoids with Haar system preserving bonding maps have limits, and corresponding direct systems of groupoid C∗-algebras. The thesis uses simplicial methods to study actions, bundles, and bibundles of higher groupoids.
problem Understanding actions, bundles, and bibundles of higher groupoids.
method Employing simplicial methods to model actions, principal bundles, and bibundles of higher groupoids.
result The simplicial definitions agree with categorification approaches and prove a theorem on differentiation of higher Lie groupoids.
New groupoids relate surface mapping classes to Thompson's groups.
problem Understanding relationships between surface mapping classes and Thompson's groups.
method Introducing ΠMG and ΩMG groupoids. result Found connections between surface mapping classes and Thompson's groups.
The paper explores connections between topological dynamics and groupoids.
problem Understanding connections between topological dynamics and groupoids.
method Investigates connections between Topological Dynamics, G-Principal Bundles, and Locally Trivial Groupoids.
result Connections between topological dynamics and groupoids are explored.
Analytic linearization and holomorphic extensions for proper groupoids.
problem Analytic linearization and holomorphic extensions of proper groupoids.
method Establish analytic linearization around invariant submanifolds and apply to holomorphic extensions.
result Proper groupoids admit holomorphic extensions.