Newly proves equivariant bundles equivalence to differential quotient stacks.
problem Equivalence of equivariant bundles and differential quotient stacks.
method Proved equivalence through Lie group G action on manifold M.
result Category equivalence of equivariant bundles and differential quotient stacks.
Stacky Lie groupoids are generalizations of Lie groupoids in which the "space of arrows" of the groupoid is a differentiable stack. In this paper, we consider actions of stacky Lie groupoids on differentiable stacks and their associated quotients. We provide a characterization of principal actions of stacky Lie groupoi…
This work introduces Lie 2-algebra structures for multiplicative sections of LA-groupoids.
problem Understanding algebraic structures of sections in LA-groupoids.
method Introducing and proving natural strict Lie 2-algebra structures on the category of multiplicative sections.
result The Lie algebra of geometric vector fields is described in various cases.
This work characterizes global quotient stacks---smooth stacks associated to a finite group acting a manifold---among smooth quotient stacks [M/G], where M is a smooth manifold equipped with a smooth proper action by a Lie group G. The characterization is described in terms of the action of the connected componen…
New geometry connects super loops to Chern character.
problem Understanding super parallel transport on quotient stacks.
method Super holonomy on loop stacks to construct Chern character.
result Global equivariant Chern character from super holonomy.
Defines differential equivariant cohomology for Lie group actions.
problem Equivariant cohomology for quotient stacks.
method Principal bundles with connections, differential cohomology.
result Chern-Weil homomorphism factors through differential equivariant cohomology.
A new mathematical approach to general covariance using stacks and Lie algebras.
problem Understanding general covariance in curved spacetime field theories.
method Using stacks and groupoids to study the quotient of metrics modulo diffeomorphism, and analyzing the tangent complex and Lie algebra actions.
result Recovering a novel expression for the stress-energy tensor in scalar field theories.
Quantum Kirwan maps between K-theories of G-varieties and GIT quotients.
problem Constructing maps between K-theories of G-varieties and their GIT quotients.
method Formal construction of maps in quantum K-theory, using equivariant and non-equivariant quantum K-theory.
result Presentation of quantum K-theory for smooth proper toric DM stacks.
This is the writeup of a lecture given at the May Wisconsin workshop on mathematical aspects of orbifold string theory. In the first part of this lecture, we review recent work on discrete torsion, and outline how it is currently understood in terms of the B field. In the second part of this lecture, we discuss the rel…
Study of foliations' geometric and topological structures.
problem Analyzing the geometric and topological properties of transversely affine foliations.
method Attach holonomy group and quotient stack, identify reparametrisations, classify them, and study the Kato-Nakayama space.
result Holonomy group controls the geometric part, while the Kato-Nakayama space captures the topological and dynamical aspects.
The paper classifies equivariant gerbe connections on Lie groups.
problem Classifying equivariant gerbe connections on Lie groups.
method Using Meinrenken's G-equivariant bundle gerbe connections and differential quotient stacks.
result Isomorphism classes of G-equivariant gerbe connections are classified by degree three differential equivariant cohomology.
Normal forms and moduli stacks for flat connections on complex manifolds.
problem Understanding singular flat connections on complex manifolds.
method Introducing homogeneous Lie groupoids and studying their representation theory to prove normal form theorems and moduli space structures.
result Moduli spaces of singular flat connections admit the structure of algebraic quotient stacks.
The paper studies higher geometric structures and connections on manifolds, constructing moduli stacks and proving equivalence criteria.
problem Classifying and understanding higher geometric structures and connections on manifolds.
method Constructing smooth higher symmetry groups, moduli stacks, and higher gauge actions; proving equivalence criteria.
result Construction and classification of moduli stacks of higher geometric data and connections.
Geometric compactification for complex structures on Lie groups.
problem Compactifying moduli stack of complex structures on Lie groups.
method Describes a geometric compactification using CR structures transverse to a real foliation.
result Extra points represent CR structures transverse to a real foliation.
Constructs equivariant cohomology models for differentiable stacks.
problem Developing cohomology theory for stacks with group actions.
method Extends classical results for smooth manifolds to differentiable stacks.
result Derives spectral sequences generalizing Bott's spectral sequence.
Study connections on Lie groupoids and stacks using Atiyah sequences.
problem No specific problem stated; general connections on Lie groupoids and stacks.
method Construct connections using Atiyah sequences associated with transversal tangential distributions.
result Detailed study and construction of connections on Lie groupoids and stacks.
Relates discrete group actions to orbit spaces as differentiable stacks.
problem Understanding dynamics of discrete groups on manifolds.
method Relating discrete group actions to orbit spaces as differentiable stacks.
result Orbit stack encodes dynamics up to conjugation and inversion.
Researchers compute differential K-theory for moduli stacks.
problem Computing differential K-theory for moduli stacks of principal G-bundles.
method Using homotopy theory of presheaves of spaces and spectra, they formulate results in terms of invariant polynomials and representation rings.
result They successfully compute the connective differential K-theory and differential cohomology of moduli stacks.
In this paper, we consider diffeological spaces as stacks over the site of smooth manifolds, as well as the "underlying" diffeological space of any stack. More precisely, we consider diffeological spaces as so-called concrete sheaves and show that the Grothendieck construction sending these sheaves to stacks has a left…
New contact structures defined on differentiable stacks.
problem Defining contact structures on differentiable stacks.
method Introducing 0-shifted and +1-shifted contact structures. result Shifted contact structures provide new insights into geometry.
Study measures on differentiable stacks, proving invariance and formulas.
problem Defining and studying measures on differentiable stacks.
method Generalizing Haefliger's approach to leaf spaces and transverse measures for foliations.
result Proved Morita invariance, a Stokes formula, and Van Est isomorphism.
Develops theory of differential graded schemes for derived stacks.
problem Creating a theory for derived stacks using dg schemes.
method Formulates dg schemes as homotopy sites, equates to stacks on dg algebras.
result Infinity category of stacks represented by dg schemes is derived schemes.
Studies geometric structures on Lie groupoids and differentiable stacks.
problem None explicitly stated in the abstract.
method Various geometric structures and connections on Lie groupoids and differentiable stacks.
result Introduces new concepts like topological groupoid extensions and gerbes over topological stacks.
The paper explores Poisson structures on differentiable stacks, developing new mathematical tools.
problem Investigating shifted Poisson structures on differentiable stacks.
method Developed new mathematical tools including Morita equivalence of quasi-Poisson groupoids and tangent/cotangent complexes.
result Shifted (+1) Poisson structures on differentiable stacks correspond to elements of the Maurer-Cartan moduli set of a specific Lie 2-algebra. Study vector fields and derivations on differentiable stacks.
problem Understanding structures on differentiable stacks.
method Introduced module structures on dgla of multiplicative vector fields and graded algebra of functions on Lie groupoids.
result Associated structure of a graded Lie-Rinehart algebra on vector fields is Morita invariant.
New neural stack and Turing Machine architectures prove stability and computational power.
problem Designing stable neural network architectures for Turing Machine simulation.
method Introducing neural stack and Turing Machine architectures, proving stability and computational equivalence.
result Differentiable nnTM with bounded neurons can simulate Turing Machine in real-time and is equivalent to UTM.
We give a new interpretation of the Faddeev-Mickelsson anomaly in certain Yang-Mills theories in terms of S^1-central extensions of Lie groupoids.
Smooth stacks of orbifolds are shown to be infinite-dimensional orbifolds.
problem Understanding the structure of Hom-stacks of orbifolds.
method Using Lie groupoids and Fréchet-Lie groupoids to represent Hom-stacks.
result Hom-stacks of orbifolds are infinite-dimensional orbifolds.
Characterizes primary operations in differential cohomology using stacks.
problem Understanding primary operations in differential cohomology.
method Characterization via stacks, explicit refinement of Steenrod squares and powers, interplay between different cohomology types.
result Developed techniques for differential cohomology, including Künneth decomposition.
This paper proves cohomology invariants for differentiable stacks.
problem Understanding cohomology of differentiable stacks.
method Simplicial approach to representations up to homotopy.
result Cohomology with coefficients in a representation up to homotopy is a Morita invariant of the underlying stack.
We develop a theory of Lie algebroids over differentiable stacks that extends the standard theory of Lie algebroids over manifolds. In particular we show that Lie algebroids satisfy descent for submersions, define the category of Lie algebroids over a differentiable stack, construct a cohomology theory for these object…
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.
This paper introduces the notions of vector field and flow on a general differentiable stack. Our main theorem states that the flow of a vector field on a compact proper differentiable stack exists and is unique up to a uniquely determined 2-cell. This extends the usual result on the existence and uniqueness of flows o…
We introduce the notion of Lusternik-Schnirelmann category for differentiable stacks and establish its relation with the groupoid Lusternik-Schnirelmann category for Lie groupoids.
We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids. Cofoliations on stacks arise from flat connections on groupoids. Connecti…
We extend Massey products from cohomology to differential cohomology via stacks, organizing and generalizing existing constructions in Deligne cohomology. We study the properties and show how they are related to more classical Massey products in de Rham, singular, and Deligne cohomology. The setting and the algebraic m…
This paper explores the relationship between gerbes over stacks and Lie groupoid extensions.
problem Exploring the relationship between gerbes over stacks and Lie groupoid extensions.
method Defines a gerbe over a stack and explores its relationship with Lie groupoid extensions.
result Establishes the relationship between gerbes over stacks and Morita equivalence classes of Lie groupoid extensions.
Extends abelian differentials to log twisted differentials with spin and hyperelliptic structures.
problem Compactify the moduli space of abelian differentials with spin and hyperelliptic structures.
method Introduce log twisted differentials and hyperelliptic differentials using stable log maps and admissible covers.
result Proves the existence of up to three connected components in the open strata of log twisted differentials.
We construct connections and characteristic forms for principal bundles over groupoids and stacks in the differentiable, holomorphic and algebraic category using Atiyah sequences associated to transversal tangential distributions.
We study S1-bundles and S1-gerbes over differentiable stacks in terms of Lie groupoids, and construct Chern classes and Dixmier-Douady classes in terms of analogues of connections and curvature.
The higher gauge field in 11-dimensional supergravity -- the C-field -- is constrained by quantum effects to be a cocycle in some twisted version of differential cohomology. We argue that it should indeed be a cocycle in a certain twisted nonabelian differential cohomology. We give a simple and natural characterization…
We use Morse theory to prove that the Lefschetz Hyperplane Theorem holds for compact smooth Deligne-Mumford stacks over the site of complex manifolds. For Z⊂X a hyperplane section, X can be obtained from Z by a sequence of deformation retracts and attachments of high-dimensional finite disc quotients. We …
Proper Lie groupoids can be desingularized to regular ones.
problem Desingularizing proper Lie groupoids to regular ones.
method Successive blow-up construction on a proper Lie groupoid.
result Desingularization of proper Lie groupoids to regular ones, arbitrarily close in Gromov-Hausdorff distance.
We generalize geometric prequantization of symplectic manifolds to differentiable stacks. Our approach is atlas-independent and provides a bijection between isomorphism classes of principal circle bundles (with or without connections) and second cohomology groups of certain chain complexes.
New concept of coisotropic structures for differentiable stacks defined.
problem Defining coisotropic structures for differentiable stacks.
method Using twisted Dirac structures and Morita equivalences.
result 1-shifted coisotropic structures transfer through Morita equivalences.
Paper shows equivalences between Dixmier-Douady bundles and differential 3-cocycles.
problem Understanding equivalences between Dixmier-Douady bundles and differential 3-cocycles.
method Focuses on the model of Dixmier-Douady bundles and provides equivalences between 2-stacks.
result Equivalence between Dixmier-Douady bundles and differential 3-cocycles of height 1.
This is a concise introduction to the theory of Lie groupoids, with emphasis in their role as models for stacks. After some preliminaries, we review the foundations on Lie groupoids, and we carefully study equivalences and proper groupoids. Differentiable stacks are geometric objects which have manifolds and orbifolds …
We give a precise and general description of gerbes valued in arbitrary crossed module and over an arbitrary differential stack. We do it using only Lie groupoids, hence ordinary differential geometry. We prove the coincidence with the existing notions by comparing our construction with non-Abelian cohomology.