In this thesis, we employ simplicial methods to study actions, principal bundles, and bibundles of higher groupoids. Roughly, we use Kan fibrations to model actions of higher groupoids, we use pairs of a Kan fibration and a special acyclic fibration to model principal bundles of higher groupoids, we use inner Kan fibra…
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48 results for “Principal bibundles”
We consider the existence of bibundles, in other words locally trivial principal spaces with commuting left and right actions. We show that their existence is closely related to the structure of the group $\Out(G)$ of outer automorphisms of . We also develop a classifying theory for bibundles. The theory is …
New Morita equivalence for diffeological groupoids defined.
problem Defining Morita equivalence for diffeological groupoids.
method Developed diffeological groupoid actions, -bundles, and -bibundles; introduced principality; defined Hilsum-Skandalis tensor product.
result Biprincipal bibundles are weakly invertible in the bicategory DiffBiBund.
Study of Morita equivalences in Lie groupoids and their symmetries.
problem Exploring Morita equivalences in Lie groupoids.
method Investigation of principal bibundles and Morita equivalences in Pfaffian groupoids.
result Introduction and analysis of Pfaffian Morita equivalence.
Stacky Lie groupsmath.DG
Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary for the existence o…
We give a complete and explicit description of the kinematical data of higher gauge theory on principal 2-bundles with the string 2-group model of Schommer-Pries as structure 2-group. We start with a self-contained review of the weak 2-category Bibun of Lie groupoids, bibundles and bibundle morphisms. We then construct…
New cohomology theory shows compact Lie group actions are Morita invariant.
problem Establishing Morita invariance for cohomology of compact Lie group actions.
method Using bibundles to transfer coefficient systems between Morita equivalent groupoids.
result Twisted Bredon-Illman cohomology is Morita invariant for compact Lie group actions.
Orbifolds as stacks?math.DG
The first goal of this survey paper is to argue that if orbifolds are groupoids, then the collection of orbifolds and their maps has to be thought of as a 2-category. Compare this with the classical definition of Satake and Thurston of orbifolds as a 1-category of sets with extra structure and/or with the "modern" defi…
Quasifold groupoids and diffeological quasifolds are studied, showing an equivalence of categories.
problem Understanding the structure and equivalence of quasifold groupoids and diffeological quasifolds.
method Examining the category of diffeological quasifolds and the bicategory of quasifold groupoids, proving an equivalence of categories under certain conditions.
result Restricting to locally invertible morphisms and effective quasifold groupoids, the orbit space functor is an equivalence of categories.
The study explores how different Grothendieck topologies and functors between categories preserve locality.
problem Exploring relationships between different Grothendieck topologies and functors.
method Using Grothendieck topologies and functors to relate categories and geometric objects.
result Objects like sheaves, groupoids, and functors are invariant under equivalences of Grothendieck topologies and certain functors.
QP perspective on Poisson-Lie T-duality topology changes.
problem Understanding Poisson-Lie T-duality through QP manifolds.
method QP manifolds and canonical transformations for symplectic reductions.
result Canonical transformations mediate Poisson-Lie T-duality.
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.
Functor connects Lie groupoid algebras to bornological structures.
problem Establishing a functorial relationship between Lie groupoid convolution algebras and bornological structures.
method Developed a monoidal functor from differentiable stacks to Morita 2-category of complete bornological algebras.
result Convolution algebras are self-induced and convolution modules are smooth.
Introduces generalized principal bundles and connections, linking them to standard gauge theories.
problem Generalized principal bundles and connections in field theories.
method Local coordinate transformation laws and horizontal lifts.
result Generalized principal connections are associated to Lie group fiber bundle connections.
Introduces principal fairness for fair decision-making.
problem Discrimination among similarly affected individuals.
method Uses principal stratification from causal inference.
result Explicitly accounts for decision impacts, not just protected attributes.
Study of semi-principal bundles using group actions and wreath products.
problem Understanding bundles with fibers as free -spaces.
method Defining semi-principal bundles, bases, and frame bundles; using wreath products and functors.
result Semi-principal bundles can be retracted to principal bundles, preserving parallel transport.
The paper classifies and determines properties of specific hypersurfaces in complex hyperbolic quadrics.
problem Analyzing Hopf hypersurfaces with constant principal curvatures in complex hyperbolic quadrics.
method Classification and determination of principal curvatures for hypersurfaces with different numbers of distinct curvatures.
result Classification and determination of principal curvatures for Hopf hypersurfaces with up to four distinct values.
Study estimates heterogeneous principal causal effects with binary treatments and intermediate variables.
problem Estimating subgroup effects within strata defined by potential values of an intermediate variable.
method Proposes a framework for estimating and forming confidence intervals for heterogeneous principal causal effects under principal ignorability assumption. Develops several estimators with varying robustness properties.
result Established large-sample theory and analyzed bias contributions of each approach.
Study on discrete surfaces with constant principal curvature for nanocarbon applications.
problem Understanding discrete geometry properties of nanocarbon materials.
method Developed discrete surface theory on 3-ary oriented trees, defined discrete principal directions, constructed examples of discrete CPC surfaces.
result Construction of discrete constant principal curvature surfaces, including discrete CPC tori.
The paper introduces a method for detecting principal communities and embedding vertices.
problem Detecting and embedding vertices in graphs with community structure.
method Principal graph encoder embedding method that detects principal communities and produces vertex embeddings.
result The method successfully detects principal communities and produces accurate vertex embeddings.
We enumerate all the principal congruence link complements in , there by answering a question of W. Thurston. Related articles: "Technical Report: All Principal Congruence Link Groups" (arXiv:1902.04722), "All Known Principal Congruence Links" (arXiv:1902.04426).
A study on how a principal can incentivize an agent to make better decisions in a repeated game.
problem Optimizing a principal's utility in a misaligned principal-agent bandit game.
method Developed nearly optimal learning algorithms for the principal's regret in multi-armed and linear contextual settings.
result The principal can iteratively learn an incentive policy to maximize her total utility.
Localized Sparse Principal Component Analysis of Multivariate Time Series in Frequency Domainstat.ME
The paper introduces a method for interpretable principal component analysis of high-dimensional time series.
problem Inconsistent and difficult-to-interpret principal component estimates in high-dimensional regimes.
method Localized sparse principal component analysis of spectral density matrices in frequency domain.
result Efficient algorithm for sparse-localized estimates of principal subspaces.
The paper solves the Integration Problem for principal connections.
problem Describing discrete connections associated with a principal connection.
method Using the Lie or derivative functor to induce connections on the principal bundle.
result For flat principal connections, the Integration Problem has a unique solution among flat discrete connections.
Principal component regression (PCR) is a two-stage procedure that selects some principal components and then constructs a regression model regarding them as new explanatory variables. Note that the principal components are obtained from only explanatory variables and not considered with the response variable. To addre…
New simulations advise caution in choosing principal components for multivariate functional data.
problem Inaccurate selection of principal components in multivariate functional data.
method Extensive simulations investigating the reliability of percentage of variance explained thresholds.
result Conventional threshold methods may fail to accurately explain overall variance in multivariate functional data.
The equivalence of principal bundles with transitive Lie groupoids due to Ehresmann is a well known result. A remarkable generalisation of this equivalence, due to Mackenzie, is the equivalence of principal bundle extensions with those transitive Lie groupoids over the total space of a principal bundle, which also admi…
Conventional principal component analysis (PCA) finds a principal vector that maximizes the sum of second powers of principal components. We consider a generalized PCA that aims at maximizing the sum of an arbitrary convex function of principal components. We present a gradient ascent algorithm to solve the problem. Fo…
Holonomies match for higher local systems and principal 2-bundles.
problem Matching holonomies for higher local systems and principal 2-bundles.
method Higher Riemann-Hilbert correspondence and principal 2-bundles.
result Holonomies coincide for both formalisms.
Isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in Finsler spacesmath.DG
Study on isoparametric and constant-curvature hypersurfaces in Finsler spaces.
problem Characterizing and constructing hypersurfaces with specific curvature properties in Finsler geometry.
method Analyzing isoparametric and constant-curvature hypersurfaces on Randers manifolds.
result Construction of a conformally flat Randers manifold with nonisoparametric hyperplanes of constant curvatures.
PCHAL and PCHAR use principal components to speed up HAL and HAR methods.
problem Computational infeasibility in high dimensions for HAL and HAR.
method Outcome-blind principal component reduction of HAL basis.
result Empirical performance comparable to HAL and HAR, with computational gains.
Principal binets generalize curvature line surfaces to square lattices and are a discrete integrable system.
problem Discretizing curvature line surfaces on square lattices.
method Showed principal binets as a multi-dimensional consistent system.
result Principal binets generalize to higher-dimensional square lattices and are integrable.
Efficient private matrix analysis algorithms for recent variants.
problem Private analysis of recent matrix updates.
method Identifying sufficient conditions on positive semidefinite matrices.
result First efficient differentially private algorithms for various matrix analysis tasks.
On n-tuple principal bundlesmath.DG
We develop the concept of a double (more generally n-tuple) principal bundle departing from a compatibility condition for a principal action of a Lie group on a groupoid.
In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation quantization. We apply it to give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. The paper also contains a putative de…
In this paper are determined the principal curvatures and principal curvature lines on canal surfaces which are the envelopes of families of spheres with variable radius and centers moving along a closed regular curve in R^3. By means of a connection of the differential equations for these curvature lines and real Ricc…
Defines hybrid systems on principal bundles and studies impact effects.
problem Understanding impact effects in hybrid mechanical systems.
method Defines hybrid systems on principal bundles, studies underlying geometry, and finds conditions for impact preservation.
result Conditions for preservation of both exterior and interior impacts by mechanical connections.
A new metric-based principal curve method learns 1D manifolds from spatial data.
problem Learning 1D manifolds from spatial data.
method Metric-based Principal Curve (MPC) approach.
result The method effectively learns the shape of 1D manifolds from synthetic and real datasets.
Behavior of principal curvatures of frontals near non-front singular points and their applicationmath.DG
Study behavior of curvatures near singular points of frontals.
problem Understanding frontals near singular points.
method Investigate principal curvatures and vectors near singular points of frontals.
result Extend Ribaucour transformations to frontals with singular points.
Let be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let be a connected complex reductive affine algebraic group equipped with a real form . We define pseudo-real principal --bundles on ; these are generalizations of re…
Principal Manifold Flowsstat.ML
PFs and iPFs learn principal manifolds for efficient density estimation.
problem Understanding the geometric structure of normalizing flows.
method Characterize flows using principal manifolds and contours.
result PFs and iPFs can learn principal manifolds and perform density estimation.
In many physical, statistical, biological and other investigations it is desirable to approximate a system of points by objects of lower dimension and/or complexity. For this purpose, Karl Pearson invented principal component analysis in 1901 and found 'lines and planes of closest fit to system of points'. The famous k…
A principal pair consists of a holomorphic principal -bundle together with a holomorphic section of an associated Kaehler fibration. Such objects support natural gauge theoretic equations coming from a moment map condition, and also admit a notion of stability based on Geometric Invariant Theory. The Hitchin--Kobaya…
We consider the classification problem and focus on nonlinear methods for classification on manifolds. For multivariate datasets lying on an embedded nonlinear Riemannian manifold within the higher-dimensional ambient space, we aim to acquire a classification boundary for the classes with labels, using the intrinsic me…
Study Lie algebroid connections on principal bundles over complex projective varieties.
problem Existence and properties of Lie algebroid connections on principal bundles.
method Definition and study of Lie algebroid valued connections on holomorphic principal G-bundles, investigation of existence criteria.
result Investigation of criteria for existence of Lie algebroid connections on principal G-bundles over smooth complex projective curves.
In this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby extending the results of Barrett, Caetano and Picken, and Schreiber and Waldof f…
Spherical Principal Curvesstat.ME
New approach for principal curves on spherical data.
problem Dimension reduction of spherical data.
method Projection of data onto a continuous curve on a sphere.
result Stationary principal curves on a sphere.
Paper introduces Atiyah sequence for Lie groupoids and studies gauge transformations.
problem Defining connections on principal 2-bundles over Lie groupoids.
method Introduced Atiyah sequence, defined strict and semi-strict connections, constructed gauge transformations.
result Existence criterion for connections on principal 2-bundles over proper, étale Lie groupoids.