Study of geodesics on Riemannian stacks, measuring distances on orbit spaces.
problem Extending Riemannian geometry to singular spaces.
method Introduce stacky metrics and study stacky curves and geodesics on Riemannian stacks.
result Establish a stacky version of Hopf-Rinow Theorem.
In this article, we derive many properties of étale stacks in various contexts, and prove that étale stacks may be characterized categorically as those stacks that arise as prolongations of stacks on a site of spaces and local homeomorphisms. Moreover, we show that the bicategory of étale differentiable stacks and loca…
We study Riemannian metrics on Lie groupoids in the relative setting. We show that any split fibration between proper groupoids can be made Riemannian, and we use these metrics to linearize proper groupoid fibrations. As an application, we derive rigidity theorems for Lie groupoids, which unify, simplify and improve si…
Study Lie 2-group actions on Riemannian groupoids, proving existence and developing geometric Killing vector fields.
problem Understanding isometric actions of Lie 2-groups on Riemannian groupoids.
method Exhibit properties, prove existence, construct bi-invariant metrics, provide infinitesimal description.
result Existence of 2-equivariant Slice Theorem and Equivariant Tubular Neighborhood Theorem.
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.
In this paper we prove that every proper Lie groupoid admits a desingularization to a regular proper Lie groupoid. When equipped with a Riemannian metric, we show that it admits a desingularization to a regular Riemannian proper Lie groupoid, arbitrarily close to the original one in the Gromov-Hausdorff distance betwee…
Riemannian metrics on orbifolds are equivalent to diffeological ones.
problem Equivalence of Riemannian and diffeological orbifolds.
method Framework of Riemannian diffeology and analysis of 2-metrics.
result Riemannian metrics on orbifolds are equivalent to diffeological ones.
Stacking is a general approach for combining multiple models toward greater predictive accuracy. It has found various application across different domains, ensuing from its meta-learning nature. Our understanding, nevertheless, on how and why stacking works remains intuitive and lacking in theoretical insight. In this …
We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a ``stack over a stack'' and the distinction between small stacks (which are algebraic objects) and large stacks (which are generalized spaces).
Constructs cohomology decompositions for symmetric stacks.
problem Cohomology of symmetric stacks.
method Constructs decompositions of cohomology, Borel--Moore homology, and vanishing cycle cohomology.
result Defines BPS cohomology and proves its equivalence to intersection cohomology for smooth stacks.
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.
Bayesian stacking improves model performance with varying model weights.
problem Improving model predictions with heterogeneous input performance.
method Bayesian hierarchical stacking with varying model weights inferred via Bayesian inference.
result Hierarchical stacking yields better predictions than linear averaging.
Paper combines machine learning and model averaging for robust parameter estimation.
problem Estimating structural parameters with partially unknown functional forms.
method Pairing double/debiased machine learning with stacking for model averaging.
result DDML with stacking is more robust to unknown functional forms than single learners.
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.
We generalize the notion of a small sheaf of sets over a topological space or manifold to define the notion of a small stack of groupoids over an étale topological or differentiable stack. We then provide a construction analogous to the étalé space construction in this context, establishing an equivalence of 2-categori…
NN-Stacking improves predictive power of regression models by adjusting stacking coefficients with features.
problem Low predictive power of linear stacking methods.
method NN-Stacking uses neural networks to estimate adaptive stacking coefficients.
result NN-Stacking leads to better predictive power, especially in large datasets.
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…
The BPS decomposition theorem splits cohomology of symmetric stacks into invariant parts.
problem Decomposing the cohomology of smooth symmetric stacks into invariant parts.
method Using cohomological Hall induction and intersection cohomology of moduli spaces.
result Establishes the BPS decomposition theorem for various symplectic stacks.
Stacked conformal prediction simplifies model validation.
problem Validating stacked predictive models efficiently.
method Meta-learner at the top of a stacked ensemble for approximate marginal validity.
result The method achieves approximate marginal validity without a separate calibration sample.
Constructs moduli stacks for quiver connections and extends non-Abelian Hodge theory.
problem Extending non-Abelian Hodge theory to moduli stacks of quiver connections.
method Formalizes and constructs moduli stacks of bundles with λ-connections over prestacks.
result Shows moduli stacks are algebraic and locally of finite presentation when base is smooth and projective.
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.
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.
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…
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.
Constructs moduli stacks of quiver bundles and applies to Higgs bundles.
problem Classifying morphisms of vector bundles over a fixed base.
method General method for constructing moduli stacks of diagrams of vector bundles indexed by a simplicial set.
result Recovery of Nakajima quiver varieties and alternate construction of moduli stacks of Higgs bundles.
Tree-SMU enables strong compositional generalization in neural networks.
problem Zero-shot generalization to novel compositions of concepts.
method Tree Stack Memory Units (Tree-SMU) with Stack Memory Units (SMU).
result Tree-SMU achieves strong empirical results on mathematical reasoning benchmarks.
New potentials found for sheaves on Calabi-Yau 4-folds.
problem Understanding sheaves on Calabi-Yau 4-folds.
method Derived Quot-stacks and Lagrangian distributions.
result Globally defined −1-shifted potentials on sheaves. This thesis is divided into four chapters. The first chapter discusses the relationship between stacks on a site and groupoids internal to the site. It includes a rigorous proof of the folklore result that there is an equivalence between the bicategory of internal groupoids and the bicategory of geometric stacks. The s…
Stacked LSTM networks improve traffic volume forecasting.
problem Accurate traffic volume prediction for better planning.
method Applying stacked Long Short-Term Memory (LSTM) networks for time series forecasting.
result Stacked LSTM networks enhance the accuracy of traffic volume predictions.
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.
Stacking improves deep neural network training efficiency.
problem Improving the efficiency of training deep neural networks.
method Proposes stacking as a form of accelerated gradient descent.
result Proves stacking provides accelerated training for certain deep linear residual networks.
Study moduli spaces of elliptic PDEs using derived C∞-geometry.
problem Representability of moduli spaces of solutions of elliptic PDEs.
method Derived C∞-geometry, stacks of relative jets, nonlinear Fredholm analysis. result Moduli stack of solutions is relatively representable by quasi-smooth derived C∞-schemes. We develop a universal framework to study smooth higher orbifolds on the one hand and higher Deligne-Mumford stacks (as well as their derived and spectral variants) on the other, and use this framework to obtain a completely categorical description of which stacks arise as the functor of points of such objects. We choo…
We propose a novel stacked generalization (stacking) method as a dynamic ensemble technique using a pool of heterogeneous classifiers for node label classification on networks. The proposed method assigns component models a set of functional coefficients, which can vary smoothly with certain topological features of a n…
For a compact manifold M and a differentiable stack \cX presented by a Lie groupoid X, we show the Hom-stack Hom(M,\cX) is presented by a Fréchet-Lie groupoid Map(M,X) and so is an infinite-dimensional differentiable stack. We further show that if \cX is an orbifold, presented by a proper étale Lie groupoid, then Map(M…
Paper proves stacking ensembling is effective and proposes a new family of stacked generalizations.
problem Lack of theoretical guarantees for stacking ensembling methods.
method Proves novel theoretical result and proposes a new family of stacked generalizations.
result Proves stacking ensembling is effective and proposes a new family of stacked generalizations.
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.
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 relate the study of actions of discrete groups over connected manifolds to that of their orbit spaces seen as differentiable stacks. We show that the orbit stack of a discrete dynamical system on a simply connected manifold encodes the dynamics up to conjugation and inversion. We also prove a generaliz…
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…
Proves a stack of G-bundles with logarithmic connections is finite type.
problem Moduli of G-bundles with logarithmic connections over curves.
method Algebraic stack analysis and finite type proof.
result Proves the moduli stack is of finite type.
Delayed-RNN approximates stacked and bidirectional RNNs.
problem Improving RNN expressiveness and representational capacity.
method Weight-constrained delayed-RNN, equivalent to stacked-RNNs, with partial acausality.
result Delayed-RNN can approximate stacked and bidirectional RNNs, outperforming them in some tasks.
NeSS combines neural and symbolic approaches for better compositional generalization.
problem Lack of compositional generalization in deep learning models.
method NeSS uses a neural network to generate traces, executed by a symbolic stack machine with sequence manipulation.
result Achieves 100% generalization performance across multiple domains.
Study of quantum decorated character stacks and their quantizations.
problem Quantization of decorated character stacks and their compatibility with cutting and gluing.
method Using stratified factorization homology, extend Fock and Goncharov's construction to include stacky points.
result Construction of categorical charts and flips on quantum decorated character stacks.
We prove two results on stacked triangulated manifolds in this paper: (a) every stacked triangulation of a connected manifold with or without boundary is obtained from a simplex or the boundary of a simplex by certain combinatorial operations; (b) in dimension d≥4, if Δ is a tight connected closed homology d…
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 …
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.
New minimal surfaces show stacking disorder in periodic structures.
problem Reproducing experimental twinning defects in periodic minimal surfaces.
method Constructing non-periodic minimal surfaces that lift to disordered stacking in 3D.
result Reproduced twinning defects in periodic minimal surfaces as stacking disorder.