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.
This thesis explores geometric stacks and Poisson manifolds, proving new results in their classification and equivalence.
problem Classifying and understanding geometric stacks and Poisson manifolds.
method Rigorous proofs and new site constructions for geometric stacks and Poisson manifolds.
result Classification and equivalence results for b-symplectic manifolds.
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.
The paper proves a new version of dimensional reduction in cohomological Donaldson-Thomas theory.
problem Proving a new version of dimensional reduction in cohomological Donaldson-Thomas theory.
method Using cohomological Donaldson-Thomas theory and loop stacks of 0-shifted symplectic stacks.
result Shows the BPS cohomology of loop stacks admits a description analogous to orbifold cohomology.
Introduces new connections in higher geometry.
problem Defining connections in higher geometry.
method Develops formal differentiation and integration of maps to derived stacks.
result Establishes new L∞-algebras of higher symmetries. New theory captures framing anomaly in gauge theory.
problem Capturing framing anomaly in gauge theory.
method Constructs a relative Crane-Yetter theory from non-semisimple data.
result Establishes invertibility property for the theory.
New self-shrinkers with multiple ends constructed by stacking planes.
problem Constructing self-shrinkers with any number of ends in 3D space.
method PDE gluing methods and Linearised Doubling methodology.
result Construction of self-shrinkers with 2J+1 ends and genus 2J(m−1). We construct and analyze minimal disc stackings with bounds on their Morse index.
problem Constructing and analyzing minimal free boundary disc stackings.
method Constructing minimal free boundary disc stackings in a three-dimensional Euclidean unit ball, proving bounds on their Morse index.
result Uniform, linear bounds on the Morse index of all such surfaces.
In this paper we will describe an approach to mirror symmetry for appropriate 1-dimensional DM stacks of arithmetic genus g≤1, called tcnc curves, which was developed by the author with Treumann and Zaslow in arXiv:1103.2462 . This involves introducing a conjectural sheaf-theoretic model for the Fukaya category …
Just like Atiyah Lie algebroids encode the infinitesimal symmetries of principal bundles, exact Courant algebroids are believed to encode the infinitesimal symmetries of S1-gerbes. At the same time, transitive Courant algebroids may be viewed as the higher analogue of Atiyah Lie algebroids, and the non-commutative a…
Latent MoS learns multiple symmetries for efficient dynamic learning.
problem Efficiently learning dynamics from limited system measurements.
method Latent Mixture of Symmetries (Latent MoS) with hierarchical architecture.
result Latent MoS outperforms baselines in interpolation and extrapolation tasks.
Develops SymGCP for tensor decompositions with general symmetry.
problem Handling symmetry in tensor decompositions for better model accuracy.
method Introduces SymGCP, a generalized CP decomposition that accounts for any subset of tensor modes' symmetry.
result SymGCP enables efficient and scalable tensor decomposition with improved model robustness and accuracy.
New methods compare Steklov eigenspaces of free boundary minimal surfaces in balls.
problem Comparing Steklov eigenspaces of free boundary minimal surfaces.
method Developed new methods to compare span of coordinate functions with Steklov eigenspace.
result Proved congruence of free boundary minimal annuli in 3D unit ball.
New optimizer designs respect symmetry, improving deep learning models.
problem Optimizers lack respect for symmetry in neural networks.
method Introduce symmetry-compatible principle for optimizer design.
result Symmetry-compatible optimizers improve model performance.
The paper generalizes TQFTs to fermionic systems and classifies SPTs and SETs.
problem Classifying fermionic SPTs and SETs with finite group symmetries.
method Formulating fermionic TQFTs, gauging SPTs, using bordism groups, and constructing anomalous boundary states.
result Explicit classification of fermionic SPTs and SETs, including new anomalous boundary states.
Haar scattering networks improve pattern recognition across various tasks.
problem Improving pattern recognition in diverse tasks like regression and classification.
method Stacking convolutional filters based on Haar wavelets followed by non-linear operators.
result Outperformed best algorithms in 4 out of 18 data classification problems.
The paper analyzes how stacking improves model stability.
problem Lack of theoretical insight into how stacking works.
method Stability analysis of learning algorithms, focusing on hypothesis stability.
result The hypothesis stability of stacking is a product of base models and combiner.
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).
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…
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.
D-branes on noncommutative spaces mimic string theory, offering new insights into mirror symmetry.
problem Exploring noncommutative mirror symmetry through D-branes on noncommutative Calabi-Yau spaces.
method Constructing noncommutative ringed spaces from local resolutions, realizing D-branes as morphisms, and defining kinetic energy.
result Dynamical D-branes on noncommutative spaces can be described by a Polyakov-like action, suggesting a bridge between string theory and noncommutative geometry.
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.
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…
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.
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.
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 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. 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.
Stacked LSTM improves weather forecasting accuracy by incorporating spatial information.
problem Improving temperature prediction accuracy in weather forecasting.
method 2-layer spatio-temporal stacked LSTM model with independent LSTM models per location in the first layer and combined hidden states in the second layer.
result The stacked LSTM model outperforms single LSTM models in most cases by utilizing spatial information.
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…
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.
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…
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…