The Pachner graph of 2-spheres is studied, focusing on subgraphs of flag and stacked 2-spheres.
problem Characterize subgraphs of the Pachner graph of 2-spheres.
method Analyzes various induced subgraphs of the Pachner graph of n-vertex triangulated 2-spheres, proving connectivity and providing bounds on the number of connected components. result The subgraph of n-vertex flag 2-spheres is connected, while the subgraph of n-vertex stacked 2-spheres has at least as many connected components as trees with specific properties. In 1987, Kalai proved that stacked spheres of dimension d≥3 are characterised by the fact that they attain equality in Barnette's celebrated Lower Bound Theorem. This result does not extend to dimension d=2. In this article, we give a characterisation of stacked 2-spheres using what we call the {\em separatio…
Unique simple spines of homotopy 2-spheres are shown to be ambiently isotopic.
problem Characterizing and comparing simple spines of homotopy 2-spheres.
method Proving ambient isotopy of simple spines representing the same homology class.
result Simple spines of homotopy 2-spheres are unique.
The 3-sphere has either 2 minimal 2-spheres or an optimal foliation by 2-spheres.
problem Proving existence of minimal 2-spheres or optimal foliations in arbitrary Riemannian 3-spheres.
method Analyzing the properties of arbitrary Riemannian metrics on 3-spheres.
result The existence of at least two minimal 2-spheres or an optimal foliation in 3-spheres with arbitrary metrics.
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.
Concrete example of 2-sphere with constant curvature 1 and closed geodesics.
problem Constructing a 2-sphere with constant curvature 1 and closed geodesics.
method Constructing a concrete example of a Finsler metric on the 2-sphere.
result A 2-sphere with constant Gauss curvature 1 and all geodesics closed.
The paper studies minimal 2-spheres of constant curvature in complex hyperquadrics using matrix theory.
problem Exploring minimal 2-spheres of constant curvature in complex hyperquadrics.
method Using singular-value decomposition of complex matrices to study the moduli space of noncongruent spheres.
result Uniqueness proven for totally real constantly curved 2-spheres.
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…
Complete obstruction found for 2-spheres in 5-manifolds to be isotopic.
problem Obstructing homotopic embeddings of 2-spheres into 5-manifolds from being isotopic.
method Using a level preserving Whitney move in codimension 3 to eliminate double points, and classical methods.
result New results for simply-connected 5-manifolds and 2-spheres with algebraic dual 3-spheres.
We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show that an Ahlfors 2-regular, linearly locally contractible metric 2-sphere is quasisymmetrically homeomorphic to the standar…
Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then apply the equation to obtain a classification of biharmonic maps in a family of r…
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 degree of certain holomorphic 2-spheres is bounded.
problem Understanding the degree of holomorphic 2-spheres in complex Grassmannians.
method Analyzing the degree of linearly full constantly curved holomorphic 2-spheres in G(2,n+2;C).
result The degree is bounded between n and 2n for specific n values.
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.
Study hyperbolic 2-spheres with cone points, describing spaces for n=3.
problem Characterize the space of hyperbolic 2-spheres with cone points.
method Analyzing the space C(a0,a1,…,an) for n=3 and n=4. result Detailed description of spaces for n=3 and examples for n=4. We study metrics on conic 2-spheres when no Einstein metrics exist. In particular, when the curvature of a conic metric is positive, we obtain the best curvature pinching constant. We also show that when this best pinching constant is approached, the conic 2-sphere has an explicit Gromov-Hausdorff limit. This is a gene…
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.
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.
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.
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.
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.
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.
A dynamic stacking method for network node classification.
problem Node classification on networks with varying topological features.
method Dynamic functional coefficients for heterogeneous classifiers.
result Significantly more accurate model for network node classification.
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.
Taut foliations map leaves to branched 2-sphere covers.
problem Understanding taut foliations in 3-manifolds.
method Map foliation leaves to branched 2-sphere covers.
result Taut foliations are characterized by such maps.
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.
Associate stacks to Poisson manifolds for equivalence checking.
problem Equivalence checking of Poisson manifolds.
method Associate stacks to Poisson manifolds using Dirac manifolds and closed 2-forms.
result Two Poisson manifolds are equivalent if their associated stacks are isomorphic.
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…
Kähler structures on products of 2-spheres have identical Chern classes.
problem Classifying Kähler structures on products of 2-spheres.
method Using complex Bott manifolds and iterated P1-bundle constructions, we classify Kähler structures up to biholomorphism via Bott diagrams. result Kähler structures on products of 2-spheres have identical Chern classes.
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.
The study proves the existence of geodesics on reversible Finsler spheres.
problem Existence of closed geodesics on Finsler 2-spheres.
method Generalization of Grayson's curve shortening flow.
result Existence of three simple closed geodesics and infinitely many closed geodesics.
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.
Complete invariant defined for doodles on a sphere.
problem Doodles on a 2-sphere.
method Coefficients in series of chord diagrams.
result Finite type invariants of order at most 2n.
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. Describes Teichmüller stack structure near Kähler points.
problem Understanding Teichmüller spaces/stacks near Kähler points.
method Analyzes local structure of Teichmüller stack near Kähler points.
result Affirmative answer to Catanese Kur=Teich question at generic Kähler points.
Proves existence of curves with constant curvature in a sphere.
problem Existence of curves with constant geodesic curvature in a Riemannian 2-sphere.
method Develops a min-max scheme for a weighted length functional.
result Proves existence for almost every prescribed curvature.
Study describes bifurcations of gradient flows on 2-sphere with holes.
problem Analyzing gradient flows on a 2-sphere with up to six singular points.
method Using separatrix diagrams to specify saddle-node and saddle connections.
result Identified all possible topological structures of bifurcations.
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.
In a previous paper, we showed that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (i.e., is tangent to a smooth variation through harmonic maps). In this paper, in contrast, we show that there are (non-full) harmonic maps from the 2-sphere to the 3-sphere and 4-sp…