Survey on tight triangulated manifolds and their properties.
problem Finding minimal triangulations of manifolds.
method Analyzing known tight triangulated manifolds and their properties.
result Many new tight triangulated manifolds have been identified.
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…
Tightness of a triangulated manifold is a topological condition, roughly meaning that any simplexwise linear embedding of the triangulation into euclidean space is "as convex as possible". It can thus be understood as a generalization of the concept of convexity. In even dimensions, super-neighborliness is known to be …
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…
The paper proves conditions for tight triangulations in 3-manifolds.
problem Characterizing tight triangulations in 3-manifolds.
method Analyzing properties of triangulations in terms of orientability, neighbourliness, and stacking.
result Triangulations of closed 3-manifolds are tight if they are orientable, neighbourly, and stacked.
It is well known that a triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is neighbourly and orientable. No such characterization of tightness was previously known …
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. For d≥2, Walkup's class K(d) consists of the d-dimensional simplicial complexes all whose vertex-links are stacked (d−1)-spheres. Kalai showed that for d≥4, all connected members of K(d) are obtained from stacked d-spheres by finitely many elementary handle additions. According to …
We give an explicit construction of vertex-transitive tight triangulations of d-manifolds for d≥2. More explicitly, for each d≥2, we construct two (d2+5d+5)-vertex neighborly triangulated d-manifolds whose vertex-links are stacked spheres. The only other non-trivial series of such tight triangulated …
Walkup's class K(d) consists of the d-dimensional simplicial complexes all whose vertex links are stacked (d−1)-spheres. According to a result of Walkup, the face vector of any triangulated 4-manifold X with Euler characteristic χ satisfies f1≥5f0−15/2χ, with equality only for $X \in {\cal …
Develops skein theory for 3-manifolds with defects, extending quantum character stacks.
problem Quantum character stacks and their applications in 3-manifolds with surface defects.
method Parabolic induction/restriction for quantum groups, quantum decorated character stacks, ideal triangulations, gluing equations.
result Knot invariants related to quantum A-polynomial, concrete computation method. We introduce the k-stellated spheres and compare and contrast them with k-stacked spheres. It is shown that for d≥2k, any k-stellated sphere of dimension d bounds a unique and canonically defined k-stacked ball. In parallel, any k-stacked polytopal sphere of dimension d≥2k bounds a unique and c…
In a recent work [2] with Datta, we introduced the mu vector (with respect to a given field) of simplicial complexes and used it to study tightness and lower bounds. In this paper, we modify the definition of mu vectors. With the new definition, most results of [2] become correct without the hypothesis of 2-neighbourli…
The study solves a 2008 problem by Kalai about infinitely many homology-spheres.
problem Proving infinitely many homology-spheres with g3=0 in dimensions higher than four. method Using handlebody decompositions and PL manifolds with specific triangulations.
result There are infinitely many homology-spheres with g3=0 in dimensions higher than four. Geometric invariant theory introduces stability conditions mirroring abelian category theory.
problem Stability conditions in geometric invariant theory.
method Axiomatic notion of central charge and stability condition on schemes and stacks.
result Introduction of stability conditions for polarized schemes and smooth projective varieties.
For d≥2, Walkup's class $\Kd$ consists of the d-dimensional simplicial complexes whose vertex-links are stacked (d−1)-spheres. Recently Lutz, Sulanke and Swartz have shown that all F-orientable triangulated d-manifolds satisfy the inequality (2f0−d−1)≥(2d+2)β1 for $d\geq …
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…
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.
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.
The paper extends parallel transport to stacks.
problem Parallel transport over stacks.
method Introducing parallel transport for principal bundles over differentiable stacks.
result Principal bundles with connections over stacks can be recovered from their parallel transport.
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…
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.
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.
New category defined for differentiable stacks.
problem No specific problem stated; focuses on new definition.
method Introducing a new category definition.
result Established relation with Lie groupoid category.
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.
Connected flip graphs for triangulations on hyperbolic surfaces.
problem Connecting triangulations on hyperbolic surfaces via flips.
method Proving connectedness of flip graphs and giving bounds on edge flips.
result Flip graphs of geometric triangulations are connected.
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.
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.
New isolated geometric triangulations found in once-punctured torus bundles.
problem Identifying isolated geometric triangulations in 3-manifolds.
method Examining ideal triangulations and their moves to find isolated geometric ones.
result Infinite family of once-punctured torus bundles with isolated geometric triangulations.
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…
Theory of Lie algebroids over stacks developed.
problem No specific problem stated; theory development for Lie algebroids over stacks.
method Extends Lie algebroid theory to differentiable stacks, defines categories, cohomology, and examples.
result Lie algebroids satisfy descent for submersions and have a cohomology theory.
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.
Study distances between triangulations on surfaces via simultaneous flips.
problem Calculating distances between triangulations on surfaces.
method Performing simultaneous flips on triangulations of finite type surfaces.
result Upper bounds on distance depend only on surface topology.
Efficient triangulations help in understanding 3-manifold boundaries.
problem Understanding boundary slopes in 3-manifolds.
method Introducing and studying boundary-efficient triangulations and inflating ideal triangulations.
result There are only finitely many boundary slopes for incompressible and \(\partial\)-incompressible surfaces in compact 3-manifolds.
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.