Geometrically interprets a duality theorem linking cochain and chain complexes.
problem Understanding a complex duality theorem in geometric terms.
method Introduces a chain isomorphism involving simplicial and cellular complexes.
result Establishes a geometric interpretation of Ranicki duality.
The paper proves a theorem about higher-dimensional expansion and its topological implications.
problem Higher-dimensional expansion properties and their topological consequences.
method Detailed proof of Gromov's Topological Overlap Theorem using cellular cochains and simplicial complexes.
result The theorem states that if a complex has strong higher-dimensional expansion properties, it has a topological overlap property.
Spatial refinement of Bar-Natan homology constructed.
problem Link invariants and their homotopy types.
method CW-spectrum construction and stable homotopy type.
result Stable homotopy type of constructed CW-spectrum is a link invariant.
Generalizes Milnor's axiomatic homology theorems to Polish spaces.
problem Characterize (co)homology of Polish spaces.
method Eilenberg-Steenrod axioms, Milnor's map excision axiom, and a generalized additivity axiom.
result Unique characterization of (co)homology of Polish spaces.
Defines a simplicial operad related to Fulton-MacPherson.
problem Equipping the symplectic cochain complex with BV algebra structure.
method Constructs an operad in Top with CW and simplicial decompositions.
result Expected isomorphism to the 2-dimensional Fulton-MacPherson operad.
Paper explores connection cochain in abelian extensions and its relation to connection forms.
problem Understanding the connection cochain in abelian extensions.
method Apply Moriyoshi's connection cochain concept to abelian extensions and relate it to connection 1-forms.
result Established the relationship between connection cochain and connection 1-forms in abelian extensions.
We describe algorithms for finding harmonic cochains, an essential ingredient for solving elliptic partial differential equations in exterior calculus. Harmonic cochains are also useful in computational topology and computer graphics. We focus on finding harmonic cochains cohomologous to a given cocycle. Amongst other …
In this paper we develop several algebraic structures on the simplicial cochains of a triangulated manifold that are analogues of objects in differential geometry. We study a cochain product and prove several statements about its convergence to the wedge product on differential forms. Also, for cochains with an inner p…
New method compares geometric and standard cup products.
problem Reconciling partially defined and fully defined cochain products.
method Vector field flow through cubulation to compare products.
result Explicit cochain level comparison between intersection and cup products.
The paper refines cohomology of VB-groupoids with homogeneous cochains.
problem Cohomology of VB-groupoids with linear structure.
method Refine cohomology by considering k-homogeneous cochains.
result Van Est theorem for k-homogeneous cochains on VB-groupoids.
The paper discusses q-deformations of the Aomoto complex.
problem Deformation of cochain complexes associated with hyperplane arrangements.
method Replaces entries of coboundary maps with q-analogues and analyzes the resulting structures. result The q-deformation can be a cochain complex under certain conditions and yields local system cohomology groups. A fundamental theorem of Wolfe isometrically identifies the space of flat differential forms of dimension m in Rn with the space of flat m-cochains, that is, the dual space of flat chains of dimension m in Rn. The main purpose of the present paper is to generalize Wolfe's theorem to the se…
The paper extends cellular algebras with relative orderings and explores their properties.
problem Generalizing cellular algebras with different partial orderings.
method Classification and construction of simple modules, characterizations, and examples.
result Examples of relative cellular algebras that are not cellular.
Extends CW topology to describe cellular homology on cell complexes.
problem Describing topological properties of cell complexes.
method Introduces descriptive cellular homology on cell complexes, extending Whitehead's CW topology.
result Descriptive cellular complex is a topology on fibres in a fibre bundle.
The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub-)linear functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio-Kirchhei…
The paper connects group extensions, cochains, and spectral sequences.
problem Understanding the relationship between group extensions and spectral sequences.
method Using connection cochains, the paper derives a formula for the extension class.
result A formula clarifies the relation among connection cochains, extension classes, and the LHS spectral sequence.
We construct a pair of transverse genuine laminations on an atoroidal 3-manifold admitting transversely orientable uniform 1-cochain. The laminations are induced by the uniform 1-cochain and they are indeed the "straightening" of the coarse laminations defined in [Ca], by using minimal surface techniques. Moreover, whe…
Paper constructs Thom-Smale complex using instantons from Morse functions.
problem Constructing Thom-Smale complex for Morse functions.
method Analytic instanton construction using eigenspaces of mapping cone Laplacian.
result Instanton complex is cochain isomorphic to Thom-Smale complex.
This paper equips Morse cochain complexes with A∞-algebra structures.
problem Equipping Morse cochain complexes with A∞-algebra structures. method Analogous to K. Fukaya's definition, this paper provides a detailed treatment of Abouzaid's approach.
result Provides a coherent and detailed treatment of Abouzaid's approach to Morse cochain complexes.
A theory for cellular BF models on decomposed spaces.
problem Constructing invariant partition functions for cellular BF models.
method Construction of cellular BF theory on cobordisms with cellular decompositions.
result Invariants under subdivisions and gluing formulas.
Paper proves all Lagrangians unobstructed if one is, using non-archimedean analytic structure.
problem Proving the existence of bounding cochains for unobstructed Lagrangians.
method Introducing non-archimedean analytic structure and using family Floer techniques.
result All Lagrangians in a connected family are unobstructed if one is.
The paper computes KV cochain differentials and their geometric implications.
problem Deformation theory of flat and torsion-free affine connections.
method Explicit computation of KV cochain differentials and their relations to geometric transformations.
result KV algebra with non-vanishing second cohomology group.
The study classifies cellular pseudomanifolds and their properties.
problem Understanding the structure of cellular pseudomanifolds.
method Analyzing the combinatorial and geometric properties of cellular pseudomanifolds.
result Complete classification of cellular pseudomanifolds with excess < 2, and progress towards excess 2.
The paper studies Morse theory on manifolds with boundaries, constructing cellular structures and estimating critical points.
problem Understanding Morse functions on manifolds with boundaries.
method Constructing a cellular structure and analyzing its algebraic properties.
result Estimation of the number of critical points of a Morse function with boundary conditions.
Paper bridges matching rules and height functions in aperiodic tilings.
problem Relationship between matching rules and height functions in aperiodic tilings.
method Cochain-first framework to establish equivalence between matching rules, Ammann bar continuity, cycle closure of 1-cochains, and height-function existence.
result Unified framework for aperiodic tilings including Penrose and canonical projection tilings.
Quantum cellular automata form a homology theory.
problem Understanding the topological structure of quantum cellular automata.
method Formal properties of coarse homology theories.
result Quantum cellular automata naturally form the degree-zero part of a coarse homology theory.
This work redefines data-centric AI by unifying categorical and cochain notions.
problem The need to rethink data notions for data-centric AI.
method Proposes unifying principles from categorical and cochain notions of data.
result Unified definition of data impacts machine learning development, implementation, and utilization.
We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A-infinity structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given a pair of smooth manifolds of the same dimension with embeddin…
Tree almost automorphism groups have a cellular action on a contractible complex.
problem Understanding the structure of tree almost automorphism groups.
method Showed cellular action on a contractible complex with specific stabilizers.
result Tree almost automorphism groups satisfy the F∞-finiteness condition. Optimizes natural frequencies of cellular composites with various microstructures.
problem Designing cellular composites with diverse microstructures for maximizing natural frequencies.
method Data-driven topology optimization with a latent-variable Gaussian process model.
result Cellular designs with multiclass microstructures achieve higher natural frequencies.
Study on instanton homology of pretzel knots and pillowcase Floer homology.
problem Investigating instanton knot homology and Floer homology for a family of pretzel knots.
method Analyzes the reduced singular instanton knot homology and computes bounding cochains in the pillowcase.
result Computed bounding cochains in the pillowcase, revealing a sharp experimental law and rigidity asymmetry.
Machine learning attacks mimic cellular decision-making, revealing new defense mechanisms.
problem Adversarial perturbations fool machine learning models, similar to how ligands prevent correct signaling in cells.
method Formal analogy between neural networks and cellular decision-making models, applying machine learning techniques to study cellular processes.
result Found two regimes in cellular decision-making models, each with a critical point that shapes the loss landscape and defense mechanisms.
For leveled spatial graphs, we find a surface embedding that allows cellular embedding.
problem Finding a surface embedding for general spatial graphs is not always possible.
method Define leveled property, decompose graph into subgraphs, and construct surface.
result For leveled spatial graphs with a small number of levels, a surface can always be found.
The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, González, and Rudyak [1009.1851] with the aim of constructing a cellular model of the configuration space of a sphere. In particular, it was shown that the classifying space (order complex) of the face poset of a totally …
We give examples of harmonic cellular maps between negatively curved manifolds which are not diffeomorphisms but are homotopic to diffeomorphisms.
Revisits Van Est theory for Lie groupoids using homotopy inverses.
problem Relating Lie groupoids and their Lie algebroids cohomology.
method Using the Perturbation Lemma from homological algebra to construct homotopy inverses.
result Constructs homotopy inverses to the van Est differentiation maps at the cochain level.
Study immersions of punctured 4-manifolds for quantum automata applications.
problem Existence of immersions between specific 4-manifolds.
method Analyzing immersions of punctured 4-manifolds to establish a partial order.
result Established a partial order on closed 4-manifolds via immersions.
The paper extends Gaussian processes to model complex interactions in cellular complexes.
problem Capturing topological inductive biases in machine learning models.
method Proposes Gaussian processes on cellular complexes, introducing novel kernels.
result Derives two novel kernels for modeling interactions between cells.
New complexes derived from any filtered cochain complex compute the same cohomology.
problem Constructing cohomologically equivalent subcomplexes from filtered cochain complexes.
method Presenting a general construction that produces subcomplexes from any filtered cochain complex of finite depth.
result The construction of subcomplexes depends only on the filtration up to isomorphism.
Let X be a topological space, and let C(X) be the complex of singular cochains on X with real coefficients. We denote by Cc(X) the subcomplex given by continuous cochains, i.e. by such cochains whose restriction to the space of simplices (endowed with the compact-open topology) defines a continuous real function. We pr…
A saliency detection method for ECT images segments cellular components without supervision.
problem Automatic segmentation of cellular components from ECT images is difficult due to structural complexity and imaging limits.
method Supervoxel over-segmentation, feature extraction, feature matrix decomposition, and computation of saliency.
result The method successfully labels most salient regions detected by a human observer and filters out background regions.
In this work we develop a cellular equivariant homology functor and apply it to prove an equivariant Euler-Poincare formula and an equivariant Lefschetz theorem.
New cochain algebra for diffeological spaces connects de Rham and singular cohomologies.
problem Incompatibility of de Rham and singular cohomologies in diffeology.
method Introduces a new singular de Rham complex and proves it quasi-isomorphic to the original de Rham complex for manifolds and spaces with singularities.
result The new cochain complex resolves the incompatibility issue in diffeology.
Invariants derived from finite group colorings of cut cellular surfaces.
problem Counting colorings of 1-cells to define invariants of cut cellular surfaces.
method Counting colorings of 1-cells with elements of a finite group, subject to a flatness condition.
result Derived invariants of cut cellular surfaces under Pachner-like moves.
Predicts cellular functions in human tissues using multi-layer networks.
problem Challenges in predicting tissue-specific cellular function.
method Hierarchy-aware unsupervised node feature learning for multi-layer networks.
result Improves prediction accuracy of cellular functions in 48 tissues.
The paper computes the cellular homology of real flag manifolds.
problem Computing the cellular homology of real flag manifolds.
method Explicit parametrizations of Schubert cells by closed balls (cubes) in R^n, using them to compute the boundary operator.
result Explicit formula for the boundary operator with refined coefficients.
We compute Steenrod squares on Khovanov homology.
problem Computing Steenrod squares on Khovanov homology.
method Stable cup-i products on cochain complexes of augmented semi-simplicial objects in the Burnside category.
result Explicit formulas for cohomology operations on Khovanov homology.
Deep RL improves cellular network fault management and performance.
problem Fault management and radio performance improvement in outdoor cellular networks.
method Deep Q-Learning for self-organizing networks fault management.
result The proposed algorithm learns to clear alarms and improve radio performance better than existing methods.