Study higher dimensional Reidemeister torsion for twist knots surgeries.
problem Asymptotic behavior of Reidemeister torsion for exceptional surgeries.
method Analyzing graph manifolds and twist knot groups, determining limits of coefficients.
result Explicit set of limits for leading coefficients in higher dimensional Reidemeister torsion.
Formula for mass in higher-dimensional graphs proves mass theorems.
problem Proving mass theorems for higher-dimensional graphs.
method Explicit formula for Gauss-Bonnet-Chern mass, applied to asymptotically flat graphical manifolds.
result Proves positive mass theorem and Penrose inequality for graphs with flat normal bundle.
The paper extends Hodge-de Rham theory to higher-dimensional Sierpinski gaskets.
problem Analyzing differential forms and Laplacians on higher-dimensional fractal structures.
method Constructing sequences of graphs approximating Sierpinski gaskets, defining k-forms, de Rham derivatives, and their duals, proving harmonic properties, and exploring 2-forms.
result Obtained a basis for the space of harmonic 1-forms on level-3 Sierpinski gasket.
We analyze higher-dimensional sliding puzzles, finding solvability patterns.
problem Solvability of higher-dimensional cubical sliding puzzles.
method Study of puzzle graphs and token movement constraints.
result Characterization of solvability regimes from stuck to fully solvable.
We define a class of Euclidean distances on weighted graphs, enabling to perform thermodynamic soft graph clustering. The class can be constructed form the "raw coordinates" encountered in spectral clustering, and can be extended by means of higher-dimensional embeddings (Schoenberg transformations). Geographical flow …
Study homology of periodic cell complexes using quotient spaces and spectral sequences.
problem Quantifying homology in periodic cell complexes.
method Finite representation of periodic cell complexes, Mayer-Vietoris spectral sequence.
result Full recovery of homology generators for d-periodic graphs. Generalized Fáry's theorem to higher dimensions.
problem No specific problem stated; generalization of Fáry's theorem.
method Proof of a higher-dimensional version of Fáry's theorem.
result Proved a generalization of Fáry's theorem in higher dimensions.
New model learns graph neural networks equivariant to various transformations.
problem Learning equivariant graph neural networks for complex transformations.
method E(n)-Equivariant Graph Neural Networks (EGNNs) that are computationally efficient and scalable.
result Achieves competitive or better performance without higher-order representations.
Random matrix models generalize to Group Field Theories (GFT) whose Feynman graphs are dual to gluings of higher dimensional simplices. It is generally assumed that GFT graphs are always dual to pseudo manifolds. In this paper we prove that already in dimension three (and in all higher dimensions), this is not true due…
A simplified proof for embedding higher-dimensional complexes into manifolds.
problem Embedding higher-dimensional complexes into manifolds with constraints.
method A short and accessible proof for the Patak-Tancer theorem.
result A simplified proof for the Heawood inequality in higher dimensions.
Study solves Dirichlet problem for higher-dimensional submanifolds.
problem Existence and uniqueness of graphical maximal submanifolds.
method General existence and uniqueness results for any codimension.
result General existence and uniqueness results for graphical maximal submanifolds of higher codimension.
Researchers found a quadratic estimate for embedding higher-dimensional simplices into sphere-connected sums.
problem Estimating the number of handles required for embedding higher-dimensional simplices into sphere-connected sums.
method Combining geometric topology, combinatorics, and linear algebra.
result Presented a quadratic estimate g≥ckn2 for embedding k-faces of n-simplex. 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.
Higher dimensional graphs can be used to colour two-dimensional geometric graphs. If G the boundary of a three dimensional graph H for example, we can refine the interior until it is colourable with 4 colours. The later goal is achieved if all interior edge degrees are even. Using a refinement process which cuts the in…
Simplified exposition of graph drawing invariants.
problem Graph drawing invariants and their applications.
method Defining and analyzing a mod2-valued self-intersection invariant.
result Elementary formulations accessible to mathematicians.
Method proves complex homeomorphic to a sphere using bisimplices.
problem Proving regular CW complexes homeomorphic to spheres.
method Discrete Morse theory and bisimplices.
result Flag bisimplicial completion of quadric complexes is contractible.
The paper provides conditions for realizing graphs and polytopes with specified edge lengths.
problem Proving the existence of planar embeddings or polyhedra with specified edge lengths.
method Practical sufficient conditions and software verification for non-self-intersecting perturbations of initial realizations.
result Existence of planar embeddings and polyhedra with specified edge lengths.
New neural architectures invariant to sign flips and basis symmetries for graph representation learning.
problem Learning invariant graph representations from eigenvectors.
method SignNet and BasisNet neural architectures that are invariant to sign flips and basis symmetries.
result Proven to be universal, approximating any continuous function of eigenvectors with desired invariances.
The paper explores invariants of graph drawings in the plane.
problem Understanding the invariants of almost embeddings of graphs in the plane.
method Proves relations between invariants, connects to homology, constructs examples.
result Constructs almost embeddings realizing some values of invariants.
Constructs integer-valued cohomology classes from graph cocycles.
problem Constructing nontrivial cohomology classes from graph cocycles.
method Gluing compactified configuration spaces to construct integer-valued classes from integer-valued graph cocycles.
result Obtains nontrivial classes from trivalent graph cocycles.
The paper creates non-isotopic but homotopic diffeomorphisms in 4-manifolds.
problem Creating non-isotopic but homotopic diffeomorphisms in 4-manifolds.
method Surgery along Θ-graphs and use of a twisted characteristic class.
result Constructs countably many non-isotopic but homotopic diffeomorphisms.
Study Higgs sections and flat sections for nonlinear harmonic bundles.
problem Equivalence of Higgs and flat sections for nonlinear harmonic bundles.
method Analyze harmonic vector bundles, generalize to sub-fibrations and morphisms.
result Vanishing of a degree obstruction for general nonlinear harmonic bundles.
Defines a transgression functor for higher-dimensional Courant algebroids.
problem None explicitly stated; focuses on definition and properties.
method Definition of transgression functor for Courant algebroids.
result Established a connection between Courant algebroids and Lie algebroids.
Minimal simplicial complexes in high dimensions always contain complex links.
problem Existence of complex links in high-dimensional embeddings.
method Demonstrated through minimal simplicial complexes in R2n. result Minimal simplicial n-complexes inevitably contain a nonsplittable two-component link. This paper reconstructs complex graph signals using kernel methods on manifolds.
problem Reconstructing complex graph signals from samples on graph vertices.
method Kernel methods on complex manifolds, embedding vertices into higher-dimensional spaces.
result Effective reconstruction of complex graph signals, outperforming conventional methods.
The paper provides a converse to linking theorems for graphs in 3-space and higher dimensions.
problem Linking properties of graphs in 3-space and higher dimensions.
method Proves a converse to specific linking theorems for graphs in 3-space and higher dimensions.
result Proves a higher-dimensional analogue of a converse to a lemma by Segal-Spież.
Graph Convolutional Networks improve performance on complex data.
problem Processing high-dimensional, graph-based data for automation.
method Enhanced existing Graph Convolutional Network models with four improvements.
result Significant performance improvements on four benchmark datasets.
Spectral clustering is a standard approach to label nodes on a graph by studying the (largest or lowest) eigenvalues of a symmetric real matrix such as e.g. the adjacency or the Laplacian. Recently, it has been argued that using instead a more complicated, non-symmetric and higher dimensional operator, related to the n…
Survey on approximability of graph embeddings and related problems.
problem Determining when graphs can be embedded in the plane without intersections.
method Criteria for approximability by embeddings and van Kampen obstruction.
result Completeness of the van Kampen obstruction for approximability by embeddings.
An observable for nonabelian, higher-dimensional forms is introduced, its properties are discussed and its expectation value in BF theory is described. This is shown to produce potential and genuine invariants of higher-dimensional knots.
A novel method visualizes higher-dimensional spaces using hyperbolic geometry.
problem Challenges in visualizing higher-dimensional spaces.
method Interactive visualization of higher-dimensional grids based on hyperbolic geometry.
result Our method shows the whole higher-dimensional space at once and avoids disadvantages of previous methods.
A new graph embedding method using Hebbian learning for improved vector representations.
problem Creating accurate vector representations for nodes in graphs.
method Hebbian learning with non-convex Gaussian mixture model for node embeddings.
result The method outperforms state-of-the-art methods on benchmark data sets and generates relevant recommendations.
We extend graph neural networks to transfer performance across different input sizes.
problem Transferability of graph neural networks across varying input dimensions.
method Introduce a general framework for transferability across dimensions, showing it corresponds to continuity in a limit space.
result Transferability of graph neural networks is driven by data and learning task, and can be ensured with design principles.
The (abelian bosonic) heterotic string effective action, equations of motion and Bianchi identity at order alpha prime in ten dimensions, are shown to be equivalent to a higher dimensional action, its derived equations of motion and Bianchi identity. The two actions are the same up to the gauge fields: the latter are a…
The paper characterizes graph manifolds using fold maps and embeddability of polyhedra.
problem Understanding the global topologies of graph manifolds.
method Using fold maps into the plane and embeddability of polyhedra in 3-manifolds.
result Characterizes graph manifolds via fold maps and polyhedra embeddability.
Graphs represent gene segment organization, revealing complex interrelationships in a scrambled genome.
problem Understanding gene segment organization and interrelationships in a scrambled genome.
method Directed graphs representing gene segments and their relationships, with graph properties mapped to higher-dimensional space for analysis.
result Emerging star-like structures indicate complex interrelationships, including segments from multiple genes interleaving or overlapping.
Improved graph-based multiclass classification for multilayer data.
problem Efficient classification of multilayer data with limited labeled examples.
method Generalized diffuse interface methods applied to multilayer graphs, using spectral decomposition and fast matrix-vector products.
result Highly scalable and efficient classification for large, high-dimensional data sets.
New higher-dimensional Schwarz and Scherk surfaces discovered.
problem Constructing complete embedded periodic minimal hypersurfaces.
method Higher dimensional generalizations of Schwarz's and Scherk's surfaces.
result Complete embedded periodic minimal hypersurfaces constructed in Rn. Paper extends eigenvalue inequality to higher dimensions.
problem Eigenvalue inequality for Steklov eigenvalues in higher dimensions.
method Extended Hersch-Payne-Schiffer trick to higher dimensional manifolds.
result Generalized inequality for Steklov eigenvalues.
We present a topological interpretation of knot and braid contact homology in degree zero, in terms of cords and skein relations. This interpretation allows us to extend the knot invariant to embedded graphs and higher-dimensional knots. We calculate the knot invariant for two-bridge knots and relate it to double branc…
Proposes a new dictionary learning method for high-dimensional graph signals.
problem Challenges of traditional sparse representation methods in high-dimensional graph signals.
method Integrates graph topology implicitly through sparse combinations of graph-wavelet functions and explicitly through graph constraints.
result Demonstrates effectiveness in high-dimensional graph signal processing.
The paper explains the topological origin of the distinction between incidence theorems over division rings and fields.
problem Understanding the distinction between incidence theorems over division rings and fields.
method Extending the surface-graph approach to noncommutative settings, the paper analyzes the topological properties of graphs embedded on surfaces of different genera.
result Theorems associated with graphs on the sphere hold over any division ring, while those on surfaces of positive genus typically hold only if the ground ring is a field.
Proposes a method to learn representations of higher-dimensional simplicial complexes.
problem Lack of methods for representing entire simplicial complexes.
method Geometric message passing schemes for end-to-end learning of simplicial complex representations.
result First method for learning representations of entire simplicial complexes.
Homology theory for qualgebras yields knotted graph and foam invariants.
problem Developing a homology theory for qualgebras.
method Constructing a classifying space from prisms and adding degenerate cells.
result Homotopy classes of maps from spheres to the classifying space of G. We provide a generalization of Bianchi's Bäcklund transformation from 2-dimensional quadrics to higher dimensional quadrics. The starting point of our investigation is the higher dimensional (infinitesimal) version of Bianchi's main four theorems on the theory of deformations of quadrics and Bianchi's treatment of the …
A new clustering method for simplicial complexes using homology.
problem Clustering simplicial complexes efficiently and accurately.
method Inspired by graph spectral clustering, the method uses sparse eigenproblems.
result Produces clusters sensitive to simplicial complex homology.
This paper concerns some stability properties of higher dimensional catenoids in $\rr^{n+1}$ with n≥3. We prove that higher dimensional catenoids have index one. We use δ-stablity for minimal hypersurfaces and show that the catenoid is n2-stable and a complete n2-stable minimal hypersurface is a …
Graph energy helps detect communities in networks better than traditional methods.
problem Detecting communities in sparse networks where traditional methods fail.
method Using graph energy based on the full spectrum of adjacency matrices.
result The difference in graph energy between a planted partition model and an Erdős--Rényi network has a distinct transition at the detectability threshold.