Geometric duality connects graph isomorphism and knot equivalence.
problem Understanding the equivalence of graph isomorphism and knot equivalence.
method Observation of geometric duality in planar graphs and links.
result The equivalence relation defined by isomorphisms of checkerboard graphs is the same as 2-isomorphisms of checkerboard graphs.
Geometric group theory explores groups through their geometric properties.
problem Understanding groups via geometric properties.
method Cayley and Schreier graphs, ping-pong lemma, quasi-isometries, growth of groups, hyperbolicity.
result Gromov's theorem on groups of polynomial growth and amenability.
Geometric deep learning predicts knot invariants.
problem Predicting knot invariants from knot data.
method Constructing a functor from knots to graphs and using graph neural networks.
result High generalization capabilities demonstrated.
Surveying connections between graph combinatorics and algebraic right-angled Artin groups.
problem Understanding the relationship between graph structures and algebraic properties of right-angled Artin groups.
method Analyzing the defining and extension graphs of right-angled Artin groups.
result Discovers connections to geometric group theory and complexity theory.
Study finds the shortest triply periodic graph spanning a cubic lattice.
problem Finding the shortest periodic graph with a fixed volume.
method Analyzes the body centred cubic lattice and the gyroid surface.
result The shortest graph is the srs network with K4 quotient. Simplified Milnor-Schwarz lemma for geometric group theory.
problem Conditions for orbit maps to be quasi-isometries.
method Succinct treatment and applications to non-Archimedean groups.
result Sharpened results on mapping class groups and quasi-isometry classification.
Root Laplacian Eigenmaps help in spectral embedding of graphs.
problem Efficient spectral embedding of graphs.
method Square root of graph-Laplacian operator.
result Improved spectral embedding techniques.
The study constructs periodic surfaces using graph theory and applies cyclically branched coverings to identify their conformal type.
problem Constructing and identifying the conformal type of periodic surfaces with a given geometric structure.
method Graph theory and cyclically branched coverings.
result Explicit cone metrics on compact Riemann surfaces can be realized as the quotient of triply periodic polyhedral surfaces.
New method reduces model selection sample complexity for geometric graphs.
problem Model selection in Gaussian Markov fields with sample deficiency.
method Introducing spatial stationarity to geometric graphs, developing information-theoretic bounds and efficient reconstruction techniques.
result Spatial stationarity leads to significant reduction in sample complexity for consistent recovery.
New technique connects graph matching complexes to Morse theory for better topology understanding.
problem Understanding the topology of matching complexes of complete graphs.
method Developed discrete Morse theory technique to analyze Mn. result Showed Mn is geometrically (νn−1)-connected, improving on previous homotopical results. The curve graph's model theory reveals its central role in surface study.
problem Why is the curve graph central in surface and mapping class group studies?
method Developed a bridge between model theory, topology, and group theory; bi-interpreted curve graph with mapping class group.
result Proved the curve graph's first-order theory is ω-stable and has quantifier elimination. This paper develops graph theory for racks and quasigroups.
problem Characterizing and realizing right quasigroups and related structures.
method Study of graph markings, Schreier graphs, and Cayley graphs.
result All right quasigroups are realizable by specific types of graphs.
Functional adapts to graph structures for machine learning applications.
problem Discretizing Mumford-Shah functionals on graphs for machine learning.
method Discretization of nonlocal approximations to Mumford-Shah functional on random geometric graphs.
result Minimizers of graph Mumford-Shah functionals converge to a continuum Mumford-Shah functional under certain conditions.
Geometric approach monitors dynamic large graphs, detects major events.
problem Monitoring dynamic large graphs is challenging due to local changes affecting global properties.
method Developed a geometric approach using Ollivier-Ricci curvature for real-time monitoring.
result Detects major events and changes via graph embedding geometry.
Lecture notes on group actions on injective spaces and Helly graphs.
problem Understanding group actions on specific metric spaces.
method Review of injective metric spaces and Helly graphs, elementary properties, constructions, and exercises.
result Presentation of various constructions of injective metric spaces and Helly graphs with interesting group actions.
New method selects diffusion scales for graph wavelets.
problem Choosing optimal diffusion scales for graph wavelets.
method Proposes an unsupervised method using information theory.
result Method selects diffusion scales for graph wavelets.
Graphs of multicurves are hyperbolic, relatively hyperbolic, or thick.
problem Characterizing graphs of multicurves based on their geometric properties.
method Proving graphs of multicurves are hyperbolic, relatively hyperbolic, or thick based on subsurface intersections.
result Geometric characterization of graphs of multicurves.
Develops methods to analyze manifold singularities using graph Laplacian.
problem Analyzing geometric properties of singularities in datasets.
method Theory and methods using the graph Laplacian to provide explicit bounds on manifold singularities.
result Explicit bounds on the graph Laplacian for functions near manifold singularities.
This study improves graph coarsening methods by preserving graph spectrum and distances.
problem Solving large-scale graph problems by working on a smaller graph.
method Developed a geometric approach using Gromov--Wasserstein distance to minimize the difference between graph distances and their coarsened versions.
result Minimizing the difference between graph distances and their coarsened versions can be achieved using the weighted kernel K-means method. New optimization algorithms on orthogonal group for machine learning.
problem Efficient optimization on the orthogonal group for machine learning tasks.
method Stochastic geometric algorithms on Lie groups.
result Strong performance on diverse machine learning tasks.
Surjectivity of Cannon-Thurston map proven for metric graph bundles.
problem Proving surjectivity of Cannon-Thurston map in metric graph bundles.
method Generalized Mj-Sardar's result to include more types of fibers.
result Continuous extension map between boundaries is surjective.
The homology of Kontsevich's commutative graph complex parameterizes finite type invariants of odd dimensional manifolds. This {\it graph homology} is also the twisted homology of Outer Space modulo its boundary, so gives a nice point of contact between geometric group theory and quantum topology. In this paper we give…
Study curves' intersections and distances, with applications in graph and group studies.
problem Understanding intersections and distances of curves.
method Using a relationship between intersection numbers and subsurface projection distances, applications in curve graphs and mapping class groups.
result Explicit quasi-constants for the relationship between intersection numbers and subsurface projection distances.
Graph-based Bayesian learning theory ensures scalable algorithms for large datasets.
problem Consistency and scalability in semi-supervised learning with graphs.
method Introduces new scaling theory for graph parameters and proves uniform spectral gaps for Markov chain Monte Carlo algorithms.
result Graph-based Markov chain Monte Carlo algorithms have a uniform spectral gap independent of unlabeled data size.
Study on right-angled Coxeter groups and their geometric properties.
problem Characterizing the coarse geometry of right-angled Coxeter groups.
method Analyzing graph properties and applying geometric group theory.
result Proves properties of right-angled Coxeter groups, including quasi-isometry and divergence.
We introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric descriptions of the "life span" or "persistence" of a homology class. As a case study, …
In this note, we consider two Riemannian metrics on a moduli space of metric graphs. Each of them could be thought of as an analogue of the Weil-Petersson metric on the moduli space of metric graphs. We discuss and compare geometric features of these two metrics with the "classic" Weil-Petersson metric in Teichmüller t…
A new method integrates forms on Riemann surfaces, leading to modular forms.
problem Integrating differential forms with poles on Riemann surfaces.
method Simple procedure to integrate differential forms with arbitrary holomorphic poles, establishing an analytic theory for integrals over configuration spaces.
result Regularized graph integrals on elliptic curves are almost-holomorphic modular forms.
Paper presents voxel graph operators for vector data models.
problem Efficient conversion and analysis of geometric models.
method Topological voxelization, graph construction, differential operator derivation.
result Discrete differential and integral operators from voxel complexes.
The paper proves index theorems for graph-based optimal control problems.
problem Optimal control problems on graphs with constraints.
method Proves Morse index theorems for a broad class of variational problems on graphs.
result Formulas compute the difference of Hessians related to different graphs or boundary conditions.
In geometric group theory one uses group actions on spaces to gain information about groups. One natural space to use is the Cayley graph of a group. The Cayley graph arguments that one encounters tend to require local finiteness, and hence finite generation of the group. In this paper, I take the theory of intersectio…
The paper investigates the distribution of systoles on arithmetic Riemann surfaces.
problem Understanding the asymptotic behavior of systoles on arithmetic Riemann surfaces.
method Combining combinatorics, group theory, and geometric group theory.
result The set of arithmetic surfaces cannot be concentrated, indicating the same for systoles.
Consider a finite connected graph possibly with multiple edges and loops. In discrete geometric analysis, Kotani and Sunada constructed the crystal associated to the graph as a standard realization of the maximal abelian covering of the graph. As an application of what the author showed in an earlier paper with Seshadr…
tf_geometric simplifies graph deep learning in TensorFlow.
problem Efficient graph deep learning in TensorFlow.
method Kernel libraries and infrastructures for GNNs.
result tf_geometric supports various graph tasks and provides efficient GNN models.
Study on communication delays in decentralized learning networks.
problem Optimizing communication latency in decentralized learning networks.
method Utilized network information theory and random geometric graph theory.
result Communication delay scales as O(n^(2-3β)/βlog n).
Generalizes Sobolev IPM for graph-based measures using Orlicz geometric structure.
problem Limitation of Le et al. (2025) framework to Lp geometry. method Generalizes Sobolev IPM through Orlicz geometric structure, employing convex functions to capture nuanced geometric relationships.
result GSI-M reduces to a simple univariate optimization problem, achieving remarkable computational efficiency.
The paper establishes inequalities for minimal graphs on manifolds with nonnegative Ricci curvature.
problem Establishing inequalities for minimal graphs on manifolds with nonnegative Ricci curvature.
method Combining Cheeger-Colding theory and geometric measure theory to derive Sobolev and Neumann-Poincaré inequalities.
result Gradient estimates and Liouville theorem for minimal graphs over manifolds with nonnegative Ricci curvature.
Geometric GNNs improve graph discrimination through GWL.
problem Discriminating geometric graphs embedded in Euclidean space.
method Proposed a geometric version of the Weisfeiler-Leman test (GWL) for geometric graphs.
result Characterized the expressive power of geometric GNNs based on physical symmetries.
Researchers prove inner product recovery is impossible in latent space models.
problem Recovering inner products in latent space models with random geometric graphs.
method Rate-distortion theory applied to Gaussian or spherical latent locations.
result Impossible to recover inner products if dimensionality exceeds nh(p), matching positive results' conditions. BGNNs model particle-boundary interactions efficiently.
problem Efficiently modeling geometric boundaries in 3D simulations.
method Introduce Boundary Graph Neural Networks (BGNNs) to dynamically modify graph structures.
result BGNNs accurately reproduce 3D granular flows without handcrafted conditions.
Researchers determined the second homology group of a specific symplectic derivation Lie algebra.
problem Determining the entire homology group of a specific symplectic derivation Lie algebra.
method Used classical representation theory of Sp(2g; Q) and weight decomposition.
result Determined H_2(\mathfrak{c}_g^{+})
This is a glossary of notions and methods related with the topological theory of collections of affine planes, including braid groups, configuration spaces, order complexes, stratified Morse theory, simplicial resolutions, complexes of graphs, Orlik--Solomon rings, Salvetti complex, matroids, Spanier--Whitehead duality…
The paper studies matrix normalization and graph balancing using a new functional and gradient descent.
problem Matrix normalization and graph balancing.
method A new functional called the non-normal energy, and gradient descent.
result Gradient descent of the non-normal energy converges to balanced graphs and preserves spectra and realness of weights.
Enhanced spectral clustering for geometric graphs improves clustering accuracy.
problem Ineffective standard spectral clustering for geometric graphs.
method Higher-order spectral clustering using higher-order eigenvectors.
result Established weak and strong consistency for Soft Geometric Block Model.
Researchers identify critical protein residues using advanced graph theory.
problem Identifying essential residues in proteins for function.
method Learning Random Geometric Graphs (RGG) with Cramer's V correlation and organic thresholding.
result Advanced RGG methods accurately identify critical residues compared to existing techniques.
The paper explores efficient graph algorithms on geometric graphs and their computational limits.
problem Efficiently solving spectral graph theory problems on geometric graphs.
method Investigates algorithms and hardness results for multiplying vectors, finding spectral sparsifiers, and solving Laplacian systems on K-graphs. result Formal limitations on subquadratic time algorithms for spectral graph theory problems on geometric graphs, including the Gaussian kernel and Neural tangent kernels.
Geometric scattering for graph data enhances feature retention and classification.
problem Tackling the generalization of scattering transforms to graph data.
method Analogous to ConvNets, we develop geometric scattering for graph data, focusing on feature stability under graph deformations.
result Extracted features retain informative variability and relations in graph data, aiding classification and exploration.
Graph dynamics link combinatorics to geometry, revealing manifold intersections and stability.
problem Understanding the geometry of graph dynamical systems with odd interactions.
method Proved geometry and stability of manifolds governed by graph homology and coverings.
result Derived upper and lower bounds on the dimension of the equilibrium set.