Study on evolving non-compact graphs by powers of Gauss curvature.
problem Evolution of non-compact graphs by powers of Gauss curvature.
method Evolution study of convex complete non-compact graphs by positive powers of Gauss curvature.
result Initial graph evolves by any positive power of Gauss curvature for all time, even if not differentiable.
The Q k Q_k Q k flow on graphs evolves them up to a time T.
problem Evolution of graphs by Q k Q_k Q k curvature flow. method Proving evolution of complete non-compact graphs up to time T.
result Complete graphs evolve by Q k Q_k Q k curvature up to time T. New graph-based method for compact n-manifolds.
problem Representing compact n-manifolds efficiently.
method Using (n+1)-colored graphs to represent compact n-manifolds.
result Established results on topology of represented manifolds.
Constructs real algebraic functions with both compact and non-compact preimages.
problem Finding real algebraic functions with specific preimage properties.
method Explicit construction of real algebraic functions.
result Demonstrates real algebraic functions on non-compact manifolds with non-compact preimages.
The paper estimates heights of constant mean curvature graphs in specific spaces.
problem Estimating heights of constant mean curvature graphs in N i l 3 \mathrm{Nil}_3 Nil 3 and P S L ~ 2 ( R ) \widetilde{PSL}_2(\mathbb{R}) P S L 2 ( R ) . method Height estimates for compact, constant mean curvature graphs in N i l 3 \mathrm{Nil}_3 Nil 3 and P S L ~ 2 ( R ) \widetilde{PSL}_2(\mathbb{R}) P S L 2 ( R ) . result Announced a structure-type result for proper graphs defined on relatively compact domains.
Graphs approximate Laplacian spectra on manifolds.
problem Approximating Laplacian spectra on complex manifolds.
method Graph Laplacians on proximity graphs.
result Spectra of graph Laplacians approximate the Laplacian spectra of manifolds.
Defined a new graph type for compact surfaces, proving its connectedness and infinite diameter.
problem Understanding the structure of arc graphs on compact surfaces.
method Defining and analyzing the prescribed arc graph A ( Σ , Γ ) \mathscr A(Σ,Γ) A ( Σ , Γ ) for compact surfaces Σ Σ Σ with boundary and relations Γ Γ Γ . result The prescribed arc graph A ( Σ , Γ ) \mathscr A(Σ,Γ) A ( Σ , Γ ) is connected and infinite-diameter, with specific conditions for Gromov hyperbolicity. Automates graph convolutional network design for semi-supervised node classification.
problem Designing optimal graph convolutional network architectures for semi-supervised node classification.
method An automatic process to define a problem-specific architecture based on graph structure.
result The proposed method outperforms existing methods in classification performance and network compactness.
Study shows long-term existence of IMCF on non-compact graphs in hyperbolic space.
problem Long-term existence of IMCF on non-compact graphs in hyperbolic space.
method Investigation of IMCF on bounded graphs over horospheres, use of cutoff functions, and development of a non-compact ODE maximum principle.
result Long time existence of IMCF on non-compact graphs in hyperbolic space.
A compact 4-dimensional manifold is a non-singular graph-manifold if it can be obtained by the glueing T^2-bundles over compact surfaces (with boundary) of negative Euler characteristics. If none of glueing diffeomorphisms respect the bundle structures, the graph-structure is called reduced. We prove that any homotopy …
Proposes methods for local clustering in attributed graphs.
problem Finding a single cluster concentrated on a specific region in a graph.
method Introduces Graph Unimodality (GU) and Attribute Unimodality (AU) measures, and LOCLU algorithm to optimize Compactness score.
result Local cluster detected by LOCLU concentrates on the region of interest and exhibits unimodal data distribution.
Every graph can be represented as a singular set of a special surface.
problem Representing any finite graph as the singular set of a compact 3D surface.
method Constructing a calibrated 3-dimensional homologically area minimizing surface with a special Lagrangian form.
result The singular set of the surface is precisely the given graph.
Smooth manifolds can be triangulated with graphs of bounded twin-width.
problem Understanding the structure of triangulations of smooth manifolds.
method Using Whitney's triangulation method and bounding the twin-width of specific graphs.
result Compact smooth manifolds have triangulations with graphs of bounded twin-width.
Graph Laplacian spectral convergence rate established for manifold learning.
problem Consistency of spectral clustering algorithms on manifold data.
method Established spectral convergence rate of graph Laplacian for a random sample of a d-dimensional compact submanifold.
result Consistency of spectral clustering algorithms via spectral convergence rate.
Let M be a graph manifold. We show that π_1M is the fundamental group of a compact nonpositively curved cube complex if and only if M is chargeless. We also prove that in that case π_1M is virtually compact special.
Study of mapping class groups on infinite graphs, focusing on their large-scale geometry.
problem Understanding the large-scale geometry of mapping class groups on infinite graphs.
method Using coarse geometry techniques, classify coarsely bounded groups and compute asymptotic dimension.
result Identify conditions for global and local coarsely bounded pure mapping class groups of infinite rank graphs.
The paper extends graph complexity calculations for certain 3-manifolds up to 14.
problem Calculating the minimum 4-colored graph complexity of compact 3-manifolds.
method Exact calculations and two-sided bounds for graph complexity.
result Exact value of graph complexity computed for an infinite family of tetrahedral manifolds.
We introduce a representation of compact 3-manifolds without spherical boundary components via (regular) 4-colored graphs, which turns out to be very convenient for computer aided study and tabulation. Our construction is a direct generalization of the one given in the eighties by S. Lins for closed 3-manifolds, which …
Groups of homotopy equivalences of graphs help realize compact subgroups.
problem Realizing compact subgroups of homotopy equivalences of graphs.
method Introduced a Polish group topology on the group of proper homotopy equivalences and proved the Nielsen Realization theorem.
result Compact subgroups of homotopy equivalences can be realized by simplicial isomorphisms of graphs.
Compact clustering in latent space improves semi-supervised learning.
problem Improving semi-supervised learning with unlabeled data.
method Dynamic graph creation over embeddings, label propagation, and Markov chain regularization.
result Compact clustering in latent space facilitates better separation and separation of labeled and unlabeled data.
In this paper, we address the following question: What does a typical compact Riemann surface of large genus look like geometrically? We do so by constructing compact Riemann surfaces from oriented 3-regular graphs. The set for such Riemann surfaces is dense in the space of all compact Riemann surfaces, namely Belyi su…
The study provides a structure theorem for a new class of noncompact 3-manifolds.
problem Understanding a new class of noncompact 3-manifolds.
method Proved a structure theorem for irreducible open graph manifolds.
result A canonical 'reduced' decomposition of irreducible open graph manifolds along embedded, incompressible 2-tori.
The Shannon theorem is extended to locally compact groups.
problem Identifying the Poisson boundary of locally compact groups.
method Random walks and Shannon-McMillan-Breiman theorem.
result Generalized criteria for identifying Poisson boundaries.
Minimal graphs over non-compact domains in 3-manifolds solved with estimates and uniqueness results.
problem Solving minimal graphs over non-compact domains in 3-manifolds with a Killing vector field.
method Killing Submersion, Dirichlet problem, Collin-Krust estimates, uniqueness results, removable singularities.
result General Collin-Krust type estimates and uniqueness results for minimal Killing graphs.
We study the existence and uniqueness problem of compact minimal vertical graphs in H n × R \mathbb{H}^n\times\mathbb{R} H n × R , n ≥ 2 n\geq 2 n ≥ 2 , over bounded domains in the slice H n × { 0 } \mathbb{H}^n\times\{0\} H n × { 0 } , with non-connected boundary having a finite number of C 0 C^0 C 0 hypersufaces homeomorphic to the sphere S n − 1 \mathbb{S}^{n-1} S n − 1 , with prescri…
Disc graphs are uniformly quasiconvex in curve graphs of surfaces.
problem Characterizing quasiconvexity in curve graphs of surfaces.
method Proof using a universal constant K without train tracks.
result Disc graphs are K-quasiconvex in curve graphs.
Study shows consistency of shallow GCNNs on sampled point clouds under manifold assumption.
problem Consistency of shallow GCNNs on sampled point clouds under manifold assumption.
method Functional analysis perspective, weakly compact product of unit balls, Sobolev regularity, frequency cutoff.
result Proves Γ Γ Γ -convergence of regularized empirical risk minimization functionals and convergence of their global minimizers. In this paper, we show that a nontrivial compact graph manifold is nonpositively curved if and only if its fundamental group virtually embeds into a right-angled Artin group. As a consequence, nonpositively curved graph manifolds have linear fundamental groups.
Non-relatively hyperbolic separating curve graph for surfaces.
problem Classifying hyperbolicity of separating curve graphs.
method Proof of non-relatively hyperbolic property.
result Separating curve graph is not relatively hyperbolic for surfaces with genus ≥ 3 and one boundary component.
This paper is concerned with lower bounds for the connectivity of graphs (one-dimensional skeleta) of triangulations of compact manifolds. We introduce a structural invariant b_M for simplicial d-manifolds M taking values in the range 0 <= b_M <= d-1. The main result is that b_M influences connectivity in the following…
The paper proves removable singularity for nonlocal minimal graphs.
problem Proving removable singularities for nonlocal minimal graphs.
method Analyzing ( s , 1 ) (s, 1) ( s , 1 ) -capacity zero compact sets to ensure graphs are minimal in the entire domain. result Nonlocal minimal graphs are removable in the entire domain if they are minimal in a set of ( s , 1 ) (s, 1) ( s , 1 ) -capacity zero. We prove that the existence of a positively defined, invariant Einstein metric m m m on a connected homogeneous space G / H G/H G / H of a compact Lie group G G G is the consequence of non-contractibility of some compact set C = X G , H Σ C=X_{G,H}^Σ C = X G , H Σ (Böhm polyhedron) introduced by C.Böhm. There is a natural continuous map of C C C onto the flag …
Proposes a deep learning approach for attributed graph clustering.
problem Suboptimal performance in graph clustering due to two-step frameworks.
method Goal-directed deep learning approach using attention networks and inner product decoders.
result Superior performance compared to state-of-the-art algorithms.
Polynomial algorithm estimates mGH distance between unweighted graphs.
problem Efficiently measuring shape difference between unweighted graphs.
method Polynomial algorithm for mGH distance estimation.
result Algorithm finds mGH distances exactly on most scale-free graphs.
Study geodesic flow on graph-related nilmanifolds, finding integrable and non-integrable cases.
problem Understanding geodesic flow on specific geometric structures.
method Construction of first integrals to show complete integrability.
result Examples of integrable and non-integrable geodesic flows.
Generalizes Kauffman's clock theorem to surfaces.
problem Proving a lattice structure on graph states in various surfaces.
method Using matchings and graph orientations, extending Propp's results.
result Two generalizations of Kauffman's theorem for more surfaces.
GOTabPFN improves tabular model performance with compact tokenization for HDLSS data.
problem Making tabular models effective for high-dimensional, low-sample size data without retraining.
method Introducing Graph-guided Ordering with Local Refinement (GO-LR) and Neuro-Inspired Subunit Compression (NSC) to create compact meta-features.
result GOTabPFN improves stability and accuracy in tabular benchmarks with compact tokenization.
GLAD learns a compact model to recover sparse graphs from data.
problem Recovering sparse conditional independence graphs from data.
method GLAD uses an Alternating Minimization (AM) algorithm as a model inductive bias and learns parameters via supervised learning.
result GLAD learns a very compact and effective model for sparse graph recovery.
Researchers describe the Gromov boundary of a graph related to surfaces.
problem Understanding the Gromov boundary of a graph associated with surfaces.
method Described a dense subset of the Gromov boundary as geodesic laminations, proving the graph satisfies a bounded geodesic image theorem.
result The boundary is not compact.
We establish existence and uniqueness of compact graphs of constant mean curvature in MxR over bounded multiply connected domains of Mx{0} with boundary lying in two parallel horizontal slices of MxR
Maps from surfaces to handlebodies are projected uniformly.
problem Mapping homotopy classes from surfaces to handlebodies.
method Uniformly Lipschitz retraction of sphere graph to arc graph.
result Retraction is uniformly bounded and close to nearest point projection.
The paper studies the connectedness of a graph's boundary for surfaces.
problem Understanding the topology of the Gromov boundary of fine curve graphs for surfaces.
method Proved a bounded geodesic image theorem, used to show linear connectivity of the Gromov boundary.
result The Gromov boundary of fine curve graphs for surfaces is linearly connected.
New trick builds hyperbolic manifolds from compact ones, proving some don't virtually fiber.
problem Proving some hyperbolic manifolds don't virtually fiber.
method Hyperbolic reflection group trick, embedding theory, manifold topology.
result Constructed Gromov hyperbolic 7-manifolds that don't virtually fiber over a circle.
The paper extends spectral results to non-abelian groups acting on compact Riemannian manifolds.
problem Determining potential functions from spectral data for non-abelian group actions.
method Generalized Legendrian relations and spectral invariants.
result Potential functions are determined by the equivariant spectrum for certain Schrödinger operators.
We prove the connectedness and calculate the diameter of the oriented graph of graftings associated to exotic complex projective structures on a compact surface S with a given holonomy representation of Fuchsian type. The oriented graph of graftings is the graph whose vertices are the equivalence classes of marked CP^1…
Study GKM actions on special manifolds with interval orbit spaces.
problem Understanding GKM actions on specific types of manifolds.
method Analyzing group diagrams and orbit spaces; describing GKM graphs.
result Necessary and sufficient conditions for GKM actions on cohomogeneity one manifolds.
In this paper we extend to non-compact Riemannian manifolds with boundary the use of two important tools in the geometric analysis of compact spaces, namely, the weak maximum principle for subharmonic functions and the integration by parts. The first one is a new form of the classical Ahlfors maximum principle whereas …
Graph Laplace operators uniquely identify metrics and densities on manifolds.
problem Identifying Riemannian metrics and sampling densities from graph Laplace operators.
method Analyzing intrinsic and extrinsic graph Laplace operators on compact Riemannian manifolds.
result Graph Laplace operators uniquely determine metrics and densities under certain conditions.