arXiv research

Proves Singer conjecture for graph manifolds with residually finite groups.

In this short note we prove the Borel conjecture for a family of aspherical manifolds that includes higher graph manifolds.

We determine the asymptotic behavior of the higher dimensional Reidemeister torsion for the graph manifolds obtained by exceptional surgeries along twist knots. We show that all irreducible SL(2;C)-representations of the graph manifold are induced by irreducible metabelian representations of the twist knot group. We al…

In this paper, we first introduce higher order Dirichlet-to-Neumann maps on graphs which can be viewed as a discrete analogue of the corresponding Dirichlet-to-Neumann maps on compact Riemannian manifolds with boundary and a higher order generalization of the Dirichlet-to-Neumann map on graphs introduced by Hua-Huang-W…

In this short note we introduce higher graph manifolds and use a version of the barycenter technique to characterize when they undergo volume collapse. In the case when the pure pieces are hyperbolic, we compute the exact value of the minimal volume. We verify the coarse Baum--Connes conjecture for these manifolds and …

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…

The study shows how to embed cusp-decomposable manifolds quasi-isometrically.

We show that after one stabilization, a strongly irreducible Heegaard splitting of suitably large genus of a graph manifold is isotopic to an amalgamation along a modified version of the system of canonical tori in the JSJ decomposition. As a corollary, two strongly irreducible Heegaard splittings of a graph manifold o…

The study constructs geometrically decomposable aspherical 4-manifolds with non-zero signature and explores their properties.

Extends graph theory to hypergraphs with manifold-valued nodes.

Paper rigorously defines Feynman graph integrals on Kähler manifolds.

Study of mean curvature flows on graphs in warped product manifolds, focusing on behavior at infinity.

The paper bounds higher Steklov eigenvalues of graphs on surfaces.

The paper presents a new method to represent directed graphs using pseudo-Riemannian manifolds.

The paper characterizes graph manifolds using fold maps and embeddability of polyhedra.

This paper reconstructs complex graph signals using kernel methods on manifolds.

We study the strong maximum principle for horizontal (p-) mean curvature operator and p-(sub)laplacian operator on subriemannian manifolds including, in particular, Heisenberg groups and Heisenberg cylinders. Under a certain Hormander type condition on vector fields, we show the strong maximum principle holds in higher…

We give an explicit formula for the Gauss-Bonnet-Chern mass of an asymptotically flat graphical manifold of arbitrary codimension and use it to prove the positive mass theorem and the Penrose inequality for graphs with flat normal bundle.

Study compares hypergraph and graph-level models for higher-order relational learning.

A new method predicts higher-order interactions in evolving graphs using simplicial complexes.

In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle $TM\oplus\wedge^nT^*M$ for an $m$-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an $(n+1)$-vector fi…

The paper creates non-isotopic but homotopic diffeomorphisms in 4-manifolds.

Study Hodge Laplacians for manifold data, improving error bounds.

Combines neural networks and probabilistic graphical models for efficient higher-order inference.

Enhanced spectral clustering for geometric graphs improves clustering accuracy.

Neural network learns from higher-order connections in molecules.

This paper presents the first use of graph neural networks (GNNs) for higher-order proof search and demonstrates that GNNs can improve upon state-of-the-art results in this domain. Interactive, higher-order theorem provers allow for the formalization of most mathematical theories and have been shown to pose a significa…

We introduce nonlinear higher-order label spreading for semi-supervised learning.

In recent years, graph neural networks (GNNs) have emerged as a powerful neural architecture to learn vector representations of nodes and graphs in a supervised, end-to-end fashion. Up to now, GNNs have only been evaluated empirically -- showing promising results. The following work investigates GNNs from a theoretical…

SpeqNets improve graph neural networks by scaling and adapting to graph sparsity.

The study proves conditions for hyperbolic isometries on fine curve graphs of higher genus surfaces.

We give a generalization of Fukaya's Morse homotopy theoretic approach for 2-loop Chern--Simons perturbation theory to 3-valent graphs with arbitrary number of loops at least 2. We construct a sequence of invariants of integral homology 3-spheres with values in a space of 3-valent graphs (Jacobi diagrams or Feynman dia…

We define two new families of invariants for (3-manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and (-1/2)-additive under trivalent vertex sum of pairs. The first of these families is closely related to both bridge number and tunnel number. The second of these families is a …

Investigates conditions for GKM fiber bundles and realizability of fiber bundles of GKM graphs.

Derives formulae for general permutation equivariant layers and presents a second order graph variational encoder.

The pants graph has proved to be influential in understanding 3-manifolds concretely. This stems from a quasi-isometry between the pants graph and the Teichmüller space with the Weil-Petersson metric. Currently, all estimates on the quasi-isometry constants are dependent on the surface in an undiscovered way. This pape…

This work proposes a geometric approach to equivariant message passing on Riemannian manifolds.

A simplified proof for embedding higher-dimensional complexes into manifolds.

New theory defines invariants for framed 3-manifolds using trivalent graphs with self-loops.

H-GAT improves stock selection by capturing complex higher-order stock relations and integrating both technical and fundamental analysis.

With higher-order neighborhood information of graph network, the accuracy of graph representation learning classification can be significantly improved. However, the current higher order graph convolutional network has a large number of parameters and high computational complexity. Therefore, we propose a Hybrid Lower …

Data samples collected for training machine learning models are typically assumed to be independent and identically distributed (iid). Recent research has demonstrated that this assumption can be problematic as it simplifies the manifold of structured data. This has motivated different research areas such as data poiso…

Generalized Fáry's theorem to higher dimensions.

The covering spectrum is a geometric invariant of a Riemannian manifold, more generally of a metric space, that measures the size of its one-dimensional holes by isolating a portion of the length spectrum. In a previous paper we demonstrated that the covering spectrum is not a spectral invariant of a manifold in dimens…

Novel TRI-GNN framework improves graph classification robustness.

Proposes a 2-WL-based graph convolution for improved graph classification.

Research shows finiteness in triangulations with girth constraints.

Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about v^v such manifolds of volume at most v, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, alm…