arXiv research

Study simplicial volume in fiber bundles with connected groups.

A notion of up and down Grover walks on simplicial complexes are proposed and their properties are investigated. These are abstract Szegedy walks, which is a special kind of unitary operators on a Hilbert space. The operators introduced in the present paper are usual Grover walks on graphs defined by using combinatoria…

Positive simplicial volume implies locally symmetric space structure.

Local Kan conditions enable differentiation of simplicial manifolds.

The study explores discrete versions of Riemannian geometry structures on manifolds.

Let $G$ be a higher-rank semisimple Lie group over a nonarchimedean local field, for example $G={\rm PGL}(n,Q_P)$. To any lattice $L$ in $G$ there is an associated simplicial complex $B_L$, given by the quotient by $L$ of the Bruhat-Tits building associated to $G$. In this paper prove that the simplicial structure $B_L…

Paper studies spaces of flattenings of simplicial spheres and their homotopy type.

Stokes-Dirac structures are infinite-dimensional Dirac structures defined in terms of differential forms on a smooth manifold with boundary. These Dirac structures lay down a geometric framework for the formulation of Hamiltonian systems with a nonzero boundary energy flow. Simplicial triangulation of the underlaying m…

We analyze oversquashing in topological message-passing using relational structures.

Hypernetworks are simplified simplicial complexes with curvature.

We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to any crossed module there is a simplicial group NC, the nerve of the 1-category defined by the crossed module and its geometric realization |NC|. Equivalence classes of principal bundles with structure group |NC| …

Geometric arguments show simplicial arrangements with few double points can't have an irreducible cubic curve dual.

Graph manifolds are manifolds that decompose along tori into pieces with a tame $S^1$-structure. In this paper, we prove that the simplicial volume of graph manifolds (which is known to be zero) can be approximated by integral simplicial volumes of their finite coverings. This gives a uniform proof of the vanishing of …

Let g be a simplicial Lie algebra with Moore complex Ng of length k. Let G be the simplicial Lie group integrating g, which is simply connected in each simplicial level. We use the 1-jet of the classifying space of G to construct, starting from g, a Lie k-algebra L. The so constructed Lie k-algebra L is actually a diff…

We show that closed aspherical manifolds supporting an affine structure, whose holonomy map is injective and contains a pure translation, must have vanishing simplicial volume. This provides some further evidence for the veracity of the Auslander Conjecture. Along the way, we provide a simple cohomological criterion fo…

BScNets expands graph learning to higher-order interactions.

Topo-MLP learns network representations without message passing.

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study …

Defines and classifies Thurston geometries and connects simplicial volume to Kodaira dimension.

Simplicial persistence measures financial market dynamics, revealing long-term structure evolution.

Tutorials on signal processing on higher-order networks like simplicial complexes and hypergraphs.

Homotopy equivalent boundaries of cube complexes are studied.

In a finite-dimensional real vector space furnished with a rational structure with respect to a subfield of the field of real numbers, every (simplicial) rational semifan is contained in a complete (simplicial) rational semifan. In this paper this result is proved constructively on use of techniques from polyhedral geo…

The Reeb space of a smooth map whose codimension is minus is the space defined as the space of all connected components of inverse images. For generic maps such as Morse functions and their higher dimensional versions, they are polyhedra whose dimensions are equal to those of the target manifolds and which have simplic…

Minimal simplicial maps constructed for spheres and manifolds.

New Markov chains defined on simplicial complexes for understanding their topology.

Simplicial complexes are increasingly used to study complex system structure and dynamics including diffusion, synchronization and epidemic spreading. The spectral dimension of the graph Laplacian is known to determine the diffusion properties at long time scales. Using the renormalization group here we calculate the s…

A discrete (finite-difference) analogue of differential forms is considered, defined on simplicial complexes, including triangulations of continuous manifolds. Various operations are explicitly defined on these forms, including exterior derivative and exterior product. The latter one is non-associative. Instead, as ant…

Paper introduces Simplet Frequency Distribution (SFD) for SCs.

Proposes SGM for modeling complex dependencies in high-dimensional systems.

Smooth structures on diffeological spaces and sheaves, resolving conjectures.

Defines a simplicial operad related to Fulton-MacPherson.

The moduli space of convex projective structures on a simplicial hyperbolic Coxeter orbifold is either a point or the real line. Answering a question of M. Crampon, we prove that in the latter case, when one goes to infinity in the moduli space, the entropy of the Hilbert metric tends to 0.

This paper explores Brunnian twin groups and their properties.

Unified framework for observables in n-plectic geometry.

TopoNTK kernel captures higher-order interactions in simplicial complexes.

It is proved that the category of simplicial complete bornological spaces over $\mathbb R$ carries a combinatorial monoidal model structure satisfying the monoid axiom. For any commutative monoid in this category the category of modules is also a monoidal model category with all cofibrant objects being flat. In particu…

Shellable tilings on simplicial complexes help understand their structure.

The paper constructs simplicial maps of any degree on spheres, solving a long-standing problem.

Proposes a probabilistic framework for stationary topological signals on simplicial complexes.

New framework uses simplicial and categorical methods to detect market inconsistencies.

Study of $p$-adic simplicial volumes and their properties.

The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in 1982. In order to study its main properties, Gromov himself initiated the dual theory of bounded cohomology, that developed into an active and independent research field. Gromov's theory of bounded cohomology was based on the use of mult…

The moment-angle complex Z_K is cell complex with a torus action constructed from a finite simplicial complex K. When this construction is applied to a triangulated sphere K or, in particular, to the boundary of a simplicial polytope, the result is a manifold. Moment-angle manifolds and complexes are central objects in…

Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical m…

New Lipschitz de Rham theorem for $L_p$-cohomology.

We define the ideal simplicial volume for compact manifolds with boundary. Roughly speaking, the ideal simplicial volume of a manifold $M$ measures the minimal size of possibly ideal triangulations of $M$ "with real coefficients", thus providing a variation of the ordinary simplicial volume defined by Gromov in 1982, t…

We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space. We show that integral foliated s…