Study of harmonic maps on 2D simplicial complexes, proving existence and regularity.
problem Existence and regularity of harmonic maps between 2D simplicial complexes.
method Extending previous work, study metrics conformal to flat or ideal hyperbolic, proving existence, uniqueness, and regularity of harmonic maps.
result Existence, uniqueness, and regularity results for harmonic maps between 2D simplicial complexes.
Constructs simplified or complexified simplicial complexes.
problem Efficiently simplifying or complexifying complex spaces.
method Embeddings of simplicial complexes into a simplicial ball with bounded degrees and low volume.
result Realizes complicated spaces as parts of a ball/sphere or gives spheres specific metrics.
Link condition for simplicial complexes to be CUB spaces.
problem Understanding when a simplicial complex is a CUB space.
method Establishing a link condition based on local lattice properties.
result The link condition generalizes Gromov's link condition for cube complexes.
The complex of domains D(S) is a geometric tool with a very rich simplicial structure, it contains the curve complex C(S) as a simplicial subcomplex. In this paper we shall regard it as a metric space, endowed with the metric which makes each simplex Euclidean with edges of length 1, and we shall discuss its coarse…
Study de Rham homomorphism for Lipschitz cohomologies on metric simplicial complexes.
problem Triviality of de Rham homomorphism kernel and non-increasing monotonicity of parameters.
method Regularization in Lipschitz de Rham calculus on metric simplicial complexes with bounded geometry.
result Explicit specification of non-trivial cohomology classes for a sequence of parameters.
Homotopy equivalent boundaries of cube complexes are studied.
problem The equivalence of different boundaries of cube complexes.
method Using a partial order on a quotient of the Roller boundary, we obtain the simplicial Roller boundary and show homotopy equivalence among the Tits, simplicial, and simplicial Roller boundaries.
result The Tits, simplicial, and simplicial Roller boundaries are homotopy equivalent.
Rust library solves complex equations on abstract simplicial complexes.
problem Solving partial differential equations on abstract simplicial complexes.
method Finite Element Exterior Calculus, intrinsic Riemannian metric, first-order Whitney basis functions.
result Verification through convergence studies on elliptic Hodge-Laplace eigenvalue and source problems.
Homotopy types of Vietoris-Rips metric thickenings of the circle confirmed.
problem Understanding the homotopy types of Vietoris-Rips metric thickenings of the circle.
method Finding quotients of the metric thickenings that preserve homotopy type and showing that the quotient spaces can be described as CW complexes.
result The Vietoris-Rips metric thickenings of the circle are homotopy equivalent to odd-dimensional spheres at the expected scale parameters.
New Lipschitz de Rham theorem for Lp-cohomology.
problem Developing a new de Rham theorem for Lp-cohomology. method Regularization procedure in Lipschitz de Rham calculus applied to metric simplicial complexes.
result Established Lipschitz de Rham theorem for Lp-cohomology. Given a sample Y from an unknown manifold X embedded in Euclidean space, it is possible to recover the homology groups of X by building a Vietoris--Rips or Čech simplicial complex on top of the vertex set Y. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular…
Paper proves Gromov's conjecture on manifolds with certain group properties.
problem Gromov's conjecture on positive scalar curvature and simplicial volume.
method Proves conjecture under a fundamental group decay property.
result Proves Gromov's conjecture for manifolds with a weakened rapid decay property.
Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one of them can be transformed to the other by contiguous factorizations of bonding…
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.
Alexander's conjecture extended to infinite simplicial complexes.
problem Alexander's conjecture for infinite simplicial complexes.
method Generalization of recent result for finite simplicial complexes.
result Alexander's conjecture holds for infinite simplicial complexes.
Let Out(Fn) be the outer automorphism group of the free group Fn. It acts properly on the outer space Xn of marked metric graphs, which is a finite-dimensional infinite simplicial complex with some simplicial faces missing. In this paper, we construct complete geodesic metrics and complete piecewise s…
Locally finite complexes with polyhedral CAT(0) metrics are arborescent.
problem Characterizing locally finite complexes with CAT(0) metrics. method Proving arborescence for complexes with polyhedral CAT(0) metrics. result Locally finite complexes with polyhedral CAT(0) metrics are arborescent. The study examines conditions for minimal volume entropy of simplicial complexes.
problem Conditions for minimal volume entropy of simplicial complexes.
method Topological conditions and growth of fundamental groups.
result Examples of simplicial complexes with zero simplicial volume and large minimal volume entropy.
Mixes higher-order simplicial complexes for data augmentation.
problem Lack of labeled data for complex systems with multiway interactions.
method Proposes mixup mechanisms for simplicial complexes, including linear and nonlinear mixup, and a convex clustering mixup.
result Synthetic simplicial complexes interpolate between existing data based on homomorphism densities.
An additional minimal simplicial n-complex contains a non-splittable link in R^(2n).
problem Constructing minimal simplicial n-complexes with a specific link property.
method Presenting a new minimal simplicial n-complex.
result An additional simplicial n-complex with the same link property.
The paper shows how to use fine shape to understand infinite-dimensional spaces.
problem Understanding infinite-dimensional metrizable spaces and their homology theories.
method Obtained results indicating fine shape is tractable and can be used for Polish spaces.
result Every Polish space is fine shape equivalent to the limit of an inverse sequence of simplicial maps.
An invariant of three-dimensional orientable manifolds is built on the base of a solution of pentagon equation expressed in terms of metric characteristics of Euclidean tetrahedra.
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…
We study the multiscale simplicial flat norm (MSFN) problem, which computes flat norm at various scales of sets defined as oriented subcomplexes of finite simplicial complexes in arbitrary dimensions. We show that the multiscale simplicial flat norm is NP-complete when homology is defined over integers. We cast the mul…
Minimal volume entropy vanishes or is positive under certain fiber growth conditions.
problem Conditions for minimal volume entropy to be zero or positive.
method Analyzes topological conditions related to fiber growth of maps.
result Examples of finite simplicial complexes with zero simplicial volume and large minimal volume entropy.
Hypernetworks are simplified simplicial complexes with curvature.
problem Representing hypernetworks geometrically for analysis.
method Hypernetworks are interpreted as posets, which are simplicial complexes with Forman Ricci curvature.
result Hypernetworks have intrinsic curvature that correlates with their Euler characteristic.
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. We determine the moduli space of metric 2-step nilpotent Lie algebras of dimension up to 6. This space is homeomorphic to a cone over a 4-dimensional contractible simplicial complex.
The simplicial complexity is an invariant for finitely presentable groups that was recently introduced by Babenko, Balacheff and Bulteau to study systolic area. The simplicial complexity κ(G) was proved to be a good approximation of the systolic area σ(G) for large values of κ(G). In this paper we compute the sim…
New simplicial complexes show unavoidable link of spheres in high dimensions.
problem Finding unavoidable links of spheres in high-dimensional spaces.
method Simple argument in piecewise linear topology and application of the van Kampen--Flores theorem.
result Existence of additional simplicial complexes with unavoidable links of spheres.
The simplicial volume of non-R^3 contractible 3-manifolds is infinite.
problem Characterizing contractible 3-manifolds based on their simplicial volume.
method Analyzing the simplicial volume of contractible 3-manifolds and open 3-manifolds.
result The Euclidean space is the unique contractible 3-manifold with vanishing minimal volume.
Characterizes simplicial complexes embedding into spheres with few vertices.
problem Characterizing simplicial complexes that embed into spheres with few vertices.
method Simple characterization using non-face families and analogy with Fáry's theorem.
result Recovery of van Kampen--Flores theorem and Erd\H os--Ko--Rado theorem.
We present criteria for establishing a triangulation of a manifold. Given a manifold M, a simplicial complex A, and a map H from the underlying space of A to M, our criteria are presented in local coordinate charts for M, and ensure that H is a homeomorphism. These criteria do not require a differentiable structure, or…
The scale and complexity of modern data sets and the limitations associated with testing large numbers of hypotheses underline the need for feature selection methods. Spectral techniques rank features according to their degree of consistency with an underlying metric structure, but their current graph-based formulation…
We consider closed simplicial and cubical n-complexes in terms of link of their (n−2)-faces. Especially, we consider the case, when this link has size 3 or 4, i.e., every (n−2)-face is contained in 3 or 4 n-faces. Such simplicial complexes with {\em short} (i.e. of length 3 or 4) links are completely classified…
Study the boundary operator property on simplicial complexes, proving essential properties for Hodge theory.
problem Characterize the boundary operator property ∂∂=0 on simplicial complexes. method Characterization in ℓ2 terms of recurrence of links, defining relative cohomology, and proving harmonic eigenforms. result Essential properties for Hodge theory, including weak decomposition and existence of harmonic eigenforms.
Extends circle pattern theorem to quasi-simplicial triangulations.
problem Characterize circle patterns on quasi-simplicial triangulated surfaces.
method Use finite covering technique to reduce problem to simplicial case, prove characterization by KAT inequalities.
result Curvature image is characterized by KAT inequalities.
New f-vectors reveal geometric Lefschetz-like decompositions of flag spheres.
problem Understanding f-vectors of balanced simplicial complexes and flag spheres. method Analyzing h-vectors and f-vectors of flag spheres and balanced simplicial complexes. result Found f-vectors leading to geometric Lefschetz-like decompositions. We introduce new simplicial complexes by using various invariants and local moves for knots, which give generalizations of the Gordian complex defined by Hirasawa and Uchida. In particular, we focus on the simplicial complex defined by using the Alexander-Conway polynomial and the Delta-move, and show that the simplici…
The study explores discrete versions of Riemannian geometry structures on manifolds.
problem Understanding the relationship between discrete structures and continuous Riemannian geometry.
method Surveying and analyzing discrete counterparts of Riemannian geometry concepts on graphs and simplicial complexes.
result Recent developments include Cheeger type inequalities for higher-dimensional simplicial complexes and Floer type constructions.
We introduce a combinatorial energy for maps of triangulated surfaces with simplicial metrics and analyze the existence and uniqueness properties of the corresponding harmonic maps. We show that some important applications of smooth harmonic maps can be obtained in this setting.
This article generalizes the work of Ballmann and Światkowski to the case of Reflexive Banach spaces and uniformly convex Busemann spaces, thus giving a new fixed point criterion for groups acting on simplicial complexes.
Let ρ:(D2)m→Im be the orbit map for the diagonal action of the torus Tm on the unit poly-disk (D2)m, Im=[0,1]m is the unit cube. Let C be a cubical subcomplex in Im. The moment-angle complex $\ma(C)$ is a Tm-invariant bigraded cellular decomposition of the subset ρ−1(C)⊂(D2)m wit…
We define the Ricci curvature on simplicial complexes by modifying the definition of the Ricci curvature on graphs, and we prove the upper and lower bounds of the Ricci curvature. These properties are generalizations of previous studies. Moreover, we obtain an estimate of the eigenvalues of the Laplacian on simplicial …
Geometrically interprets a duality theorem linking cochain and chain complexes.
problem Understanding a complex duality theorem in geometric terms.
method Introduces a chain isomorphism involving simplicial and cellular complexes.
result Establishes a geometric interpretation of Ranicki duality.
V. V. Fedorchuk has recently introduced dimension functions K-dim \leq K-Ind and L-dim \leq L-Ind, where K is a simplicial complex and L is a compact metric ANR. For each complex K with a non-contractible join |K| * |K| (we write |K| for the geometric realisation of K), he has constructed first countable, separable com…
Alternative proof of simplicial volume bound using area-minimizing sets.
problem Bounding simplicial volume of manifolds with restricted ball volumes.
method Area-minimizing separating sets method.
result Explicit constants for simplicial volume bound derived.
The study bounds distances in simplicial complexes and defines new invariants for 3-manifolds and handlebody-knots.
problem Estimating distances in simplicial complexes associated with low-dimensional manifolds.
method Obtained bounds on distances in simplicial complexes using topological conditions on vertices and curve complexes. Defined new invariants for 3-manifolds and handlebody-knots using splitting distances.
result Splitting distances in simplicial complexes are bounded from below under stabilizations, leading to converging invariants.
We construct a metric simplicial complex which is an almost isometric model of the moduli space M(S) of Riemann surfaces. We then use this model to compute the "tangent cone at infinity" of M(S): it is the topological cone on the quotient of the complex of curves C(S) by the mapping class group of S, endowed with an ex…