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.
This paper is devoted to a coordinate-free approach to several classic geometries such as hyperbolic (real, complex, quaternionic), elliptic (spherical, Fubini-Study), and lorentzian (de Sitter, anti de Sitter) ones. These geometries carry a certain simple structure that is in some sense stronger than the riemannian st…
Paper analyzes Hit-and-Run's convergence rates and applies similar methods to randomized Kaczmarz.
problem Quantifying advantages of Hit-and-Run's coordinate-free property.
method Sharp estimates via coupling methods and mixing time bounds.
result Ballistic and superdiffusive convergence rates in certain settings.
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.
New classification of complex hypersurfaces in 3D.
problem Classifying simply-transitive Levi non-degenerate hypersurfaces in C3. method Novel Lie algebraic approach, new coordinate-free formula for quartic tensor.
result Unique non-tubular model with geometric relations to planar equi-affine geometry.
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.
These lecture notes are based on [arXiv: math/0702714, 0907.4469, 0907.4470]. We introduce and study basic aspects of non-Euclidean geometries from a coordinate-free viewpoint.
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.
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. 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.
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.
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 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.
Solves differentiation for Lie ∞-groups using formal groupoids.
problem Differentiation of Lie ∞-groups.
method Develops homotopy theory of formal ∞-groupoids and analyzes Dold-Kan adjunction for cosimplicial algebras.
result Differentiation functor from finite-dimensional Lie ∞-groups to finite-type Lie ∞-algebras is homotopically well-behaved.
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.
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.
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.
Solved Cheeger inequalities for simplicial complexes, combining topological and graph theoretic methods.
problem Extend Cheeger inequalities to simplicial complexes and their higher order Laplacians.
method Combining constructions from simplicial topology, signed graphs, Gromov filling radii, and interpolating between 1-Laplacians and 2-Laplacians.
result Developed a general theory for p-Laplacians on simplicial complexes and proved Cheeger-type inequalities.
Finite simplicial complexes dominate certain manifolds with a bounded number of simplices.
problem Understanding the finite domination of manifolds by simplicial complexes.
method Proving that a manifold can be dominated by the n-skeleton of a finite simplicial complex with a bounded number of simplices. result The total number of simplices in the n-skeleton is bounded above by a constant depending only on n and the embolic volume of the manifold. Constructs a simplicial cell decomposition of complex projective space for n ≥ 2.
problem Finding a simplicial cell decomposition for complex projective space.
method Starting with a standard crystallisation of the 2-sphere, constructing a simplicial subdivision, and quotienting by the Sym(n) action.
result Explicit construction of a simplicial cell decomposition of complex projective space for n ≥ 2.
We give a necessary and sufficient condition for a non-degenerate symmetric 3-differential with nonzero Blaschke curvature on a complex surface to be locally representable as a product of three closed holomorphic 1-forms. We give two versions of this condition corresponding to different choices of coordinates, one of w…
Integral filling volume of mapping tori grows sublinearly with complexity.
problem Characterizing mapping classes with vanishing integral filling volume.
method Analyzing Dehn twists and mapping tori, using simplicial volume and complexity.
result Integral simplicial volume of mapping tori grows sublinearly with respect to the monodromy power.
Paper introduces Simplet Frequency Distribution (SFD) for SCs.
problem Frequency analysis of simplets in large SCs.
method Developed SFD vector and uniform sampling-based algorithm.
result Validated theoretical bounds with experiments.
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…
Let G be a higher-rank semisimple Lie group over a nonarchimedean local field, for example G=PGL(n,QP). To any lattice L in G there is an associated simplicial complex BL, given by the quotient by L of the Bruhat-Tits building associated to G. In this paper prove that the simplicial structure $B_L…
In this paper, we investigate a relation between finite graphs, simplicial flag complexes and right-angled Coxeter groups, and we provide a class of reconstructible finite graphs. We show that if Γ is a finite graph which is the 1-skeleton of some simplicial flag complex L which is a homology manifold of dimension …
We outline a novel clustering scheme for simplicial complexes that produces clusters of simplices in a way that is sensitive to the homology of the complex. The method is inspired by, and can be seen as a higher-dimensional version of, graph spectral clustering. The algorithm involves only sparse eigenproblems, and is …
We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse of finite spaces induces a simplicial …
We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of usi…
Graph conditions ensure matching arc complexes are connected and hyperbolic.
problem Conditions for connectedness and hyperbolicity of matching arc complexes.
method Conditions on finite simplicial graphs guaranteeing connectedness and hyperbolicity of matching arc complexes.
result Conditions on finite simplicial graphs ensure connectedness and hyperbolicity of matching arc complexes.
There is a canonical way to associate two simplicial complexes K, L to any relation R⊂X×Y. Moreover, the geometric realizations of K and L are homotopy equivalent. This was studied in the fifties by C.H. Dowker. In this article we prove a Galois-type correspondence for relations R⊂X×Y when…
New Markov chains defined on simplicial complexes for understanding their topology.
problem Understanding the topology of simplicial complexes and hypergraphs.
method Defining new Markov chains on simplicial complexes and studying their properties.
result The generator of the new Markov chain is the upper Laplacian, and the Markov chain is positive recurrent.
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.
Proposes SGM for modeling complex dependencies in high-dimensional systems.
problem Limited pairwise interactions in PGMs for high-dimensional systems.
method Simplicial Gaussian model (SGM) using discrete Hodge theory and independent random components.
result Maximum-likelihood inference algorithm for parameter recovery and conditional dependence structure.