Proves existence and regularity of energy-minimizing maps between ideal hyperbolic simplicial complexes.
problem Existence and regularity of energy-minimizing maps between ideal hyperbolic simplicial complexes.
method Proves existence and regularity results for energy minimizing maps between ideal hyperbolic 2-dimensional simplicial complexes.
result Establishes existence and regularity of energy-minimizing maps between ideal hyperbolic simplicial complexes.
Defines ideal simplicial volume for manifolds with boundary, showing it's bounded by classical volume.
problem Measuring the complexity of manifolds with boundary.
method Introduces ideal simplicial volume, compares it to classical volume, and computes specific examples.
result Ideal simplicial volume is bounded by classical volume and can be strictly smaller for certain manifolds.
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.
New proof shows extendable shellability for simple complexes.
problem Proving extendable shellability for specific simplicial complexes.
method Considering chordal graph structure and linear quotients.
result All d-dimensional complexes with d+3 vertices are extendably shellable. We study the ideal triangulation graph T(S) of a punctured surface S of finite type. We show that if S is not the sphere with at most three punctures or the torus with one puncture, then the natural map from the extended mapping class group of S into the simplicial automorphism group of T(S) is an isomorphism…
Researchers prove the arc complexes of decorated hyperbolic polygons are balls.
problem Understanding the structure of decorated hyperbolic polygons.
method Combinatorial approach using pseudo-manifolds and shellability.
result Arc complexes of decorated hyperbolic polygons are closed piecewise linear balls.
This paper studies deformations of hyperbolic surfaces with special structures.
problem Infinitesimal deformations of hyperbolic surfaces with boundary and ideal vertices.
method Description of the admissible cone of deformations in terms of the arc complex.
result Realization of the admissible cone and its faces as arc complexes for specific surface families.
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 M be a hyperbolic n-manifold whose cusps have torus cross-sections. In arXiv:0901.0056, the authors constructed a variety of nonpositively and negatively curved spaces as "2π-fillings" of M by replacing the cusps of M with compact "partial cones" of their boundaries. These 2π-fillings are closed pseudomanifolds, an…
By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to…
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.
We investigate the representation theory of the polynomial core of the quantum Teichmuller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S. Our main result is that irreducible finite-dimensional represent…
The paper explores continuous analogues of combinatorial structures.
problem Understanding continuous analogues of finite combinatorial structures.
method Systematic study of continuous analogues of posets, polytopes, and matroids.
result New insights and results in continuous combinatorics.
Abstract Szegedy walks on simplicial complexes are studied, revealing connections to combinatorial and geometric properties.
problem Investigating spectral structures of abstract Szegedy walks on simplicial complexes.
method Introduced modified Grover walks on simplicial complexes, focusing on orientations of simplices.
result Strong relationships between the spectrum of discriminants and combinatorial/geometry/topology properties of simplicial complexes.
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.
Defined Ricci curvature on simplicial complexes and proved bounds.
problem No specific problem stated; generalization of graph Ricci curvature to simplicial complexes.
method Modified Ricci curvature definition for simplicial complexes and proved bounds.
result Upper and lower bounds of Ricci curvature on simplicial complexes.
Paper calculates simplicial complexity of surface groups and proves stability under free product.
problem Stability of simplicial complexity under free product with free groups.
method Computing simplicial complexity for surface groups and proving stability.
result Stability of simplicial complexity under free product with free groups.
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.
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.
This paper studies covariant derivatives for Lie groupoids with representation-valued forms.
problem Understanding covariant derivatives for Lie groupoids with representation-valued forms.
method The paper explores two approaches: linear connections and multiplicative Ehresmann connections, both yielding geometrically richer curved double complexes.
result The horizontal exterior covariant derivative D is a key finding, generalizing the well-known operator from principal bundles. H. Masur and J. Smillie proved precisely which singularity index lists arise from pseudo-Anosov mapping classes. In search of an analogous theorem for outer automorphisms of free groups, Handel and Mosher ask: Is each connected, simplicial, (2r-1)-vertex graph the ideal Whitehead graph of a fully irreducible outer auto…
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.
A new clustering method for simplicial complexes using homology.
problem Clustering simplicial complexes efficiently and accurately.
method Inspired by graph spectral clustering, the method uses sparse eigenproblems.
result Produces clusters sensitive to simplicial complex homology.
We produce a one-parameter family of coordinates {Ψh}h∈R of the decorated Teichmüller space of an ideally triangulated punctured surface (S,T) with negative Euler characteristic, which is a deformation of Penner's simplicial coordinate \cite{P1}. If h⩾0, the decorated Teichmüller space in…
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 on embedding simplicial complexes in high dimensions with homological constraints.
problem Embedding simplicial complexes in Rd+1 with homological conditions. method Homological obstruction to embedding and deriving upper bounds on top-dimensional faces.
result Existence of an obstruction allowing upper bounds on top-dimensional faces.
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.
We show how to construct, for each r≥3, an ageometric, fully irreducible φ∈Out(Fr) whose ideal Whitehead graph is the complete graph on 2r−1 vertices. This paper is the second in a series of three where we show that precisely eighteen of the twenty-one connected, simplicial, five-vertex graphs are ideal …
Develops manifold calculus for simplicial complexes.
problem Approximating functors from simplicial complexes to topological spaces.
method Adapting manifold calculus to simplicial complexes and proving an approximation theorem.
result Functors can be approximated by polynomial functors under certain conditions.
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.
Algorithm approximates point clouds with simplicial complexes.
problem Approximating point clouds with simplicial complexes.
method Two-stage iterative fitting procedure that generalizes k-means clustering followed by simplex deletion.
result Dimension reduction of point clouds achieved.
Discrete version of Liouville's theorem for simplicial complexes.
problem Finding equivalent simplicial complexes under discrete conformal equivalence.
method Proving an analogous statement for simplicial complexes, considering combinatorial equivalence and scale factors associated with vertices.
result All discretely conformally equivalent simplicial complexes are combinatorially equivalent.
Geometrically solves differentiating simplicial manifolds.
problem Differentiating simplicial manifolds.
method Establishes a normal form theorem, identifies a differentiating ideal, proves quotient semi-freeness, interprets as Chevalley-Eilenberg algebra of higher Lie algebroid.
result Introduces higher van Est map and proves van Est isomorphism 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…
Estimates simplicial volume for complex hyperbolic surfaces.
problem Bounding the simplicial volume of complex hyperbolic surfaces.
method Estimates Gromov norm and uses top dimensional class in Hc4. result Explicit upper bound for simplicial volume.
The paper finds Chern-Simons forms for specific classes in simplicial de Rham complex.
problem None explicitly stated; focuses on finding forms.
method Exhibiting Chern-Simons forms of characteristic classes in simplicial de Rham complex.
result Chern-Simons forms for specific characteristic classes identified.
Study on expected topology of random subcomplexes in subdivided finite simplicial complexes.
problem Understanding the expected topology of random subcomplexes in subdivided finite simplicial complexes.
method Analysis of successive barycentric subdivisions and study of expected Betti numbers, average Morse inequalities, and Euler characteristic.
result Asymptotic upper and lower bounds for the expected Betti numbers of random subcomplexes.
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.
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.
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 construct {\it quantum hyperbolic invariants} (QHI) for triples (W,L,ρ), where W is a compact closed oriented 3-manifold, ρ is a flat principal bundle over W with structural group $PSL(2,\mc)$, and L is a non-empty link in W. These invariants are based on the Faddeev-Kashaev's {\it quantum dilogarithms},…
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 give parallel constructions of an invariant R(W,f), based on the classical Rogers dilogarithm, and of quantum hyperbolic invariants (QHI), based on the Faddeev-Kashaev quantum dilogarithms, for flat PSL(2,C)-bundles f over closed oriented 3-manifolds W. All these invariants are explicitely computed as a sum or state…