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.
Simplicial spheres have a weak Lefschetz property in characteristic 2.
problem Proving the weak Lefschetz property for simplicial spheres.
method Using bistellar moves to show the property is preserved.
result The weak Lefschetz property is preserved by bistellar moves for PL-spheres.
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.
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.
Study of p-adic simplicial volumes and their properties.
problem Understanding simplicial volumes over p-adic seminormed rings. method Definition and study of p-adic simplicial volumes, homology bounds, and computation of volumes for surfaces. result Established homology bounds and computed p-adic simplicial volumes for surfaces. 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.
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.
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.
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.
Affine manifolds with specific properties have zero simplicial volume.
problem Understanding the simplicial volume of affine manifolds.
method Analyzing the holonomy map and using cohomological criteria.
result Affine manifolds with injective holonomy containing a pure translation have vanishing simplicial volume.
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…
Innovates volume entropy semi-norm, proving equivalence to simplicial volume.
problem Equivalence of volume entropy and simplicial volume in real homology.
method Introduces volume entropy semi-norm and proves its equivalence to simplicial volume.
result Equivalence of volume entropy semi-norm and simplicial volume in every dimension.
Sphere-bases for simplicial and cubical complexes are constructed and analyzed.
problem Constructing and analyzing geometric properties of sphere-bases for simplicial and cubical complexes.
method Algorithmically-specified family of k+1-simplices or k+1-cubes are used to form the boundaries of sphere-bases.
result Geometric properties of constructed sphere-bases are investigated.
Shellable tilings on simplicial complexes help understand their structure.
problem Understanding the structure of simplicial complexes through tilings.
method Proving the existence of shellable h-tilings on finite simplicial complexes after stellar subdivisions.
result The h-vector of a tiling is determined by the critical vector, with palindromic properties for closed triangulated manifolds.
Study essentiality and simplicial volume of manifolds fibered over spheres.
problem When manifolds fibered over spheres are essential or have positive simplicial volume.
method Analyzing mapping tori and fiber bundles over spheres, using results on macroscopic dimension and characteristic classes.
result Mapping tori of odd-dimensional manifolds with non-zero simplicial volume are essential, while fiber bundles over spheres of dimension d > 1 have zero simplicial 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 discuss some additivity properties of the simplicial volume for manifolds with boundary: we give proofs of additivity for glueing amenable boundary components and of superadditivity for glueing amenable submanifolds of the boundary, and we discuss doubling of 3-manifolds.
We introduce the stable presentation length of a finitely presented group. The stable presentation length of the fundamental group of a 3-manifold can be considered as an analogue of the simplicial volume. We show that the stable presentation length have some additive properties like the simplicial volume, and the simp…
In the spirit of topological entropy we introduce new complexity functions for general dynamical systems (namely groups and semigroups acting on closed manifolds) but with an emphasis on the dynamics induced on simplicial complexes. For expansive systems remarkable properties are observed. Known examples are revisited …
simpcomp is an extension (a so called package) to GAP, the well known system for computational discrete algebra. The package enables the user to compute numerous properties of (abstract) simplicial complexes, provides functions to construct new complexes from existing ones and an extensive library of triangulations of …
New inequalities link surface bundle properties.
problem Understanding surface bundles over surfaces.
method Developed three inequalities involving signature, simplicial volume, and Euler characteristic.
result Three inequalities linking surface bundle properties.
The paper studies connectivity properties of Morse complexes as simplicial complexes grow.
problem Understanding connectivity of Morse complexes as simplicial complexes evolve.
method Bestvina-Brady Morse theory applied to a generalized Morse complex.
result Proves M(Δ) becomes arbitrarily highly connected as Δ grows. Minimal simplicial maps constructed for spheres and manifolds.
problem Constructing minimal simplicial maps of specific degrees.
method Triangulations and degree constructions for manifolds and spheres.
result Minimal triangulations for degree d self-maps of Sn−1imesS1. Study on simplicial volume and Euler characteristic of aspherical manifolds.
problem Whether vanishing simplicial volume implies vanishing Euler characteristic.
method Various strategies for both affirmative and negative answers, context with other problems, and comparative analysis of additivity properties.
result Found counterexamples among aspherical spaces that are homology equivalent to manifolds but not manifolds themselves.
Maps preserve edges between curve graphs of surfaces with certain properties.
problem Proving homeomorphism between surfaces based on curve graphs.
method Using simplicial properties of rigid expansions.
result Edge-preserving maps imply homeomorphic surfaces.
Discrete exterior calculus shows natural properties of wedge product and averaging.
problem Naturalness of discrete exterior calculus operations.
method Showed naturalness of discrete wedge product and averaging interpretation.
result Discrete wedge product is natural and equals Wilson's cochain product.
The study explores convex unions and completions in simplicial pseudomanifolds, revealing unexpected behavior.
problem Understanding the behavior of convex unions in simplicial pseudomanifolds.
method Generalization to simplicial pseudomanifolds, considering PL homeomorphisms and edge subdivisions.
result Unexpected behavior in convex unions and completions, including empty contraction spaces and large/small contraction spaces.
Proposes a probabilistic framework for stationary topological signals on simplicial complexes.
problem Complex data structures require new models and tools.
method Generalizes stationarity to topological signals on simplicial complexes.
result Defines topological power spectral density (PSD) for stationary signals.
The 4-dimensional abstract Kummer variety K^4 with 16 nodes leads to the K3 surface by resolving the 16 singularities. Here we present a simplicial realization of this minimal resolution. Starting with a minimal 16-vertex triangulation of K^4 we resolve its 16 isolated singularities - step by step - by simplicial blowu…
We show that the de Rham theorem, interpreted as the isomorphism between distributional de Rham cohomology and simplicial homology in the dual dimension for a simplicial decomposition of a compact oriented manifold, is a straightforward consequence of elementary properties of currents. The explicit construction of this…
Researchers calculate spectral dimension of complex networks using renormalization group theory.
problem Understanding diffusion properties in complex systems.
method Renormalization group theory applied to graph Laplacians of simplicial complexes.
result Spectral dimension decreases with randomness in topological structure.
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. 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.
Gromov's theory of multicomplexes aids in bounded cohomology and simplicial volume studies.
problem Homotopy invariants of manifolds and their properties.
method Construction and study of multicomplexes, homotopy theory, completeness condition.
result Complete proofs of Gromov's Mapping and Vanishing Theorems.
Reviews g-theorem and hard Lefschetz theorem for face rings.
problem Understanding and proving the g-theorem and hard Lefschetz theorem for face rings.
method Perturbations of maps, biased Poincaré pairings, cobordism argument, edge-contractions.
result Alternative and alternative arguments for the Lefschetz property.
Tutorials on signal processing on higher-order networks like simplicial complexes and hypergraphs.
problem Processing complex data structures with polyadic relationships.
method Introduction to simplicial complexes and hypergraphs, Fourier analysis, signal denoising, interpolation, embeddings, neural networks.
result Multi-relational operators like the Hodge Laplacian for simplicial complexes and tensor representations for hypergraphs.
New simplicial complex for infinite-type surfaces shows graph properties.
problem Characterizing infinite-type surfaces using graph theory.
method Constructing grand arc graph and analyzing its properties.
result Grand arc graph is infinite-diameter and δ-hyperbolic under certain conditions.
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…
Study on vanishing cohomology groups in random simplicial complexes.
problem Determining when cohomology groups vanish in random simplicial complexes.
method Analysis of binomial random (k+1)-uniform hypergraphs and their downward-closure. result Vanishing of cohomology groups with coefficients in F2 has a sharp threshold. Criteria ensure manifold triangulation without differentiable structure.
problem Ensuring a manifold can be triangulated without differentiable structure.
method Local coordinate chart criteria for homeomorphism verification.
result Criteria guarantee triangulation of manifolds without Delaunay property.
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.
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…
Unified framework for observables in n-plectic geometry.
problem Quantization of extended objects in higher geometric contexts.
method Develops a semi-simplicial set model for observables, using a Grassmann variable to encode submanifold codimensions.
result Establishes a categorified pre-n-Hilbert space and a quantization scheme matching multisymplectic geometry.
This paper explores Brunnian twin groups and their properties.
problem Alexander-Markov correspondence for isotopy classes of immersed circles.
method Diagrammatic representations and combinatorial structures.
result Brunnian twin groups are free groups when more than two strands are involved.
Simplicial persistence measures financial market dynamics, revealing long-term structure evolution.
problem Understanding the long-term structure evolution of financial markets.
method Simplicial persistence, null models, TMFG filtering, thresholding, generative process analysis.
result More liquid markets exhibit slower persistence decay, suggesting higher fragility to systemic shocks.
In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a compl…
Product of shellable complexes yields shellable triangulations under tameness conditions.
problem Understanding shellability in products of simplicial complexes.
method Definition and proof of shellability properties for products of complexes under tameness conditions.
result Product of shellable complexes yields shellable triangulations under certain conditions.
We propose a general notion of algebraic gauge theory obtained via extracting the main properties of classical gauge theory. Building on a recent work on transferring curved A∞-structures we show that, under certain technical conditions, algebraic gauge theories can be transferred along chain contractions. Sp…