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.
Local Kan conditions enable differentiation of simplicial manifolds.
problem Differentiating simplicial manifolds into Lie algebroids.
method Expanding a technique for higher Lie groupoids to simplicial manifolds.
result Derivation of a method to differentiate simplicial manifolds into higher Lie algebroids.
Simplicial learning improves classification by generating compact sparse representations.
problem Difficulty in distinguishing classes on the same subspace.
method Evolutionary simplicial learning approach to sparse representations.
result Evolutionary simplicial learning outperforms other methods in multi-class classification.
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.
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.
The thesis uses simplicial methods to study actions, bundles, and bibundles of higher groupoids.
problem Understanding actions, bundles, and bibundles of higher groupoids.
method Employing simplicial methods to model actions, principal bundles, and bibundles of higher groupoids.
result The simplicial definitions agree with categorification approaches and prove a theorem on differentiation of higher Lie groupoids.
Unified probabilistic foundation for fuzzy simplicial sets in dimensionality reduction.
problem Lack of clear probabilistic interpretation in fuzzy simplicial sets.
method Introducing a probabilistic framework explaining fuzzy simplicial sets as marginals of probability measures on simplicial sets.
result Unified probabilistic theoretical foundation for fuzzy simplicial sets.
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.
New techniques prove bounded acyclicity results for semi-simplicial sets.
problem Homological stability of semi-simplicial sets and groups.
method Combinatorial techniques in simplicial bounded cohomology.
result Slope-1/2 stability results for certain groups. 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. Spectral simplicial theory improves feature selection for complex data.
problem Complex data sets and high-dimensional feature spaces require efficient feature selection methods.
method Extends spectral techniques to abstract simplicial complexes, incorporating topological data analysis.
result Spectral simplicial methods provide a unified approach for feature selection in multi-modal genomic data.
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.
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…
3-manifolds' volumes match stable integral values.
problem Determining 3-manifold volumes accurately.
method Integral foliated simplicial volume and ergodic theory.
result 3-manifolds' volumes equal stable integral simplicial volumes.
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…
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…
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.
New method extends discrete Morse theory to simplicial complexes.
problem Discrete Morse theory on simplicial complexes.
method Morse shellings and compatible discrete Morse functions.
result Triangulated surfaces and manifolds have Morse shellable triangulations.
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.
Study shows small transcendental simplicial volumes exist on manifolds.
problem Understanding the limits of simplicial volumes on manifolds.
method Examined closed manifolds and transcendental numbers.
result Found manifolds with arbitrarily small transcendental simplicial volumes.
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.
Algorithm finds characteristic maps over complex shapes.
problem Finding non-singular characteristic maps over wedged simplicial complexes.
method Constructive puzzle algorithm based on linear algebra.
result Algorithm performs well compared to other methods.
Study simplicial volume in fiber bundles with connected groups.
problem Understanding simplicial volume in fiber bundles.
method Investigated fiber bundles with connected structure groups.
result Simplicial volume of total space matches trivial bundle under certain conditions.
The simplicial volume introduced by Gromov provides a topologically accessible lower bound for the minimal volume. Lafont and Schmidt proved that the simplicial volume of closed, locally symmetric spaces of non-compact type is positive. In this paper, we present a generalization of this result to certain non-compact lo…
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| …
The study limits how many parts regular simplicial partitions can overlap.
problem Bounding the intersection number of regular simplicial partitions.
method Analyzing the properties of regular simplicial partitions.
result Established a maximum limit for the intersection number.
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.
Upper bound found for rank gradient using simplicial volume.
problem Rank gradient of fundamental groups.
method Stable integral simplicial volume.
result Upper bound established for rank gradient.
We provide sharp lower bounds for the simplicial volume of compact 3-manifolds in terms of the simplicial volume of their boundaries. As an application, we compute the simplicial volume of several classes of 3-manifolds, including handlebodies and products of surfaces with the interval. Our results provide the firs…
Integral foliated simplicial volume vanishes for manifolds with S1-action.
problem Integral foliated simplicial volume of manifolds with S1-action. method Geometric construction of Yano's proof for ordinary simplicial volume combined with parametrised uniform boundary condition for S1. result Integral foliated simplicial volume of aspherical manifolds with S1-action vanishes. We provide an action for gauge theories discretized on simplicial meshes, inspired by finite element methods. The action is discretely gauge invariant and we give a proof of consistency. A discrete Noether's theorem that can be applied to our setting, is also proved.
Upper bounds for fundamental groups via foliated simplicial volume.
problem Bounding fundamental groups of manifolds.
method Integral foliated simplicial volume approach.
result Integral foliated simplicial volume gives upper bounds for fundamental groups.
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.
Study shows simplicial volumes add for certain manifold gluings.
problem Understanding simplicial volumes in manifold gluings.
method Proves additivity of locally finite simplicial volume and Lipschitz simplicial volume for connected sums and specific boundary components.
result Additivity of simplicial volumes in dimension 3 and above for certain gluings.
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.
Positive simplicial volume implies locally symmetric space structure.
problem Understanding simplicial volume in locally homogeneous spaces.
method Analyzing properties of locally homogeneous Riemannian manifolds.
result Closed locally homogeneous manifolds with positive simplicial volume are locally symmetric.
Formula connects foliated simplicial volume with group cost.
problem Calculating integral foliated simplicial volume.
method Ergodic decomposition formula for simplicial volume.
result Integration formula linking foliated simplicial volume and group cost.
Computes cohomology of smooth cubic surfaces using simplicial resolution.
problem Computing the cohomology of smooth cubic surfaces.
method Vassiliev's simplicial resolution method.
result Cohomology isomorphic to that of P2 modulo embedding. Paper studies spaces of flattenings of simplicial spheres and their homotopy type.
problem Existence and uniqueness of differentiable structures on simplicial spheres.
method Analyzes spaces of flattenings of simplicial spheres and their homotopy type.
result Spaces of flattenings have the homotopy type of the orthogonal group.
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.
Efficiently sparsifies simplicial complexes using local densities of states.
problem Prohibitive computational requirements for dense simplicial complexes.
method Probabilistic sparsification using local densities of states and kernel-ignoring decomposition.
result Approximates the spectrum of the original SC with a sparser surrogate SC.
The paper constructs simplicial maps of any degree on spheres, solving a long-standing problem.
problem Constructing simplicial maps of any degree on spheres.
method Using connected sums and facet orientations, the paper develops a method to construct maps of any prescribed degree.
result The paper answers a question posed by Ryabichev and constructs simplicial maps of degree d for large d. Investigates simplicial volume over finite fields and compares it with other coefficients.
problem Examines simplicial volume over Fp coefficients. method Analyzes simplicial volume and gradient invariants over Fp coefficients, comparing with other coefficient rings. result Compares simplicial volumes and Betti numbers over different coefficient rings.
New results on relative simplicial volume using bounded acyclicity.
problem Understanding relative simplicial volume in bounded cohomology.
method Equivariant nerve pairs, relative classifying spaces, and small relative amenable category.
result Vanishing results for ℓ2-Betti numbers and mapping degrees. We prove that cubical simplicial volume of oriented closed 3-manifolds is equal to one fifth of ordinary simplicial volume.
Study shows simplicial volume of certain fiber bundles is zero.
problem Understanding simplicial volume in fiber bundles with nonpositive curvature.
method Proved simplicial volume is zero for specific fiber bundles.
result Simplicial volume of certain fiber bundles is zero.
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.
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.