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Large simplicial complexes: universality, randomness, and ampleness
The paper surveys recent progress in understanding geometric, topological and combinatorial properties of large simplicial complexes, focusing mainly...
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Multiscale transforms for signals on simplicial complexes
Our previous multiscale graph basis dictionaries/graph signal transforms—Generalized Haar-Walsh Transform (GHWT); Hierarchical Graph Laplacian Eigen...
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Topological representations of simplicial complexes and their applications
Topological models can be used to represent complex systems which originate in the real life world. The aim of this paper is to show the equivalence...
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The homology of random simplicial complexes in the multi-parameter upper model
We study random simplicial complexes in the multi-parameter upper model. In this model simplices of various dimensions are taken randomly and...
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Embedding Dimensions of Simplicial Complexes on Few Vertices
We provide a simple characterization of simplicial complexes on few vertices that embed into the d -sphere. Namely, a simplicial complex on
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Harmonic maps between 2-dimensional simplicial complexes: conformal and singular metrics
We study metrics on two-dimensional simplicial complexes that are conformal either to flat Euclidean metrics or to the ideal hyperbolic metrics...
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On Discrete Gradient Vector Fields and Laplacians of Simplicial Complexes
Discrete Morse theory, a cell complex-analog to smooth Morse theory allowing homotopic tools in the discrete realm, has been developed over the past...
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Partitions of Vertices and Facets in Trees and Stacked Simplicial Complexes
For stacked simplicial complexes, (special subclasses of such are: trees, triangulations of polygons, stacked polytopes with their triangulations),...
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Simplicial Spanning Trees in Random Steiner Complexes
A spanning tree T in a graph G is a sub-graph of G with the same vertex set as G which is a tree. In 1981, McKay proved an asymptotic result...
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Polyhedral Products for Connected Sums of Simplicial Complexes
AbstractWe investigate how the homotopy type of a polyhedral product changes under the operation of taking the connected sum of two simplicial...
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Large simple d-cycles in simplicial complexes
We show that the size of the largest simple d -cycle in a simplicial d -complex K is at least a square root of K ’s density. This is a higher...
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Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes
We prove the following quantitative Borsuk–Ulam-type result (an equivariant analogue of Gromov’s Topological Overlap Theorem): Let X be a free...
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Ample simplicial complexes
Motivated by potential applications in network theory, engineering and computer science, we study r -ample simplicial complexes. These complexes can...
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Triangulations of polygons and stacked simplicial complexes: separating their Stanley–Reisner ideals
A triangulation of a polygon has an associated Stanley–Reisner ideal. We obtain a full algebraic and combinatorial understanding of these ideals and...
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Simplicial branching random walks
We study a model of branching random walks on simplicial complexes, which can be seen as a natural generalization of random walks on graphs....
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Expected invariants of simplicial complexes obtained from random point samples
In this paper, we study the expectation values of topological invariants of the Vietoris–Rips complex and Čech complex for a finite set of sample...
