Simplicial approximation theorem
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Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplexes Open
Collective behavior in large ensembles of dynamical units with nonpairwise interactions may play an important role in several systems ranging from brain function to social networks. Despite recent work pointing to simplicial structure, i.e…
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Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes Open
Simplicial complexes are generalized network structures able to encode interactions occurring between more than two nodes. Simplicial complexes describe a large variety of complex interacting systems ranging from brain networks to social a…
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Weighted growing simplicial complexes Open
Simplicial complexes describe collaboration networks, protein interaction networks, and brain networks and in general network structures in which the interactions can include more than two nodes. In real applications, often simplicial comp…
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Random walks on simplicial complexes and harmonicsâ \n Open
In this paper, we introduce a class of random walks with absorbing states on simplicial complexes. Given a simplicial complex of dimension d, a random walk with an absorbing state is defined which relates to the spectrum of the kâdimensi…
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Discrete topological complexity Open
We introduce a notion of discrete topological complexity in the setting of simplicial complexes, using only the combinatorial structure of the complex and replacing the concept of homotopy by that of contiguous simplicial maps. We study th…
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Integral approximation of simplicial volume of graph manifolds Open
Graph manifolds are manifolds that decompose along tori into pieces with a\ntame $S^1$-structure. In this paper, we prove that the simplicial volume of\ngraph manifolds (which is known to be zero) can be approximated by integral\nsimplicia…
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Strong Collapse for Persistence Open
We introduce a fast and memory efficient approach to compute the persistent homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by …
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A Notion of Harmonic Clustering in Simplicial Complexes Open
We outline a novel clustering scheme for simplicial complexes that produces\nclusters of simplices in a way that is sensitive to the homology of the\ncomplex. The method is inspired by, and can be seen as a higher-dimensional\nversion of, …
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Simplicial Complexity: piecewise linear motion planning in robotics Open
Using the notion of contiguity of simplicial maps, we adapt Farber's topological complexity to the realm of simplicial complexes. We show that, for a finite simplicial complex $K$, our discretized concept recovers the topological complexit…
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The amazing world of simplicial complexes Open
Defined by a single axiom, finite abstract simplicial complexes belong to the simplest constructs of mathematics. We look at a a few theorems.
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Stable integral simplicial volume of 3‐manifolds Open
We show that non-elliptic prime 3-manifolds satisfy integral approximation for the simplicial volume, that is, that their simplicial volume equals the stable integral simplicial volume. The proof makes use of integral foliated simplicial v…
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Principled Simplicial Neural Networks for Trajectory Prediction Open
We consider the construction of neural network architectures for data on simplicial complexes. In studying maps on the chain complex of a simplicial complex, we define three desirable properties of a simplicial neural network architecture:…
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Transcendental simplicial volumes Open
We show that there exist closed manifolds with arbitrarily small transcendental simplicial volumes. Moreover, we exhibit an explicit family of (transcendental) real numbers that are not realised as the simplicial volume of a closed manifol…
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Combinatorics and Representation Theory for Generalized Permutohedra I: Simplicial Plates Open
In this paper, we announce results from our thesis, which studies for the first time the categorification of the theory of generalized permutohedra. The vector spaces in the categorification are tightly constrained by certain continuity re…
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Extending Homotopy Type Theory with Strict Equality Open
In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of semi-simplicial types. More generally, it is …
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Higher-order connection Laplacians for directed simplicial complexes Open
Higher-order networks encode the many-body interactions existing in complex systems, such as the brain, protein complexes, and social interactions. Simplicial complexes are higher-order networks that allow a comprehensive investigation of …
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Simplicial Complex Representation Learning Open
Simplicial complexes form an important class of topological spaces that are frequently used in many application areas such as computer-aided design, computer graphics, and simulation. Representation learning on graphs, which are just 1-d s…
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Simplicial quantum contextuality Open
We introduce a new framework for contextuality based on simplicial sets, combinatorial models of topological spaces that play a prominent role in modern homotopy theory. Our approach extends measurement scenarios to consist of spaces (rath…
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Dist2Cycle: A Simplicial Neural Network for Homology Localization Open
Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher…
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Minimal volume entropy of simplicial complexes Open
This article deals with topological assumptions under which the minimal volume entropy of a closed manifold, and more generally of a finite simplicial complex, vanishes or is positive. In the first part of the article, we present complemen…
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Enrichment strategies for the simplicial linear finite elements Open
In this paper, we introduce a new class of finite elements by enriching the standard simplicial linear finite element in Rd with additional functions which are not necessarily polynomials. We provide necessary and sufficient conditions on …
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Persistence Diagrams for Efficient Simplicial Complex Reconstruction Open
Topological descriptors have been shown to be useful for summarizing and differentiating shapes. Related work uses persistence diagrams and Euler characteristic curves to differentiate between shapes and quantifies the number of descriptor…
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Computing simplicial representatives of homotopy group elements Open
A central problem of algebraic topology is to understand the homotopy groups of a topological space X. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group of a g…
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Semi-Simplicial Set Models for Distributed Knowledge Open
International audience
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The uniform face ideals of a simplicial complex Open
We define the uniform face ideal of a simplicial complex with respect to an ordered proper vertex colouring of the complex. This ideal is a monomial ideal which is generally not squarefree. We show that such a monomial ideal has a linear r…
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Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes Open
Higher-order networks have emerged as a powerful framework to model complex systems and their collective behavior. Going beyond pairwise interactions, they encode structured relations among arbitrary numbers of units through representation…
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Red–green refinement of simplicial meshes in $d$ dimensions Open
The local red–green mesh refinement of consistent, simplicial meshes in $d$ dimensions is considered. We give a constructive solution to the green closure problem in arbitrary dimension $d$. Suppose that $\mathcal {T}$ is a simplicial mesh…
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Construction of simplicial complexes with prescribed degree-size sequences Open
We study the realizability of simplicial complexes with a given pair of integer sequences, representing the node degree distribution and the facet size distribution, respectively. While the $s$-uniform variant of the problem is $\mathsf{NP…
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Convolutional Learning on Simplicial Complexes Open
We propose a simplicial complex convolutional neural network (SCCNN) to learn data representations on simplicial complexes. It performs convolutions based on the multi-hop simplicial adjacencies via common faces and cofaces independently a…
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Extendability of simplicial maps is undecidable Open
We present a short proof of the Čadek-Krčál-Matoušek-Vokřínek-Wagner result from the title (in the following form due to Filakovský-Wagner-Zhechev). For any fixed even $l$ there is no algorithm recognizing the extendability of the identity…