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The Topological Signal Processing (TSP) framework has been recently developed to analyze signals defined over simplicial complexes, i.e. topological spaces represented by finite sets of elements that are closed under inclusion of subsets…
We present a discrete Morse-theoretic method for proving that a regular CW complex is homeomorphic to a sphere. We use this method to define bisimplices, the cells of a class of regular CW complexes we call bisimplicial complexes. The…
Shellings of simplicial complexes have long been a useful tool in topological and algebraic combinatorics. Shellings of a complex expose a large amount of information in a helpful way, but are not easy to construct, often requiring deep…
Hypergraphs have seen widespread application in network and data science communities in recent years. We present a survey of recent work to construct auxiliary structures from hypergraphs -- specifically simplicial, relative, and chain…
We formalize the simulation paradigm of cryptography in terms of category theory and show that protocols secure against abstract attacks form a symmetric monoidal category, thus giving an abstract model of composable security definitions in…
In this article we undertake a study of extension complexity from the perspective of formal languages. We define a natural way to associate a family of polytopes with binary languages. This allows us to define the notion of extension…
Motivated by Gr\"obner basis theory for finite point configurations, we define and study the class of "standard complexes" associated to a matroid. Standard complexes are certain subcomplexes of the independence complex that are invariant…
We study word series and extended word series, classes of formal series for the analysis of some dynamical systems and their discretizations. These series are similar to but more compact than B-series. They may be composed among themselves…
We introduce the Insertion Chain Complex, a higher-dimensional extension of insertion graphs, as a new framework for analyzing finite sets of words. We study its topological and combinatorial properties, in particular its homology groups,…
In this paper, we consider a simple class of stratified spaces -- 2-complexes. We present an algorithm that learns the abstract structure of an embedded 2-complex from a point cloud sampled from it. We use tools and inspiration from…
Our work is concerned with simplicial complexes that describe higher-order interactions in real complex systems. This description allows to go beyond the pairwise node-to-node representation that simple networks provide and to capture a…
Decision-making in complex systems often relies on machine learning models, yet highly accurate models such as XGBoost and neural networks can obscure the reasoning behind their predictions. In operations research applications,…
We investigate the collapsibility of systolic finite simplicial complexes of arbitrary dimension. The main tool we use in the proof is discrete Morse theory. We shall consider a convex subcomplex of the complex and project any simplex of…
This paper describes an abstract machine for linguistic formalisms that are based on typed feature structures, such as HPSG. The core design of the abstract machine is given in detail, including the compilation process from a high-level…
The goal of this paper is to generalize some of the existing toolkit of combinatorial algebraic topology in order to study the homology of abstract chain complexes. We define shellability of chain complexes in a similar way as for cell…
Points of interest on a map such as restaurants, hotels, or subway stations, give rise to categorical point data: data that have a fixed location and one or more categorical attributes. Consequently, recent years have seen various set…
It is increasingly common for data to possess intricate structure, necessitating new models and analytical tools. Graphs, a prominent type of structure, can encode the relationships between any two entities (nodes). However, graphs neither…
We introduce the multineighbor complex of a graph, which is a simplicial complex in which a simplex is a subset of the graph with a sufficient number of mutual neighbors. We investigate the asymptotic homological properties of such…
We study the Fraisse limit of the class of all finite simplicial complexes. Whilst the natural model-theoretic setting for this class uses an infinite language, a range of results associated with Fraisse limits of structures for finite…
We provide dual algorithms for sampling the space of abstract simplicial complexes on a fixed number of vertices. We develop a generative and descriptive sampler designed with heuristics to help balance the combinatorial multiplicities of…