Related papers: Dualities in Multiparameter Persistence
We define persistent homology groups over any set of spaces which have inclusions defined so that the corresponding directed graph between the spaces is acyclic, as well as along any subgraph of this directed graph. This method…
In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is the Hecke algebra of a reductive $p$-adic group, to the case where it is the Hecke algebra…
One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and…
The following article is an application of commutative algebra to the study of multiparameter persistent homology in topological data analysis. In particular, the theory of finite free resolutions of modules over polynomial rings is applied…
In this paper, we investigate some characterizations of dual continuous frames and give some results about them. Also, we refer to the method of constructing a family of duals through a fixed dual and show there exists a one-to-one…
The Rips filtration over a finite metric space and its corresponding persistent homology are prominent methods in TDA to summarise the shape of data. Crucial to their use is the bottleneck stability result. A generalisation of the Rips…
Determining whether two graphs are isomorphic is a fundamental problem with practical applications in areas such as molecular chemistry or social network analysis, yet it remains a challenging task, with exact solutions often being…
Multiparameter persistence modules can be uniquely decomposed into indecomposable summands. Among these indecomposables, intervals stand out for their simplicity, making them preferable for their ease of interpretation in practical…
We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and…
Let $K$ be a finite simplicial, cubical, delta or CW complex. The persistence map $\mathrm{PH}$ takes a filter $f:K \rightarrow \mathbb{R}$ as input and returns the barcodes $\mathrm{PH}(f)$ of the associated sublevel set persistent…
The study of persistent homology has contributed new insights and perspectives into a variety of interesting problems in science and engineering. Work in this domain relies on the result that any finitely-indexed persistence module of…
Persistent homology (PH) studies the topology of data across multiple scales by building nested collections of topological spaces called filtrations, computing homology and returning an algebraic object that can be vizualised as a…
For a coherent filtered D-module we show that the dual of each graded piece over the structure sheaf is isomorphic to a certain graded piece of the ring-theoretic local cohomology complex of the graded quotient of the dual of the filtered…
In topological data analysis, we want to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to simply random noise. While progress has been made on…
We provide a formula (see Theorem 1.5) for the Matlis dual of the injective hull of $R/\mathfrak{p}$ where $\mathfrak p$ is a one dimensional prime ideal in a local complete Gorenstein domain $(R,\mathfrak{m})$. This is related to results…
When filtering a topological space by a single parameter, the theory of quiver representations provides a complete framework for decomposing the resulting persistence module to obtain its barcode. This is achieved by interpreting the…
The classical persistence algorithm computes the unique decomposition of a persistence module implicitly given by an input simplicial filtration. Based on matrix reduction, this algorithm is a cornerstone of the emergent area of topological…
Motivated by recent progress in topological data analysis, we establish a Matlis duality between injective hulls and flat covers of persistence modules. This extends to a duality between minimal flat and minimal injective resolutions. We…
We give bounds for dimension 0 persistent homology and codimension 1 homology of Vietoris--Rips, alpha, and cubical complex filtrations from finite sets related by enrichment (adding new elements), sparsification (removing elements), and…
We compare closed and rigid monoidal categories. Closedness is defined by the tensor product having a right adjoint: the internal hom functor. Rigidity, on the other hand, generalises the duality of finite-dimensional vector spaces. In the…