Related papers: Notes on Pointwise Finite-Dimensional $2$-Paramete…
We prove that every finite dimensional representation of a finite group over a field of characteristic p admits a finite resolution by p-permutation modules. The proof involves a reformulation in terms of derived categories.
We classify up to equivalence all finite-dimensional irreducible representations of PSL2(Z) whose restriction to the commutator subgroup is diagonalizable.
Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…
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…
Dey and Xin (J.Appl.Comput.Top., 2022, arXiv:1904.03766) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators…
We prove a result that can be applied to determine the finite-dimensional simple Poisson modules over a Poisson algebra and apply it to numerous examples. In the discussion of the examples, the emphasis is on the correspondence with the…
Multiparameter persistent homology has emerged as a powerful generalization of topological data analysis, capable of encoding multivariate filtrations. However, the algebraic complexity of multiparameter persistence modules, marked by wild…
Motivated by the need to relate the biparameter persistence module induced by a pair of scalar functions with the monoparameter persistence modules induced by each function separately, we introduce a construction that defines a kind of…
We study the multi-dimensional persistence of Carlsson and Zomorodian and obtain a finer classification based upon the higher tor-modules of a persistence module. We propose a variety structure on the set of isomorphism classes of these…
We introduce a notion of strong periodicity of a module over a finite-dimensional algebra over a field. We prove that the existence of such modules over certain idempotent algebras is both a necessary and sufficient condition for the…
Persistence modules serve as the algebraic foundation for topological data analysis, typically studied as representations of posets over a field. This article extends the structural and decomposition theory of persistence modules to the…
We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are…
In 2009, Chazal et al. introduced $\epsilon$-interleavings of persistence modules. $\epsilon$-interleavings induce a pseudometric $d_I$ on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of…
Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a…
This article shows that the unconditional stability of the Dual-Finite Volume Method, which is at least valid for linear problems, is not true for generic nonlinear differential equations including the PMEs unless the coefficient appearing…
The Harder-Narasimhan types are a family of discrete isomorphism invariants for representations of finite quivers. Previously (arXiv:2303.16075), we evaluated their discriminating power in the context of persistence modules over a finite…
In this paper, we extend bottleneck stability to the setting of one dimensional constructible persistences module valued in any small abelian category.
A persistence module is a functor $f: \mathbf{I} \to \mathsf{E}$, where $\mathbf{I}$ is the poset category of a totally ordered set. This work introduces saecular decomposition: a categorically natural method to decompose $f$ into simple…
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…
We study persistence modules defined on commutative ladders. This class of persistence modules frequently appears in topological data analysis, and the theory and algorithm proposed in this paper can be applied to these practical problems.…