2020/08/26 by Jacek Brodzki, Brodzki, Jacek, Matthew Burfitt +3
Computer Science · Mathematics · Neuroscience · #06A07 (Secondary) #55N31 (Primary) 16G20 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Topological and Geometric Data Analysis #Tryptophan and brain disorders #math.AT #math.CO #math.RT #msc:06A07 #msc:16G20 #msc:55N31
paper · pdf · doi:10.48550/arxiv.2008.11532
arxiv created 2020/08/26 · openalex publication_date 2020/08/26 · arxiv updated 2020/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory. In this paper we consider the interesting special case of multiparameter persistence in zero dimensions which can be regarded as a form of multiparameter clustering. In particular, we consider the multiparameter persistence modules of the zero-dimensional homology of filtered topological spaces when they are finitely generated. Under certain assumptions, we characterize such modules and study their decompositions. In particular we identify a natural class of representations that decompose and can be extended back to form zero-dimensional multiparameter persistence modules. Our study of this set of representations concludes that despite the restrictions, there are still infinitely many classes of indecomposables in this set.