Related papers: Persistent homology of groups
We prove that for any prime $p$ the finite $p$-groups of fixed coclass have only finitely many different mod-$p$ cohomology rings between them. This was conjectured by Carlson; we prove it by first proving a stronger version for groups of…
Persistent homology is one of the most active branches of Computational Algebraic Topology with applications in several contexts such as optical character recognition or analysis of point cloud data. In this paper, we report on the formal…
Persistent homology (PH) is a recently developed theory in the field of algebraic topology to study shapes of datasets. It is an effective data analysis tool that is robust to noise and has been widely applied. We demonstrate a general…
Let p be an odd prime. Let G be a p-local finite group over the extraspecial p-group p_+^{1+2}. In this paper we study the cohomology and the stable splitting of their p-complete classifying space BG.
We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, and give motivation as for the advantages of…
We determine the integral homology of PSL4(Z) in degrees at most 5 and determine its p-part in higher degrees for the primes p>=5. Our method applies to other arithmetic groups; as illustrations we include descriptions of the integral…
In this paper, we extend the notion of directed clique complex to quivers and introduce an associated homology theory. By applying this construction to biquandle coloring quivers, we obtain new invariants of links. We then introduce a…
Persistent homology, a powerful mathematical tool for data analysis, summarizes the shape of data through tracking topological features across changes in different scales. Classical algorithms for persistent homology are often constrained…
The mod-p cohomology ring of a non-trivial finite p-group is an infinite dimensional, finitely presented graded unital algebra over the field with p elements, with generators in positive degrees. We describe an effective algorithm to test…
We prove a general representation stability result for polynomial coefficient systems which lets us prove representation stability and secondary homological stability for many families of groups with polynomial coefficients. This gives two…
The persistence barcode (equivalently, the persistence diagram), which can be obtained from the interval decomposition of a persistence module, plays a pivotal role in applications of persistent homology. For multi-parameter persistent…
We develop a theory of persistent homology for directed simplicial complexes which detects persistent directed cycles in odd dimensions. We relate directed persistent homology to classical persistent homology, prove some stability results,…
A continuous cohomology theory for topological quandles is introduced, and compared to the algebraic theories. Extensions of topological quandles are studied with respect to continuous 2-cocycles, and used to show the differences in second…
In topological data analysis, persistent homology characterizes robust topological features in data and it has a summary representation, called a persistence diagram. Statistical research for persistence diagrams have been actively…
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently…
The concept of topological persistence, introduced recently in computational topology, finds applications in studying a map in relation to the topology of its domain. Since its introduction, it has been extended and generalized in various…
Persistent homology and zigzag persistent homology are techniques which track the homology over a sequence of spaces, outputting a set of intervals corresponding to birth and death times of homological features in the sequence. This paper…
Topological Data Analysis has grown in popularity in recent years as a way to apply tools from algebraic topology to large data sets. One of the main tools in topological data analysis is persistent homology. This paper uses undergraduate…
Persistent homology is a powerful tool for characterizing the topology of a data set at various geometric scales. When applied to the description of molecular structures, persistent homology can capture the multiscale geometric features and…
Persistent Topology studies topological features of shapes by analyzing the lower level sets of suitable functions, called filtering functions, and encoding the arising information in a parameterized version of the Betti numbers, i.e. the…