Related papers: A continuum of K\"unneth theorems for persistence …
We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module…
The classical K\"{u}nneth formula in algebraic topology describes the homology of a product space in terms of that of its factors. In this paper, we prove K\"{u}nneth-type theorems for the persistent homology of the categorical and tensor…
Given finite metric spaces $(X, d_X)$ and $(Y, d_Y)$, we investigate the persistent homology $PH_*(X \times Y)$ of the Cartesian product $X \times Y$ equipped with the sum metric $d_X + d_Y$. Interpreting persistent homology as a module…
Persistent homology was shown by Carlsson and Zomorodian to be homology of graded chain complexes with coefficients in the graded ring $\kk[t]$. As such, the behavior of persistence modules -- graded modules over $\kk[t]$ is an important…
We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent…
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…
For any additive functor from modules (or, more generally, from an abelian category with enough projectives or injectives), we construct long sequences tying up together the derived functors, the satellites, and the stabilizations of the…
We establish two structural results for Moore homology of ample groupoids. First, for every ample groupoid $\mathcal{G}$ and every discrete abelian coefficient group $A$, we prove a universal coefficient theorem relating the homology groups…
This thesis addresses the theory of topological spaces and the foundations of persistence theory. We will discuss chain complexes and the associated simplicial homology groups, as well as their relationship with singular homology theory.…
We reformulate the persistent (co)homology of simplicial filtrations, viewed from a more algebraic setting, namely as the (co)homology of a chain complex of graded modules over polynomial ring $K[t]$. We also define persistent (co)homology…
We present a parallelizable algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then…
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…
It is well-known that the cohomology ring has a richer structure than homology groups. However, until recently, the use of cohomology in persistence setting has been limited to speeding up of barcode computations. Some of the recently…
We define a class of multiparameter persistence modules that arise from a one-parameter family of functions on a topological space and prove that these persistence modules are stable. We show that this construction can produce…
We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence diagrams, which, in particular, implies that they satisfy…
Beyond H\"{o}lder's type, this paper mainly concerns the persistence and remaining regularity of an individual frequency-preserving KAM torus in a finitely differentiable Hamiltonian system, even allows the non-integrable part being…
We develop persistent homology in the setting of filtrations of (Cech) closure spaces. Examples of filtrations of closure spaces include metric spaces, weighted graphs, weighted directed graphs, and filtrations of topological spaces. We use…
Persistent homology is a fundamental tool in Topological Data Analysis. The associated algebraic structure is the persistence module, a sequence of vector spaces connected by linear maps. Persistence modules admit a complete and…
We prove homology stability for elementary and special linear groups over rings with many units improving known stability ranges. Our result implies stability for unstable Quillen K-groups and proves a conjecture of Bass. For commutative…
In this paper, we introduce a persistent (co)homology theory for Cayley digraph grading. We give the algebraic structures of Cayley-persistence object. Specifically, we consider the module structure of persistent (co)homology and show the…