Related papers: Persistent Cup-Length
A method is presented for the distributed computation of persistent homology, based on an extension of the generalized Mayer-Vietoris principle to filtered spaces. Cellular cosheaves and spectral sequences are used to compute global…
Operations on the cohomology of spaces are important tools enhancing the descriptive power of this computable invariant. For cohomology with mod 2 coefficients, Steenrod squares are the most significant of these operations. Their effective…
Persistent homology is currently one of the more widely known tools from computational topology and topological data analysis. We present in this note a brief survey on the evolution of the subject. The goal is to highlight the main ideas,…
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…
We study the categorical framework for the computation of persistent homology, without reliance on a particular computational algorithm. The computation of persistent homology is commonly summarized as a matrix theorem, which we call the…
Such modern applications of topology as data analysis and digital image analysis have to deal with noise and other uncertainty. In this environment, topological spaces often appear equipped with a real valued function. Persistence is a…
We study the grokking phenomenon through the lens of topology. Using persistent homology on point clouds derived from the embedding matrices of a range of models trained on modular arithmetic with varying primes, we identify a clear and…
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…
A fundamental challenge in multiparameter persistent homology is the absence of a complete and discrete invariant. To address this issue, we propose an enhanced framework that realizes a holistic understanding of a fully commutative…
In the first part of this paper, we consider, in the context of an arbitrary hyperplane arrangement, the map between compactly supported cohomology to the usual cohomology of a local system. A formula (i.e., an explicit algebraic de Rham…
Long lived topological features are distinguished from short lived ones (considered as topological noise) in simplicial complexes constructed from complex networks. A new topological invariant, persistent homology, is determined and…
This paper aims to discuss a method of quantifying the 'shape' of data, via a methodology called topological data analysis. The main tool within topological data analysis is persistent homology; this is a means of measuring the shape of…
In Homotopy Type Theory, cohomology theories are studied synthetically using higher inductive types and univalence. This paper extends previous developments by providing the first fully mechanized definition of cohomology rings. These rings…
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…
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…
This article has two goals. First, we hope to give an accessible introduction to persistent equivariant cohomology. Given a topological group $G$ acting on a filtered space, persistent Borel equivariant cohomology measures not only the…
The theory of multidimensional persistence captures the topology of a multifiltration -- a multiparameter family of increasing spaces. Multifiltrations arise naturally in the topological analysis of scientific data. In this paper, we give a…
We show that recollements of module categories give rise to homomorphisms between the associated Hochschild cohomology algebras which preserve the strict Gerstenhaber structure, i.e., the cup product, the graded Lie bracket and the squaring…
We compute cup product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function…
High order networks are weighted hypergraphs col- lecting relationships between elements of tuples, not necessarily pairs. Valid metric distances between high order networks have been defined but they are difficult to compute when the…