Related papers: Persistent Leray's spectral sequence

In this paper we introduce a statistic, the persistent homology transform (PHT), to model surfaces in $\mathbb{R}^3$ and shapes in $\mathbb{R}^2$. This statistic is a collection of persistence diagrams - multiscale topological summaries…

Persistent homology provides a robust methodology to infer topological structures from point cloud data. Here we explore the persistent homology of point clouds embedded into a probabilistic setting, exploiting the theory of point…

In this work, we explore links between natural homology and persistent homology for the classification of directed spaces. The former is an algebraic invariant of directed spaces, a semantic model of concurrent programs. The latter was…

We define shriek map for a finite codimensionnal embedding of fibration. We study the morphisms induced by shriek maps in the Leray-Serre spectral sequence. As a byproduct, we get two multiplicative spectral sequences of algebra wich…

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 study homotopy theory of the category of spectral sequences with respect to the class of weak equivalences given by maps which are quasi-isomorphisms on a fixed page. We introduce the category of extended spectral sequences and show that…

Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly…

We present a closed model structure for the category of pro-spectra in which the weak equivalences are detected by stable homotopy pro-groups. With some bounded-below assumptions, weak equivalences are also detected by cohomology as in the…

We construct a new type of spectral sequences for the mixed Hodge structures on the cohomology of locally symmetric varieties. These spectral sequences converge to the edge components in the Hodge triangles, and the E1-terms are expressed…

This paper contains an expository account of persistent homology and its usefulness for topological data analysis. An alternative foundation for level-set persistence is presented using sheaves and cosheaves.

Motivated by the study of persistence modules over the real line, we investigate the category of linear representations of a totally ordered set. We show that this category is locally coherent and we classify the indecomposable injective…

The purpose of this paper is to examine the calculational consequences of the duality for BPR<n> proved by the first author and Meier in 1607.02332, making them explicit in the special case n=3. We give an explicit description of the…

Beam splitters are not-free operations with regard to quantum coherence. As a consequence, they can create coherence from both coherent and incoherent states. We investigate the increase in coherence produced by cascades of beam splitters.…

Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional…

The aim of this paper is to study the cohomology theory of Reynolds Lie algebras equipped with derivations and to explore related applications. We begin by introducing the concept of Reynolds LieDer pairs. Subsequently, we construct the…

In this short note, we present a persistence module approach to directed cohomology, dual to the directed homology introduced by the author in a previous article. We lay out the first properties of directed cohomology and in particular of…

This paper presents a mathematically rigorous framework for brain-inspired representation learning founded on the interplay between persistent topological structures and cohomological flows. Neural computation is reformulated as the…

Persistent homology has recently emerged as a powerful technique in topological data analysis for analyzing the emergence and disappearance of topological features throughout a filtered space, shown via persistence diagrams. Additionally,…

A spectral sequence for the computation of the Hochschild cohomology of a coconnective dga over a field is presented. This spectral sequence has a similar flavour to the spectral sequence constructed by Cohen, Jones and Yan for the…

We give a direct description of the category of sheaves on Lichtenbaum's Weil-\'etale site of a number ring. Then we apply this result to define a spectral sequence relating Weil-\'etale cohomology to Artin-Verdier \'etale cohomology.…