Related papers: Persistent Local Systems
In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals…
We consider small perturbations of expanding maps induced by skew-product mappings whose base dynamics are not invertible necessarily. Adopting a previously developed perturbative spectral approach, we show stability of the densities of the…
In this paper we provide an explicit connection between level-sets persistence and derived sheaf theory over the real line. In particular we construct a functor from 2-parameter persistence modules to sheaves over $\mathbb{R}$, as well as a…
In this paper we give a metric construction of a tree which correctly identifies connected components of superlevel sets of $\mathbb{R}$-valued continuous functions $f$ on $X$ and show that it is possible to retrieve the $H_0$-persistent…
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter…
For nearly three decades, spatial games have produced a wealth of insights to the study of behavior and its relation to population structure. However, as different rules and factors are added or altered, the dynamics of spatial models often…
We prove a homological stability theorem for moduli spaces of simply-connected manifolds of dimension $2n > 4$, with respect to forming connected sum with $S^n \times S^n$. This is analogous to Harer's stability theorem for the homology of…
We provide abstract conditions which imply the existence of a robustly invariant neighbourhood of a global section of a fibre bundle flow. We then apply such a result to the bundle flow generated by an Anosov flow when the fibre is the…
We prove a general homological stability theorem for certain families of groups equipped with product maps, followed by two theorems of a new kind that give information about the last two homology groups outside the stable range. (These…
We study a topology on a space of functions, called sticking topology, with the property to be the weakest among the topologies preserving continuity. In suitable frameworks, this topology preserves borelianity, local integrability, right…
A method to apply and visualize persistent homology of time series is proposed. The method captures persistent features in space and time, in contrast to the existing procedures, where one usually chooses one while keeping the other fixed.…
We propose a homology theory for locally compact spaces with ends in which the ends play a special role. The approach is motivated by results for graphs with ends, where it has been highly successful. But it was unclear how the original…
Given a relatively minimal non locally trivial fibred surface f: S->B, the slope of the fibration is a numerical invariant associated to the fibration. In this paper we explore how properties of the general fibre of $f$ and global…
We investigate the relations between algebraic structures, spectral invariants, and persistence modules, in the context of monotone Lagrangian Floer homology with Hamiltonian term. Firstly, we use the newly introduced method of filtered…
Data quality is crucial for the successful training, generalization and performance of machine learning models. We propose to measure the quality of a subset concerning the dataset it represents, using topological data analysis techniques.…
In this paper we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction…
We study the persistent homology of both functional data on compact topological spaces and structural data presented as compact metric measure spaces. One of our goals is to define persistent homology so as to capture primarily properties…
Topological complexity is a homotopy invariant that measures the minimal number of continuous rules required for motion planning in a space. In this work, we introduce persistent analogs of topological complexity and its cohomological lower…
Stable fold maps are fundamental tools in a generalization of the theory of Morse functions on smooth manifolds and its application to studies of geometric properties of smooth manifolds. Round fold maps were introduced as stable fold maps…
Recently, bipath persistent homology has been proposed as an extension of standard persistent homology, along with its visualization (bipath persistence diagram) and computational methods. In the setting of standard persistent homology, the…