Related papers: Persistence for Circle Valued Maps
Let $E_i$ be an oriented circle bundle over a closed oriented aspherical $n$-manifold $M_i$ with Euler class $e_i\in H^2(M_i;\mathbb{Z})$, $i=1,2$. We prove the following: (i) If every finite-index subgroup of $\pi_1(M_2)$ has trivial…
This paper develops the idea of homology for 1-parameter families of topological spaces. We express parametrized homology as a collection of real intervals with each corresponding to a homological feature supported over that interval or,…
Classification of large and dense networks based on topology is very difficult due to the computational challenges of extracting meaningful topological features from real-world networks. In this paper we present a computationally tractable…
The persistence barcode is a well-established complete discrete invariant for finitely generated persistence modules [5] [1]. Its definition, however, does not extend to multi- dimensional persistence modules. In this paper, we introduce a…
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…
In standard persistent homology, a persistent cycle born and dying with a persistence interval (bar) associates the bar with a concrete topological representative, which provides means to effectively navigate back from the barcode to the…
We propose a new way of thinking about one parameter persistence. We believe topological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between…
Persistent homology is a common technique in topological data analysis providing geometrical and topological information about the sample space. All this information, known as topological features, is summarized in persistence diagrams, and…
We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The set of such diagrams has an interleaving…
The integral variation map and algebraic monodromy of isolated plane curve singularities are important homological invariants of the singularity which are still far from being completely understood. This work provides effective ways of…
Persistent homology is a branch of computational algebraic topology that studies shapes and extracts features over multiple scales. In this paper, we present an unsupervised approach that uses persistent homology to study divergent behavior…
Persistent homology, an algebraic method for discerning structure in abstract data, relies on the construction of a sequence of nested topological spaces known as a filtration. Two-parameter persistent homology allows the analysis of data…
An Important tool in the field topological data analysis is known as persistent Homology (PH) which is used to encode abstract representation of the homology of data at different resolutions in the form of persistence diagram (PD). In this…
We study the dynamics of the five-parameter quadratic family of volume-preserving diffeomorphisms of R^3. This family is the unfolded normal form for a bifurcation of a fixed point with a triple-one multiplier and also is the general form…
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…
I introduce the concept of a persistence diagram (PD) bundle, which is the space of PDs for a fibered filtration function (a set $\{f_p: \mathcal{K}^p \to \mathbb{R}\}_{p \in B}$ of filtrations that is parameterized by a topological space…
We study the space of oriented genus g subsurfaces of a fixed manifold M, and in particular its homological properties. We construct a "scanning map" which compares this space to the space of sections of a certain fibre bundle over M…
We strengthen the usual stability theorem for Vietoris-Rips (VR) persistent homology of finite metric spaces by building upon constructions due to Usher and Zhang in the context of filtered chain complexes. The information present at the…
The persistent homology of a stationary point process on ${\bf R}^N$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops,…
We consider each of the three classes of representations of cyclic groups that arise in the study of rational sphere maps. We study the possible number of terms for invariant polynomials with non-negative coefficients that are constant on…