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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…
We first study the higher version of the relative topological complexity by using the homotopic distance. We also introduced the generalized version of the relative topological complexity of a topological pair on both the Schwarz genus and…
A suitable measure for the similarity of shapes represented by parameterized curves or surfaces is the Fr\'echet distance. Whereas efficient algorithms are known for computing the Fr\'echet distance of polygonal curves, the same problem for…
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…
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently…
In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…
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…
This article introduces an algorithm to compute the persistent homology of a filtered complex with various coefficient fields in a single matrix reduction. The algorithm is output-sensitive in the total number of distinct persistent…
The theory of persistence modules is an emerging field of algebraic topology which originated in topological data analysis. In these notes we provide a concise introduction into this field and give an account on some of its interactions…
Persistent homology enables fast and computable comparison of topological objects. However, it is naturally limited to the analysis of topological spaces. We extend the theory of persistence, by guaranteeing robustness and computability to…
We show that both Lusternik-Schnirelmann category and topological complexity are particular cases of a more general notion, that we call homotopic distance between two maps. As a consequence, several properties of those invariants can be…
We consider the homeomorphic classification of finite-dimensional continua as well as several related equivalence relations. We show that, when $n \geq 2$, the classification problem of $n$-dimensional continua is strictly more complex than…
Given a function $f: \Xspace \to \Rspace$ on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of $f$. In addition, we…
Effective resistance is a distance between vertices of a graph that is both theoretically interesting and useful in applications. We study a variant of effective resistance called the biharmonic distance. While the effective resistance…
A graph is said to be distance-hereditary if the distance function in every connected induced subgraph is the same as in the graph itself. We prove that the ordinary Weisfeiler-Leman algorithm correctly tests the isomorphism of any two…
Persistent homology is a mathematical tool used for studying the shape of data by extracting its topological features. It has gained popularity in network science due to its applicability in various network mining problems, including…
We present a generalization of the induced matching theorem and use it to prove a generalization of the algebraic stability theorem for $\mathbb{R}$-indexed pointwise finite-dimensional persistence modules. Via numerous examples, we show…
Starting with a similarity function between objects, it is possible to define a distance metric on pairs of objects, and more generally on probability distributions over them. These distance metrics have a deep basis in functional analysis,…
Computing an optimal cycle in a given homology class, also referred to as the homology localization problem, is known to be an NP-hard problem in general. Furthermore, there is currently no known optimality criterion that localizes classes…
Persistence diagrams, combining geometry and topology for an effective shape description used in pattern recognition, have already proven to be an effective tool for shape representation with respect to a certainfiltering function.…