Related papers: Moderately Discontinuous Homology
Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of real intervals. Recent work of Edelsbrunner, Jablonski,…
Two important invariants of directed graphs, namely magnitude homology and path homology, have recently been shown to be intimately connected: there is a 'magnitude-path spectral sequence' or 'MPSS' in which magnitude homology appears as…
Uniformly finite homology is a coarse homology theory, defined via chains that satisfy a uniform boundedness condition. By construction, uniformly finite homology carries a canonical $\ell^\infty$-semi-norm. We show that, for uniformly…
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…
The persistent homology transform (PHT) represents a shape with a multiset of persistence diagrams parameterized by the sphere of directions in the ambient space. In this work, we describe a finite set of diagrams that discretize the PHT…
Topological data analysis has been recently applied to investigate stylistic signatures and trends in musical compositions. A useful tool in this area is Persistent Homology. In this paper, we develop a novel method to represent a weighted…
In many applications concerning the comparison of data expressed by $\mathbb{R}^m$-valued functions defined on a topological space $X$, the invariance with respect to a given group $G$ of self-homeomorphisms of $X$ is required. While…
We propose a notion of graph convergence that interpolates between the Benjamini--Schramm convergence of bounded degree graphs and the dense graph convergence developed by L\'aszl\'o Lov\'asz and his coauthors. We prove that spectra of…
We consider the inverse problem of reconstructing inhomogeneities by performing a finite number of scattering measurements of acoustic type in the time-harmonic setting. We set up the reconstruction as a fully discrete variational problem…
Will Merry computed Rabinowitz Floer homology above Mane's critical value in terms of loop space homology by establishing an Abbondandolo-Schwarz short exact sequence. The purpose of this article is to provide an alternative proof of…
We show existence of an infinitesimally invariant measure $m$ for a large class of divergence and non-divergence form elliptic second order partial differential operators with locally Sobolev regular diffusion coefficient and drift of some…
Computational topologists recently developed a method, called persistent homology to analyze data presented in terms of similarity or dissimilarity. Indeed, persistent homology studies the evolution of topological features in terms of a…
The purpose of this paper is twofold: 1. we prove the triangulability of smooth orbifolds with corners, generalizing the same statement for orbifolds. 2. based on 1, we propose a new homology theory. We call it geometric homology theory…
In our pursuit of finding a zero for a monotone and Lipschitz continuous operator $M : \R^n \rightarrow \R^n$ amidst noisy evaluations, we explore an associated differential equation within a stochastic framework, incorporating a correction…
We study pairs of Dirichlet forms related by an intertwining order isomorphisms between the associated $L^2$-spaces. We consider the measurable, the topological and the geometric setting respectively. In the measurable setting, we deal with…
We study a condensed version of the \'etale homotopy type of a scheme, which refines both the usual \'etale homotopy type of Friedlander-Artin-Mazur and the pro\'etale fundamental group of Bhatt-Scholze. In the first part of this paper, we…
The germ of an algebraic variety is naturally equipped with two different metrics up to bilipschitz equivalence. The inner metric and the outer metric. One calls a germ of a variety Lipschitz normally embedded if the two metrics are…
It is, by now, classical that lattices in higher rank semisimple groups have various rigidity properties. In this work, we add another such rigidity property to the list: uniform stability with respect to the family of unitary operators on…
Let $(M, \partial M)$ be a compact 3-manifold with boundary which admits a complete, convex co-compact hyperbolic metric. For each hyperbolic metric $g$ on $M$ such that $\dr M$ is smooth and strictly convex, the induced metric on $\dr M$…
The magnitude homology, introduced by R. Hepworth and S. Willerton, offers a topological invariant that enables the study of graph properties. Hypergraphs, being a generalization of graphs, serve as popular mathematical models for data with…