Related papers: On $C^0$-persistent homology and trees

We can view the Lipschitz constant as a height function on the space of maps between two manifolds and ask (as Gromov did nearly 30 years ago) what its ``Morse landscape'' looks like: are there high peaks, deep valleys and mountain passes?…

Geometry of networks endowed with a causal structure is discussed using the conventional framework of equilibrium statistical mechanics. The popular growing network models appear as particular causal models. We focus on a class of tree…

Motivated by analogies with basic density theorems in analytic number theory, we introduce a notion (and variations) of the homological density of one space in another. We use Weil's number field/ function field analogy to predict…

We endow the set of persistence diagrams with the strong topology (the topology of countable direct limit of increasing sequence of bounded subsets considered in the bottleneck distance). The topology of the obtained space is described.…

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…

A self-dual harmonic 2-form on a 4-dimensional Riemannian manifold is symplectic where it does not vanish. Furthermore, away from the form's zero set, the metric with the 2-form give a compatible almost complex structure and thus…

We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph $H$ as a minor to graphs excluding $H$ as a topological subgraph. We prove that for a fixed $H$, every graph excluding $H$ as a topological…

Working with group homomorphisms, a construction of manifolds is introduced to preserve homology groups. The construction gives as special cases Qullien's plus construction with handles obtained by Hausmann, the existence of one-sided…

We use topological data analysis to study "functional networks" that we construct from time-series data from both experimental and synthetic sources. We use persistent homology with a weight rank clique filtration to gain insights into…

Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features,…

Persistence diagrams are efficient descriptors of the topology of a point cloud. As they do not naturally belong to a Hilbert space, standard statistical methods cannot be directly applied to them. Instead, feature maps (or representations)…

Let $n \geq 2$. We prove a homological stability theorem for the diffeomorphism groups of $(4n+1)$-dimensional manifolds, with respect to forming the connected sum with $(2n-1)$-connected, $(4n+1)$-dimensional manifolds that are stably…

We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered…

We define a new topological summary for data that we call the persistence landscape. Since this summary lies in a vector space, it is easy to combine with tools from statistics and machine learning, in contrast to the standard topological…

Using ideas from algebraic topology and statistical mechanics, we generalize Kirchhoff's network and matrix-tree theorems to finite CW complexes of arbitrary dimension. As an application, we give a formula expressing Reidemeister torsion as…

We use the upper and lower potential functions and Bowen's formula estimating the Hausdorff dimension of the limit set of a regular semigroup generated by finitely many $C^{1+\alpha}$-contracting mappings. This result is an application of…

We extend in this paper several results of E. Kirchberg, S. Wassermann and the author dealing with continuous fields of C*--algebras to the semi-continuous case. We provide a new characterisation of separable lower semi-continuity…

High order networks are weighted hypergraphs col- lecting relationships between elements of tuples, not necessarily pairs. Valid metric distances between high order networks have been defined but they are difficult to compute when the…

We investigate the set of (real Dolbeault classes of) balanced metrics $\Theta$ on a balanced manifold $X$ with respect to which a torsion-free coherent sheaf $\mathcal{E}$ on $X$ is slope stable. We prove that the set of all such $[\Theta]…

We show that the Hrushovski-\fraisse limit of certain classes of trees lead to strictly superstable theories of various U-ranks. In fact, for each $ \alpha\in\omega+1\backslash\{0\} $ we introduce a strictly superstable theory of U-rank $…