Related papers: Homology by metric currents
The suggested approach makes it possible to produce a consistent description of motions of a physical system. It is shown that the concept of force fields defining the systems dynamics is equivalent to the choice of the corresponding metric…
Two problems are addressed for the path of certain stochastic processes: a) do they define currents? b) are these currents of a classical type? A general answer to question a) is given for processes like semimartingales or with Lyons-Zheng…
We show that the cohomology of flat currents on definable pseudomanifolds in polynomially bounded o-minimal structures is isomorphic to its intersection cohomology in the top perversity.
A metric space is said to be all-set-homogeneous if any of its partial isometries can be extended to a genuine isometry. We give a classification of a certain subclass of all-set-homogeneous length spaces.
In quantum mechanics, measurements are dynamical processes and thus they should be capable of inducing currents. The symmetries of the Hamiltonian and measurement operator provide an organizing principle for understanding the conditions for…
In this paper we define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an $n$-dimensional cube to a fixed metric space. We…
Recently, a new embedding/compactness theorem for integral currents in a sequence of metric spaces has been established by the second author. We present a version of this result for locally integral currents in a sequence of pointed metric…
We define the relative Dolbeault homology of a complex manifold with currents via a \v{C}ech approach and we prove its equivalence with the relative \v{C}ech-Dolbeault cohomology as defined in [T. Suwa, \v{C}ech-Dolbeault cohomology and the…
We define an algebraic setup of homology for hypergraphs, which defaults to simplicial homology in the case of graphs, and study its basic properties. As part of our study we define algebraic spanning trees of hypergraphs, along with…
In some sense, the world is composed of shapes and words, of continuous things and discrete things. The recognition and study of continuous objects in the form of shapes occupies a significant part of the effort of unraveling many geometric…
Persistent homology is a popular tool in Topological Data Analysis. It provides numerical characteristics of data sets which reflect global geometric properties. In order to be useful in practice, for example for feature generation in…
In this paper we examine two basic topological properties of partial metric spaces, namely compactness and completeness. Our main result claims that in these spaces compactness is equivalent to sequential compactness. We also show that…
This survey summarizes recent progress on the flat chain conjecture, which asserts the equivalence between metric currents and flat chains with finite mass in the Euclidean space. In particular, we focus on recent work showing that the…
Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…
We classify the metric spaces that can be approximated by finite homogeneous ones.
Two non-empty sets $A,B$ of a metric space $(X,d)$ are called parallel if $d(a,B)=d(A,B)=d(A,b)$ for any points $a\in A$ and $b\in B$. Answering a question posed on Mathoverflow, we prove that for a cover $\mathcal C$ of a metrizable space…
We introduce the metric fundamental class for metric spaces that are homeomorphic to compact, non-orientable, smooth manifolds with (possibly empty) boundary. This is an integer rectifiable current that provides an analytic representation…
Every geodesic current on a hyperbolic surface has an associated dual space. If the current is a lamination, this dual embeds isometrically into a real tree. We show that, in general, the dual space is a Gromov hyperbolic metric tree-graded…
This paper investigates spaces equipped with a family of metric-like functions satisfying certain axioms. These functions provide a unified framework for defining topology, uniformity, and diffeology. The framework is based on a family of…
Classical particle mechanics on curved spaces is related to the flow of ideal fluids, by a dual interpretation of the Hamilton-Jacobi equation. As in second quantization, the procedure relates the description of a system with a finite…