arXiv research

Given a finite set of points in $\mathbb R^n$ and a radius parameter, we study the Čech, Delaunay-Čech, Delaunay (or Alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four…

New cohomology theory for diffeological spaces developed.

Study of metric spaces and group actions using Vietoris-Rips and Čech complexes.

Study on Čech-de Rham obstruction in diffeological spaces.

The paper bridges diffeological bundle theory with higher topos theory.

The purpose of this paper is to present a ``Cech-De Rham'' model for the cohomology of leaf spaces. This model lends itself to the construction of characteristic classes (in the cohomology of classifying spaces) by explicit geometrical constructions which are immediate extensions of the standard constructions for manif…

Given a compact geodesic space $X$ we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of $X$ to obtain what we call a persistence. This paper contains the theory describing such persistence: properties of the set of critical points, their preci…

The paper studies cohomology of sheaf complexes and proves a relative de Rham theorem.

The paper extends vector bundle theory to non-Hausdorff manifolds.

Defines sheaves and Čech cohomology for diffeological spaces and classifies principal bundles.

Defines relative Dolbeault homology and proves its equivalence with Čech-Dolbeault cohomology.

Study on Čech cohomology of Morse boundaries in hyperbolic manifolds.

New TDA approach using Finsler metrics.

Plateau's problem is to find a surface with minimal area spanning a given boundary. In 1960, Reifenberg and Adams developed a definition for "span" using Čech homology, and variants of this definition have been used ever sense. However, limitations of Čech homology resulted in the lack of a natural definition for a bou…

Study of homology commutativity in separable metrizable spaces.

Introduces fine shape theory to simplify shape and antishape invariants.

Study quantum aspects of 1-form symmetries using BV-BRST cohomology and gerbes.

Study on homology of random Čech complexes on manifolds with boundary.

In this paper we construct Cech cohomology groups that form a Gysin-type long exact sequence for principal torus bundles. This sequence is modeled on a de Rham cohomology sequence published in earlier work by Bouwknegt, Hannabuss and Mathai, which was developed to compute the global properties of T-duality in the prese…

Study cohomology of Lie algebroids over algebraic spaces using derived functors and Čech cohomology.

Study on critical faces convergence in a Poisson point process.

Study quantum aspects of 1-form symmetries using BV-BRST cohomology.

A line pattern in a free group $F$ is defined by a malnormal collection of cyclic subgroups. Otal defined a decomposition space $\mathcal{D}$ associated to a line pattern. We provide an algorithm that computes a presentation for the Čech cohomology of $\mathcal{D}$, thought of as a $F$-module. This answers a relative v…

New cohomology functors refine classical invariants of homotopy types.

Algorithm finds smooth primitives for exact forms.

In this paper we introduce principal 2-bundles and show how they are classified by non-abelian Cech cohomology. Moreover, we show that their gauge 2-groups can be described by 2-group-valued functors, much like in classical bundle theory. Using this, we show that, under some mild requirements, these gauge 2-groups poss…

We consider spectral sequences in smooth generalized cohomology theories, including differential generalized cohomology theories. The main differential spectral sequences will be of the Atiyah-Hirzebruch (AHSS) type, where we provide a filtration by the Cech resolution of smooth manifolds. This allows for systematic st…

In this paper we study the local description of spaces of forms on transitive Lie algebroids. We use this local description to introduce global structures like metrics, $\ast$-Hodge operation and integration along the algebraic part of the transitive Lie algebroid (its kernel). We construct a Čech-de Rham bicomplex wit…

In this paper we show that every rational cohomology class of type $(p,p)$ on a compact Kähler manifold can be representated as a differential $(p,p)$-form given by an explicit formula involving a Čech cocycle. First we represent Chern characters of smooth vector bundles by Čech cocycles with values in the sheaf of dif…

Method constructs 2-bundles over homogeneous spaces.

Study topological invariants of complexes for Riemannian manifolds.

Homotopy equivalence shown between complex and thickened versions of manifolds.

Let $G$ be a connected affine algebraic group over $\mathbb{C}$, $G \to X$ be an open immersion of $G$-varieties, $Z = X-G$ and $i: Z \to X$ be the inclusion. Let $α\in H^*(G,\mathbb{C})$ be primitive. We give a method to compute the image of $α$ in $H^*(Z, i^!\mathbb{C}_X)$, using a lift of $α$ along the first edge ma…

Global theory of relative invariants and equivariant line bundles established.

We introduce a new homological machine for the study of secondary geometric invariants. The objects, called spark complexes, occur in many areas of mathematics. The theory is applied here to establish the equivalence of a large family of spark complexes which appear naturally in geometry, topology and physics. These co…

Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.

Paper introduces DP TDA for near-optimal private persistence diagrams.

Given a set of points that sample a shape, the Rips complex of the data points is often used in machine-learning to provide an approximation of the shape easily-computed. It has been proved recently that the Rips complex captures the homotopy type of the shape assuming the vertices of the complex meet some mild samplin…

Theory of 2-vector bundles for smooth manifolds developed.

In this paper we study the homology of a random Cech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M. In particular, we focus on the phase transition for "homological connectivity" where the homology of the complex becomes isomorphic to that of M. The results presented in this pape…

Canonical quantization of abelian BF-type topological field theory coupled to extended sources on generic d-dimensional manifolds and with curved line bundles is studied. Sheaf cohomology is used to construct the appropriate topological extension of the action and the topological flux quantization conditions, in terms …

We develop semistrict higher gauge theory from first principles. In particular, we describe the differential Deligne cohomology underlying semistrict principal 2-bundles with connective structures. Principal 2-bundles are obtained in terms of weak 2-functors from the Cech groupoid to weak Lie 2-groups. As is demonstrat…

We introduce families of decorations of a same topological space, as well as a family of sheaves over such decorated spaces. Making those families a directed system leads to the concept of emerald over a space. For the configuration space X_N of N points in the plane, connecting points of the plane with chords is a dec…

We describe the geometrical ladder of equations for Abelian bundles and gerbes, as well as higher generalisations, in terms of the cohomology of an operator that combines de Rham and Cech cohomology.

Spaces containing compact subsets with polyhedral complements are studied.

Given a sample $Y$ from an unknown manifold $X$ embedded in Euclidean space, it is possible to recover the homology groups of $X$ by building a Vietoris--Rips or Čech simplicial complex on top of the vertex set $Y$. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular…

The paper reduces normal curvature and enhances homology recovery via embedded submanifolds.

Constructs hyperbolic reflection groups with 3D limit sets.