Related papers: Persistent sheaf cohomology
Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira, after J. Curry has made the first link between persistent homology and sheaves. We prove the isometry theorem in…
In this paper, we provide two different resolutions of structural sheaves of projectivized tangent bundles of smooth complete intersections. These resolutions allow in particular to obtain convenient (and completely explicit) descriptions…
We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence…
Persistence modules have a natural home in the setting of stratified spaces and constructible cosheaves. In this article, we first give explicit constructible cosheaves for common data-motivated persistence modules, namely, for modules that…
The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of $\mathbb{R}$-valued functions, the result was later cast in a…
We reinterpret algebraic de Rham cohomology for a possibly singular complex variety X as sheaf cohomology in the site of smooth schemes over X with Voevodsky's h-topology. Our results extend to the algebraic de Rham complex as well. Our…
Persistent homology provides a robust methodology to infer topological structures from point cloud data. Here we explore the persistent homology of point clouds embedded into a probabilistic setting, exploiting the theory of point…
We introduce layer systems for proving generalizations of the modularity of confluence for first-order rewrite systems. Layer systems specify how terms can be divided into layers. We establish structural conditions on those systems that…
We survey old and new results on the existence of moduli spaces of semistable coherent sheaves both in algebraic and in complex geometry.
We introduce a new notion of persistence modules endowed with operators. It encapsulates the additional structure on Floer-type persistence modules coming from the intersection product with classes in the ambient (quantum) homology, along…
To a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on…
We study sheaves of differential forms and their cohomology in the h-topology. This allows to extend standard results from the case of smooth varieties to the general case. As a first application we explain the case of singularities arising…
It is well-known that the cohomology of symmetric quandles generates robust cocycle invariants for unoriented classical and surface links. Expanding on the recently introduced module-theoretic generalized cohomology for symmetric quandles,…
Predictive coding (PC) replaces global backpropagation with local optimization over weights and activations. We show that linear PC networks admit a natural formulation as cellular sheaves: the sheaf coboundary maps activations to edge-wise…
We give a self-contained treatment of the theory of persistence modules indexed over the real line. We give new proofs of the standard results. Persistence diagrams are constructed using measure theory. Linear algebra lemmas are simplified…
A crucial step in the analysis of persistent homology is the transformation of data into an appropriate topological object (in our case, a simplicial complex). Modern packages for persistent homology often construct Vietoris--Rips or other…
The paper begins by overviewing the basic facts on geometric exceptional collections. Then, we derive, for any coherent sheaf $\cF$ on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to…
Mathematical morphology (MM) is a powerful and widely used framework in image processing. Through set-theoretic and discrete geometric principles, MM operations such as erosion, dilation, opening, and closing effectively manipulate digital…
We study sheaves on holomorphic spaces of loops and apply this to the study of the complex, defined in \cite{BdSHK}, governing deformations of the \emph{Poisson vertex algebra} structure on the space of holomorphic loops into a Poisson…
In topological data analysis, we want to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to simply random noise. While progress has been made on…