Detecting the dimension of a hidden manifold from a point sample has become an important problem in the current data-driven era. Indeed, estimating the shape dimension is often the first step in studying the processes or phenomena associated to the data. Among the many dimension detection algorithms proposed in various fields, a few can provide theoretical guarantee on the correctness of the es...

Detecting the dimension of a hidden manifold from a point sample has become an important problem in the current data-driven era. Indeed, estimating the shape dimension is often the first step in studying the processes or phenomena associated to the data. Among the many dimension detection algorithms proposed in various fields, a few can provide theoretical guarantee on the correctness of the es...

In this article we consider examples of Fredholm modules over the group C*-algebras of the infinite dihedral and discrete Heisenberg groups. In the first case we exhibit sufficient Fredholm modules to generate the entire K-homology of the algebra. We then turn our attention to the six term cyclic exact sequence in K-homology for a crossed product by Z dual to the Pimsner-Voiculescu sequence for...

We define a deformation of the triply graded Khovanov-Rozansky homology link $L$ depending on choice parameters $y_c$ for each component $L$, which satisfies link-splitting properties similar to Batson-Seed invariant. Keeping as formal variables yields valued in modules over $\mathbb{Q}[x_c,y_c]_{c\in \pi_0(L)}$. conjecture that this invariant restores missing $Q\leftrightarrow TQ^{-1}$ symmetr...

Recently, multi-scale notions of local homology (a variant of persistent homology) have been used to study the local structure of spaces around a given point from a point cloud sample. Current reconstruction guarantees rely on constructing embedded complexes which become difficult in high dimensions. We show that the persistence diagrams used for estimating local homology, can be approximated u...

We develop some aspects of the homological algebra persistence modules, in both one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. consider module sheaf tensor product Hom bifunctors well their derived functors, Tor Ext, give explicit computations for interval a classification injective, projective, flat state Kunneth theo...

Let A be a noetherian local commutative ring and let M be a suitable complex of A-modules. This paper proves that M is a dualizing complex for A if and only if the trivial extension A ⋉M is a Gorenstein Differential Graded Algebra. As a corollary follows that A has a dualizing complex if and only if it is a quotient of a Gorenstein local Differential Graded Algebra. Let A be a noetherian local ...

In this paper we prove a conjecture of B. Shoikhet. This conjecture states that the tangent isomorphism on homology, between the Poisson homology associated to a Poisson structure on R d and the Hochschild homology of its quantized star-product algebra , is an isomorphism of modules over the (isomorphic) respective cohomology algebras. As a consequence, we obtain a version of the Duflo isomorph...

This article treats the problem of deriving the reflector of a semi-abelian category A onto a Birkhoff subcategory B of A. Basing ourselves on Carrasco, Cegarra and Grandjeán’s homology theory for crossed modules, we establish a connection between our theory of Baer invariants with a generalization—to semi-abelian categories— of Barr and Beck’s cotriple homology theory. This results in a semi-a...