We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A, B and D. This extends and explains the “splitting basis” for the homology of the partition lattice given in [20], thus answering a question asked by R. Stanley. More explicitly, the following general technique is presented and utilized. Let A be a central and essential hype...

As a step towards establishing homotopy-theoretic foundations for topological data analysis (TDA), we introduce and study homotopy interleavings between filtered spaces. These are homotopy-invariant analogues of interleavings, objects commonly used in TDA to articulate stability inference theorems. Intuitively, whereas strict interleaving spaces <inline-formula content-type="ma...

This paper presents a new hybrid split-and-merge image segmentation method based on computational geometry and topology using persistent homology. The algorithm uses edge-directed topology to initially split the image into a set of regions based on the Delaunay triangulations of the points in the edge map. Persistent homology is used to generate three types of regions: p-persistent regions, p-t...

when the past observations are correlated with future observations and their correlation is significant, the time series has long memory. in this paper the contagion effect of volatilities, with consideration of long-run effect, is investigated. the basic model is bekk (1, 1) and fbekk (1,d,1), model extended long-run memory parameter (d) is considered and estimated. furthermore in this paper p...

We construct an action of a Lie algebra on the homology groups of moduli spaces of stable sheaves on K3 surfaces under some technical conditions. This is a generalization of Nakajima’s construction of sl2-action on the homology groups [N6]. In particular, for an A, D, E-configulation of (−2)-curves, we shall give a collection of moduli spaces such that the associated Lie algebra acts on their h...

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...

and moreover that HH2(O/R) = 0. These two facts greatly facilitate the computation of the cyclic homology HC∗(O/R) ([3]). This paper presents non-commutative analogues of the main results of [3]. In particular, L is to be replaced by a central simple algebra D over K, while O is taken to be a maximal R-order in D. Of course, O is not, in general, uniquely defined by R and D, but it turns out th...

We study the homology of a filtered d-dimensional simplicial complex K as a single algebraic entity and establish a correspondence that provides a simple description over fields. Our analysis enables us to derive a natural algorithm for computing persistent homology over an arbitrary field in any dimension. Our study also implies the lack of a simple classification over non-fields. Instead, we ...

The concept of an adapted homology basis for a prime order conformal automorphism of a compact Riemann surface originated in [6, 7, 8, 9] and is extended to arbitrary finite groups of conformal automorphisms in [12]. Here we compute some examples of adapted homology bases for some groups of automorphisms. The method is to begin by apply the Schreier-Reidemeister rewriting process and the Schrei...

In [28], Lawrence Roberts, extending the work of Ozsváth and Szabó in [23], showed how to associate to a link, L, in the complement of a fixed unknot, B ⊂ S, a spectral sequence whose E term is the Khovanov homology of a link in a thickened annulus defined in [2], and whose E term is the knot Floer homology of the preimage of B inside the double-branched cover of L. In [6], we extended [23] in ...