The chain complexes underlying Floer homology theories typically carry a real-valued filtration, allowing one to associate to each Floer homology class a spectral number defined as the infimum of the filtration levels of chains representing that class. These spectral numbers have been studied extensively in the case of Hamiltonian Floer homology by Oh, Schwarz, and others. We prove that the spe...

Persistent homology is a mathematical tool from topological data analysis. It performs multi-scale analysis on a set of points and identifies clusters, holes, and voids therein. These latter topological structures complement standard feature representations, making persistent homology an attractive feature extractor for artificial intelligence. Research on persistent homology for AI is in its i...

Abstract This paper is intended to introduce a filtration analysis of sampled maps based on persistent homology, providing new method for reconstructing the underlying maps. The key idea extend definition homology induced correspondences using framework quiver representations. Our given by most direct summands uniquely determine homology. We provide stability theorems this process and show that...

Let Q be a finite group acting on a group G as a group of automorphisms. This action induces an action of Q on the standard bar complex C•(G) for computing the homology of G; denote the subcomplex of invariants by C•(G) Q. In this paper we study the homology of the complex C•(G) Q. The resulting homology groups are denoted H • (G). Before summarizing the results, a brief history is in order. A ...

Let B be a commutative algebra and A B-algebra (determined by an homomorphism ?:B?A). M. D. Staic introduced Hochschild like cohomology H•((A,B,?);A) called secondary cohomology, to describe the non-trivial deformations of A. J. Laubacher et al later obtained natural construction new chain complex C¯•(A,B,?) in process introducing cyclic (co)homology. In this paper, we establish connection betw...

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 prove that every [Formula: see text]H-thin link has no text]-torsion for text] in its Khovanov homology. Together with previous results by Lee [The support of the Khovanov’s invariants alternating knots, preprint (2002), arXiv:math.GT/0201105; An endomorphism invariant, Adv. Math. 197(2) (2005) 554–586, arXiv:math.GT/0210213] and author [Torsion homology, Fund. 225 (2014) 343–364, arXiv:math...

We define persistent homology groups over any set of spaces which have inclusions defined so that the underlying graph between the spaces is directed and acyclic. This method simultaneously generalizes standard persistent homology, zigzag persistence and multidimensional persistence to arbitrary directed acyclic graphs, and it also allows the study of arbitrary families of topological spaces or...