Related papers: Computing Connection Matrices of Conley Complexes …
The connection matrix is a powerful algebraic topological tool from Conley index theory that captures relationships between isolated invariant sets. Conley index theory is a topological generalization of Morse theory in which the connection…
Connection matrices are one of the central tools in Conley's approach to the study of dynamical systems, as they provide information on the existence of connecting orbits in Morse decompositions. They may be considered a generalisation of…
Connection matrices are a generalization of Morse boundary operators from the classical Morse theory for gradient vector fields. Developing an efficient computational framework for connection matrices is particularly important in the…
Reducing a chain complex (whilst preserving its homotopy-type) using algebraic Morse theory gives the same end-result as Gaussian elimination, but AMT does it only on certain rows/columns and with several pivots (in all matrices…
In this work we announce the Maple package conley to compute connection and C-connection matrices. conley is based on our abstract homological algebra package homalg. We emphasize that the notion of braids is irrelevant for the definition…
At the intersection of Topological Data Analysis (TDA) and machine learning, the field of cellular signal processing has advanced rapidly in recent years. In this context, each signal on the cells of a complex is processed using the…
This is a self-contained tour of the Conley index and connection matrices. The starting point is Conley's fundamental theorem of dynamical systems. There is a short stop at the necessary topological background, before we proceed to the…
We introduce the notion of a template for discrete Morse theory. Templates provide a memory efficient approach to the computation of homological invariants (e.g., homology, persistent homology, Conley complexes) of cell complexes. We…
Cartan-Eilenberg systems play an prominent role in the homological algebra of filtered and graded differential groups and (co)chain complexes in particular. We define the concept of Cartan-Eilenberg systems of abelian groups over a poset.…
E. Sk\"oldberg's Morse Theory from an Algebraic Viewpoint and M. J\"ollenbeck's Algebraic Discrete Morse Theory and Applications to Commutative Algebra, which is the algebraic generalization of R. Forman's discrete Morse Theory for Cell…
We apply the homomorphism complex construction to partially ordered sets, introducing a new topological construction based on the set of maximal chains in a graded poset. Our primary objects of study are distributive lattices, with special…
In this paper, we develop the notion of a Morse sequence, which provides an alternative approach to discrete Morse theory, and which is both simple and effective. A Morse sequence on a finite simplicial complex is a sequence composed solely…
Topological data analysis can reveal higher-order structure beyond pairwise connections between vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and…
A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. By [11], a hypergraph gives an associated simplicial complex. By [4], the embedded homology of a hypergraph is the homology of the infimum chain…
We study Morse theory on noncompact manifolds equipped with exhaustions by compact pieces, defining the Morse homology of a pair which consists of the manifold and related geometric/homotopy data. We construct a collection of Morse data…
A theorem is established where the computation of the discrete Conley index for zero dimensional basic sets is given with respect to the dynamical information contained in the associated structure matrices. A classification of the reduced…
We study the problem of computing the homology of the configuration spaces of a finite cell complex $X$. We proceed by viewing $X$, together with its subdivisions, as a subdivisional space--a kind of diagram object in a category of cell…
We present a parallelizable algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then…
Morse decompositions partition the flows in a vector field into equivalent structures. Given such a decomposition, one can define a further summary of its flow structure by what is called a connection matrix.These matrices, a generalization…
In the context of discrete Morse theory, we introduce Morse frames, which are maps that associate a set of critical simplexes to all simplexes. The main example of Morse frames are the Morse references. In particular, these Morse references…