We introduce an algorithm for construction of the Morse hierarchy, i.e. a hierarchy of Morse decompositions of a piecewise constant vector field on a surface driven by stability of the Morse sets with respect to perturbation of the vector field. Our approach builds upon earlier work on stable Morse decompositions, which can be used to obtain Morse sets of user-prescribed stability. More stable ...

A brief overviewof Forman’s discrete Morse theory is presented, from which analogues of the main results of classical Morse theory can be derived for discrete Morse functions, these being functions mapping the set of cells of a CW complex to the real numbers satisfying some combinatorial relations. The discrete analogue of the strong Morse inequality was proved by Forman for finite CW complexes...

The connection matrix theory for Morse decompositions is introduced. The connection matrices are matrices of maps between the homology indices of the sets in the Morse decomposition. The connection matrices cover, in a natural way, the homology index braid of the Morse decomposition and provide information about the structure of the Morse decomposition. The existence of connection matrices of M...

The propose a novel algorithm that compute a skeletal graph and thus capture the topology of an object Topology is an important attribute of an object which describes how different parts of an object surface are connected to each other. The method is based on capturing the topology of a modified reeb graph by tracking the critical points of a distance function. The algorithm for constructing th...

The travel time brachistochrone curves in a general relativistic framework are timelike curves, satisfying a suitable conservation law with respect to a an observer field, that are stationary points of the travel time functional. In this paper we develop a global variational theory for brachistochrones joining an event p and the worldline of an observer γ in a stationary spacetime M. More speci...

For a labelled tree on the vertex set [n] := {1, 2, . . . , n}, define the direction of each edge ij to be i → j if i < j. The indegree sequence of T can be considered as a partition λ ⊢ n − 1. The enumeration of trees with a given indegree sequence arises in counting secant planes of curves in projective spaces. Recently Ethan Cotterill conjectured a formula for the number of trees on [n] with...

In [13] a homology theory –Morse-Conley-Floer homology– for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology theory. As a preliminary we investigate functoriality in Morse homology. Functoriality for Morse homology of closed manifolds is known [1, 2, 3, 8, 14], but the proofs use isom...

1. I n t roduc t i on Discrete Morse theory was developed by Forman [9, 11] as a combinatorial analog to the classical smooth Morse theory. Applications to questions in combinatorial topology and related fields are numerous: e.g., Babson et al. [3], Forman [10], Batzies and Welker [4], and Jonsson [20]. I t turns out that the topologically relevant information of a discrete Morse function f on ...