Morse theory is a powerful tool in its applications to computational topology, computer graphics and geometric modeling. It was originally formulated for smooth manifolds. Recently, Robin Forman formulated a version of this theory for discrete structures such as cell complexes. It opens up several categories of interesting objects (particularly meshes) to applications of Morse theory. Once a Mo...

We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call "Morse 2-functions," and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is "fiber-connected," and to avoid local extrema over one-dimensional submanifolds of the range, in which case the Morse 2-function is "indefinite." This is founda...

Using graph representations a new class of computable topological invariants associated with a tame real or angle valued map were recently introduced, providing a theory which can be viewed as an alternative to MorseNovicov theory for real or angle valued Morse maps. The invariants are ”barcodes” and ”Jordan cells”. From them one can derive all familiar topological invariants which can be deriv...

The new 3and 4-manifold invariants recently constructed by Ozsváth and Szabó are based on a Floer theory associated with Heegaard diagrams. The following notes try to give an accessible introduction to their work. In the first part we begin by outlining traditional Morse theory, using the Heegaard diagram of a 3-manifold as an example. We then describe Witten’s approach to Morse theory and how ...

In this paper, we regard a space-time block of video data as a piecewise-linear 3-manifold, and we interpret video segmentation as the computation of the Morse-Smale complex for the block. In the generic case, this complex is a decomposition of space-time data into 3-dimensional cells shaped like crystals, and separated by quadrangular faces. The vertices of these are Morse critical points. In ...

This report provides theoretical justification for the use of discrete Morse theory for the computation of homology and persistent homology, an overview of the state of the art for the computation of discrete Morse matchings and motivation for an interest in these computations, particularly from the point of view of topological data analysis. Additionally, a new simulated annealing based method...

Morse theory has been considered a powerful tool in its applications to computational topology, computer graphics and geometric modeling. It was originally formulated for smooth manifolds. Recently, Robin Forman formulated a version of this theory for discrete structures such as cell complexes. It opens up several categories of interesting objects (particularly meshes) to applications of Morse ...

In classical Morse theory the number and type (index) of critical points of a smooth function on a manifold are related to topological invariants of that manifold through the Morse inequalities. There the index of a critical point is the number of negative eigenvalues that the Hessian matrix has on that tangent plane. Here deenitions of \critical point" and \index" are given that are suitable f...