An Exposition Of Discrete Morse Theory And Applications
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ORGANIZED COLLAPSE
Applied topology is a modern subject which emerged in recent years at a crossroads of many methods, all of them topological in nature, which were used in a wide variety of applications in classical mathematics and beyond. Within applied topology, discrete Morse theory came into light as one of the main tools to understand cell complexes arising in different contexts, as well as to reduce the complexity of homology calculations. The present book provides a gentle introduction into this beautiful theory. Using a combinatorial approach--the author emphasizes acyclic matchings as the central objec.
An Exposition of Discrete Morse Theory and Applications
The classical Morse theory is a powerful tool to study topological properties of a smooth manifold by examining critical points of some differentiable functions on that manifold. Robin Forman developed a discrete variant of Morse theory by adapting it on abstract simplicial complexes that resulted in a new theory with wide applications in other fields of mathematics, computer science, data science, and others. In this thesis, we present Forman's construction of discrete Morse theory, as well as its main theorems such as the Collapse theorem, discrete Morse inequalities, the theorem for cancelling critical simplices, and some additional topics. We also discuss some applications of discrete Morse theory with a major focus on the concept of persistence in topological data analysis. While many results exist in the literature, we wrote our own proofs, added more details and explanations, and adapted it to our own settings with a strong topological flavor whenever possible.
An Invitation to Morse Theory
Author: Liviu Nicolaescu
language: en
Publisher: Springer Science & Business Media
Release Date: 2011-12-02
This self-contained treatment of Morse theory focuses on applications and is intended for a graduate course on differential or algebraic topology, and will also be of interest to researchers. This is the first textbook to include topics such as Morse-Smale flows, Floer homology, min-max theory, moment maps and equivariant cohomology, and complex Morse theory. The reader is expected to have some familiarity with cohomology theory and differential and integral calculus on smooth manifolds. Some features of the second edition include added applications, such as Morse theory and the curvature of knots, the cohomology of the moduli space of planar polygons, and the Duistermaat-Heckman formula. The second edition also includes a new chapter on Morse-Smale flows and Whitney stratifications, many new exercises, and various corrections from the first edition.