ON DISCRETE MORSE THEORY
Project no. 478 · Status: Kapanmış · 2017
Summary
Discrete Morse theory devoloped by Robin Forman is one of the practical methods, by which the topology of a simplicial complex is analysed. The article discussed in this thesis is R.Forman’s paper titled ?A User’s Guide to Discrete Morse Theory?. Discrete Morse theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces, homology computation, denoising, mesh compression, and topological data analysis. Since Morse theory is best stated by the language of CW-complexes, this topic will be explained. Definitions, theorems which are the basics of the subject and definition of Morse function will be stated in this language and some examples will be given. However, the needed topological properties can also be obtained by the gradient vector fields which is the other structure has similar characteristic property as discrete Morse function. Being more useful than discrete Morse function, this structure will be explained, and some examples will be presented. At most, combinatorial point of view, Hasse diagrams, and the method of cancelling critical points which can transform a Morse function to a more useful case will be stated and Morse homology will be explained.