Thesis (Ed.D.)--Boston University.

This thesis presents a general background on discrete Morse theory, as developed by Robin Forman, as well as an introduction to computability and computational complexity. Since general point-set data equipped with a smooth structure can admit a triangulation, discrete Morse theory finds numerous applications in data analysis which can range from traffic control to geographical interpretation. Currently, there are various methods which convert point-set data to simplicial complexes or piecewise-smooth manifolds; however, this is not the focus of the thesis. Instead, this thesis will show that the Morse homology of such data is computable in the classical sense of Turing decidability, bound the complexity of finding the Morse homology of a given simplicial complex, and provide a measure for when this is more efficient than simplicial homology. / Master of Science / With the growing prevalence of data in the technological world, there is an emerging need to identify geometric properties (such as holes and boundaries) to data sets. However, it is often fruitless to employ an algorithm if it is known to be too computationally expensive (or even worse, not computable in the traditional sense). However, discrete Morse theory was originally formulated to provide a simplified manner of calculating these geometric properties on discrete sets. Therefore, this thesis outlines the general background of Discrete Morse theory and formulates the computational cost of computing specific geometric algorithms from the Discrete Morse perspective.

Orientadora: Profa. Dra. Mariana Rodrigues da Silveira / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , 2014. / Neste trabalho apresentamos um estudo das Teorias de Morse e Morse-Bott no contexto de sistemas dinâmicos. Consideramos uma variedade Riemanniana suave e fechada M de dimensão finita. Dada f : M ! R uma função de Morse-Smale, associamos a f o complexo de cadeia de Morse-Smale-Witten, que recupera a homologia da variedade M (Teorema de Homologia de Morse). Mais geralmente, qualquer função de Morse- Bott-Smale f :M !R pode ser associada ao complexo de cadeia de Morse-Bott-Smale, que é um multicomplexo que se reduz ao complexo de cadeia de Morse-Smale-Witten quando f é uma função de Morse. O Teorema de Homologia de Morse-Bott mostra que a homologia deste multicomplexo também coincide com a homologia de M sua prova tem como caso particular uma prova para o Teorema da Homologia de Morse. / In this work we present a study of Morse and Morse-Bott theories in the context of dynamical systems. We consider a Riemannian smooth, closed n-dimensional manifold M. Given a Morse-Smale function f :M !R, we associate f to the Morse-Smale-Witten chain complex, which recovers the homology of the manifold M (Morse Homology Theorem). More generally, any Morse-Bott-Smale function f :M !R can be associated to the Morse-Bott-Smale chain complex, which is a multicomplex that coincides with the Morse-Smale-Witten complex when f is a Morse function. The Morse-Bott Homology Theorem shows that the homology of thismulticomplex also coincides with the homology of M and its proof has as a particular case a proof for the Morse Homology Theorem.

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2009. / Title from title screen (site viewed March 2, 2010). PDF text: 1 v. (ca. 130 p.). UMI publication number: AAT 3379832. Includes bibliographical references. Also available in microfilm and microfiche formats.

O índice de Conley é uma ferramenta utilizada no estudo de sistemas dinâmicos. Em particular, as decomposições de Morse combinadas com uma apropriada versão do índice de Conley e uma correspondente equação de Morse freqüentemente nos permitem obter resultados de multiplicidade de soluções. Neste trabalho, apresentamos a teoria do índice de Conley e a equação de Morse associada a uma decomposição de Morse e aplicamos os resultados em equações diferenciais ordinárias / The Conley index is a well known tool used in the analysis of dynamical systems. In particular, Morse decompositions combined with an appropriate version of the Conley index and a corresponding Morse equation, often allow us to obtain multiplicity results for solutions. In this work we introduce the Conley index theory and the Morse equation relative to a Morse decomposition and apply the results to ordinary differential equations

Yau, Sin Wa. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 78-80). / Abstract also in Chinese. / Abstract --- p.i / Acknowledgements --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Morse Theory --- p.3 / Chapter 2.1 --- Definition --- p.3 / Chapter 2.2 --- Existence of Morse functions --- p.4 / Chapter 2.3 --- Properties of Morse functions --- p.7 / Chapter 2.4 --- The Morse homology --- p.16 / Chapter 2.4.1 --- Counting the number of flow lines with sign --- p.18 / Chapter 2.4.2 --- The Morse complex and the Morse homology --- p.19 / Chapter 2.5 --- The Morse inequality --- p.27 / Chapter 3 --- The Novikov homology --- p.29 / Chapter 3.1 --- The Novikov complex --- p.29 / Chapter 3.2 --- Relates the Novikov homology to the singular homology --- p.39 / Chapter 3.3 --- Properties of the Novikov homology --- p.43 / Chapter 3.4 --- The Novikov inequality and some applications --- p.51 / Chapter 4 --- Comparion with classical Morse theory --- p.55 / Chapter 5 --- Applications to knots and links --- p.58 / Chapter 5.1 --- Regular Morse functions --- p.58 / Chapter 5.2 --- The Morse-Novikov number --- p.74 / Chapter 5.3 --- The Universal Novikov homology --- p.76 / Bibliography --- p.78

Thèses (M.Sc.)--Université de Sherbrooke (Canada), 2004. / Titre de l'écran-titre (visionné le 20 juin 2006). Publié aussi en version papier.

Thesis (Ph. D.)--Columbia University, 1942. / Without thesis note. "First edition." Bibliography: p. 381-420.

Thesis (Ph. D.)--Columbia University, 1942. / Without thesis note. "First edition." Bibliography: p. 381-420.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1954. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 92-95).