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

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1991. / Includes bibliographical references (p. 106-111).

No description available.

No description available.

Kwong, Kwok Kun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 105-109). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgements --- p.iii / Introduction --- p.1 / Chapter 1 --- Morse Theory --- p.4 / Chapter 1.1 --- Introduction --- p.4 / Chapter 1.2 --- Morse Homology --- p.11 / Chapter 2 --- Symplectic Fixed Points and Arnold Conjecture --- p.24 / Chapter 2.1 --- Introduction --- p.24 / Chapter 2.2 --- The Variational Approach --- p.29 / Chapter 2.3 --- Action Functional and Moduli Space --- p.30 / Chapter 2.4 --- Construction of Floer Homology --- p.42 / Chapter 3 --- Fredholm Theory --- p.46 / Chapter 3.1 --- Fredholm Operator --- p.47 / Chapter 3.2 --- The Linearized Operator --- p.48 / Chapter 3.3 --- Maslov Index --- p.50 / Chapter 3.4 --- Fredholm Index --- p.57 / Chapter 4 --- Floer Homology --- p.75 / Chapter 4.1 --- Transversality --- p.75 / Chapter 4.2 --- Compactness and Gluing --- p.76 / Chapter 4.3 --- Floer Homology --- p.88 / Chapter 4.4 --- Invariance of Floer Homology --- p.90 / Chapter 4.5 --- An Isomorphism Theorem --- p.98 / Chapter 4.6 --- Further Applications --- p.103 / Bibliography --- p.105

No description available.

We introduce a new class of Conley-Morse chain maps for the purpose of comparing the qualitative structure of flows across multiple scales. Conley index theory generalizes classical Morse theory as a tool for studying the dynamics of flows. The qualitative structure of a flow, given a Morse decomposition, can be stored algebraically as a set of homology groups (Conley indices) and a boundary map between the indices (a connection matrix). We show that as long as the qualitative structures of two flows agree on some, perhaps coarse, level we can construct a chain map between the corresponding chain complexes that preserves the relations between the (coarsened) Morse sets. We present elementary examples to motivate applications to data analysis.

Univ., Diss.--Stockholm, 2005. / Includes bibliographical references (p. [361] - 369) and index.

Typescript. Includes vita and abstract. Includes bibliographical references (leaves 163-164) Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

The introduction of supersymmetry has led to great progress in the study of quantum field theories. Notably, with supersymmetry, properties of a quantum field theory can be computed with higher precision than what would otherwise be possible. In this project, we investigate supersymmetry in the context of quantum mechanics. In particular, we show how the Witten index is insensitive to the details of the supersymmetric quantum mechanical system, making it a robust quantity when considering variations in the system’s parameters. Explicit calculations of the supersymmetric ground states are carried out to identify what determines the Witten index. The concept of superpotential is introduced and we relate Morse theory to the Witten index by identifying the superpotential as a Morse function. Moreover, we consider supersymmetric quantum mechanics on compact orientable Riemann manifolds. We show how the structure of supersymmetric quantum mechanics has a close connection to topological properties of the target manifolds. Specifically, the Witten index is shown to be the Euler characteristic, a topological invariant.