Thesis--University of Wisconsin--Madison. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves [145-147]).

by Leung Kwok Kei. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 90-94). / Chapter 0 --- Introduction --- p.1 / Chapter 1 --- Comparison Theorems --- p.4 / Chapter 1.1 --- Introduction --- p.4 / Chapter 1.2 --- Bishop Comparison Theorem --- p.5 / Chapter 1.3 --- Hessian Comparison Theorem --- p.13 / Chapter 1.4 --- Applications of Laplacian comparison theorem --- p.16 / Chapter 2 --- "The first eigenvalue, gradient estimates and related inequalities" --- p.21 / Chapter 2.1 --- Lower bounds of the first eigenvalue --- p.21 / Chapter 2.2 --- Gradient estimate and Harnack inequality --- p.29 / Chapter 2.3 --- Mean value inequality --- p.33 / Chapter 2.4 --- Isoperimetric inequalities and Sobolev inequalities --- p.40 / Chapter 3 --- Harmonic mappings --- p.47 / Chapter 3.1 --- Definitions and notations --- p.47 / Chapter 3.2 --- Existence results --- p.50 / Chapter 3.3 --- Bochner technique --- p.54 / Chapter 3.4 --- Strong rigidity theorems --- p.58 / Chapter 3.5 --- Superrigidity and harmonic maps --- p.71 / Appendix --- p.75 / Chapter A --- Bochner formula and its integral form --- p.76 / Chapter A.1 --- Bochner formula --- p.76 / Chapter A.2 --- Reilly's formula and its applications --- p.82 / Bibliography --- p.90

by Wu Man Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 60-62). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Harmonic functions with linear growth --- p.3 / Chapter 2.1 --- A sharp estimate for dim H1 (M) --- p.3 / Chapter 2.2 --- Linear growth harmonic functions on Kahler manifolds --- p.8 / Chapter 3 --- Harmonic functions of polynomial growth --- p.21 / Chapter 3.1 --- Harmonic sections of polynomial growth --- p.21 / Chapter 3.2 --- Harmonic functions on manifolds with Sobolev in- equality --- p.34 / Chapter 4 --- Harmonic functions on manifolds with nonnegat --- p.ive / sectional curvature --- p.43 / Bibliography --- p.60

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

Wu, Lijiang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 102-105). / Abstracts in English and Chinese. / Chapter 0 --- Introduction --- p.6 / Chapter 1 --- Background knowledge --- p.9 / Chapter 1.1 --- Comparison theorems --- p.9 / Chapter 1.2 --- Bochner techniques --- p.13 / Chapter 1.3 --- Eigenvalue estimates for Laplacian operator --- p.14 / Chapter 1.4 --- Spectral theory for Schrodinger operator on Rieman- nian manifolds --- p.16 / Chapter 2 --- Vanishing theorems --- p.20 / Chapter 2.1 --- Liouville type theorem for Lp subharmonic functions --- p.20 / Chapter 2.2 --- Generalized type of vanishing theorem --- p.21 / Chapter 3 --- Finite dimensionality results --- p.28 / Chapter 3.1 --- Three types of integral inequalities --- p.28 / Chapter 3.2 --- Weak Harnack inequality --- p.34 / Chapter 3.3 --- Li's abstract finite dimensionality theorem --- p.37 / Chapter 3.4 --- Applications of the finite dimensionality theorem --- p.42 / Chapter 4 --- Ends of Riemannian manifolds --- p.48 / Chapter 4.1 --- Green's function --- p.48 / Chapter 4.2 --- Ends and harmonic functions --- p.53 / Chapter 4.3 --- Some topological applications --- p.72 / Chapter 5 --- Splitting theorems --- p.79 / Chapter 5.1 --- Splitting theorems for manifolds with non-negative Ricci curvature --- p.79 / Chapter 5.2 --- Splitting theorems for manifolds of Ricci curvature with a negative lower bound --- p.83 / Chapter 5.3 --- Manifolds with the maximal possible eigenvalue --- p.93 / Bibliography --- p.102

Thesis (Ph. D.)--Oregon State University, 1980. / Typescript (photocopy). Includes bibliographical references. Also available on the World Wide Web.

Symplectically aspherical manifolds rst appeared in the work of Floer where he proved a version of the Arnold Conjecture. Since then their topological properties have been studied. One of the results of the present thesis are new characterisations of symplectically aspherical manifolds. Namely, we prove in Chapter 4 that a closed symplectic manifold is symplectically aspherical if and only if one of the following conditions hold: Its universal cover can be symplectically embedded into the standard sym- plectic Euclidean space. Its fundamental group is large (see De nition 4.7 for the de nition of large- ness). The latter condition has a well known counterpart in algebraic geometry. It has been conjectured by Shafarevich (see [32]) that a closed algebraic variety has a Stein universal cover if and only if its fundamental group is large (in the algebraic sense). A manifold is Stein if it is holomorphically embedded into the standard complex vector space. Thus the characterisations of symplectically aspherical manifolds mentioned above prove a symplectic analog of the Shafarevich conjecture. One may ask whether the universal cover of a symplectically aspherical manifold is Stein. In Chapter 5, we construct an example of a closed symplectically aspherical manifold whose universal cover is not Stein. This is the second main result of the thesis. In Chapter 6, we focus on the properties of the fundamental group of symplectically aspherical manifolds. The main result here is to relate the Flux group of a symplecti- cally aspherical manifold with its geometric properties. More precisely, we prove that if the Flux group is nontrivial then the manifold is not symplectically hyperbolic. It is not known that if a nontrivial free product of groups can be realised as the fundamental group of a closed symplectically aspherical manifold. In Chapter 5, we construct an example of a closed symplectically aspherical manifold whose universal cover is not Stein. This is the second main result of the thesis.In Chapter 6, we focus on the properties of the fundamental group of symplectically aspherical manifolds. The main result here is to relate the Flux group of a symplecti- cally aspherical manifold with its geometric properties. More precisely, we prove that if the Flux group is nontrivial then the manifold is not symplectically hyperbolic. It is not known that if a nontrivial free product of groups can be realised as the fundamental group of a closed symplectically aspherical manifold. In Chapter 5, we investigate a more general problem to obtain some partial results. The remaining parts of the thesis introduce symplectic and symplectically aspherical manifolds and review what is known in the subject.

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There are basically two types of variables in population modelling, global and local variables. The former describes the behavior of the entire population while the latter describes the behavior of individuals within this population. The description of the population using local variables is more detailed, but it is also computationally costly. In many cases to study the dynamics of this population, it is sufficient to focus only on global variables. In applied sciences, to achieve this, the method of aggregation of variables is used. One of methods used to mathematically justify variables aggregation is the centre manifold theory. In this dissertation we provide detailed proofs of basic results of the centre manifold theory and discuss some examples of applications in population modelling. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2009.

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