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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.

The object of my thesis is to investigate projectively related metrics, that is, metrics whose Levi-Civita connections admit exactly the same family of unparametrised geodesics on 4-dimensional manifolds with positive de nite or neutral (+;+;;) signatures. The general idea is to study the relationship between projectively related metrics and the holonomy types of each metric. The main technique presented in the work requires a certain classification of the curvature map which has been developed by G. S. Hall and D. P. Lonie in the case of Lorentz signature. In chapter 1, some of the background theory will be given. This will include an introduction to bivector algebra, a revision of the Riemann curvature tensor and holonomy theory and, in particular, the fundamental equations for projective related metrics. A brief historical and bibliographical review is also given. The subsequent chapter gives the details of projective related metrics of positive definite signature. In x2.1, the structure of so(4) is described with an emphasis on the canonical decomposition of bivectors and then the subalgebras of so(4) follow. In x2.2., the problem of projective related metrics can be solved case by case decided by holonomy types. In many of these cases, the connections are found to be necessarily equal. A few cases with nontrivial projectively related metrics have been found by only in the rather special case of curvature class D, and the metrics are given in the appendices. An extension of this method to spaces of neutral signature (+;+;;) is made in chapter 3. The rst part of the chapter discusses the algebraic structure of a 4-dimensional vector space with such a metric. In contrast to metrics of the other two signatures (positive definite and Lorentz), this metric admits totally null planes. The structure of the Lie algebra so(2; 2) can be described through the action on totally null planes. The classification of all subalgebras of so(2,2) is then obtained in terms of self-dual and anti-self-dual bivectors. In most holonomy types and curvature classes, the problem has only trivial solutions. Nontrivial projectively related metrics can be found for four holonomy types with curvature class D and two holonomy types with curvature class A.

published_or_final_version / abstract / toc / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

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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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