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This thesis is concerned with some algebraic and topological aspects of group actions on groupoids, w-groupoids and crossed complexes. One of our main aims is to obtain information on the homotopy groups of orbit spaces. Let A be a groupoid. w-groupold or crossed complex with an action of a group G. The algebraic part of the thesis concentrates on the orbit objects which are universal for G-morphisms into objects with trivial action. Algebraic descriptions are given for orbit groupolds and crossed complexes. Topological considerations arise as follows. We consider the fundamental groupoid of a space in dimension one, and the homotopy crossed complex of a filtered space in higher dimensions. When the space is equipped with a suitable G-actlon there is an action induced on the algebraic invariant. We prove that, under suitable conditions, the fundamental groupoid or homotopy crossed complex of the orbit space is the orbit object of the corresponding invariant of the space. In these cases the algebraic descriptions of orbit objects give information on certain relative homotopy groups of the orbit space. Finally we consider spaces equipped with a cover by subspaces, and various related groupoids. An application of G-groupoids is given to presentations of groups of homeomorphisms.

One of the great outstanding problems of theoretical physics is the quantisation of gravity, and an associated description of quantum spacetime. It is often argued that, at short distances, the manifold structure of spacetime breaks down and is replaced by some sort of algebraic structure. Noncommutative geometry is a possible candidate for the mathematics of this structure. However, physical theories on noncommutative spaces are still essentially classical and need to be quantised. We present a path integral formalism for quantising gravity in the form of the spectral action. Our basic principle is to sum over all Dirac operators. The approach is demonstrated on two simple finite noncommutative geometries (the two-point space and the matrix geometry M(_2)(C)) and a circle. In each case, we start with the partition function and calculate the graviton propagator and Greens functions. The expectation values of distances are also evaluated. We find on the finite noncommutative geometries, distances shrink with increasing graviton excitations, while on a circle, they grow. A comparison is made with Rovelli's canonical quantisation approach, and with his idea of spectral path integrals. We also briefly discuss the quantisation of a general Riemannian manifold. Included, is a comprehensive overview of the homological aspects of noncommutative geometry. In particular, we cover the index pairing between K-theory and K-homology, KK-theory, cyclic homology/cohomology, the Chern character and the index theorem. We also review the various field theories on noncommutative geometries.

This thesis investigates the dynamics of lump-like objects in non-integrable field theories, whose stability is due to topological considerations. The work concerns three different low dimensional ((1 + 1)- and (2 + l)-dimensional) systems and addresses the questions of how the topology and metric structure of physical space, the quantum mechanics of the basic field quanta and intersoliton interactions affect soliton dynamics. In chapter 2 a sine-Gordon system in discrete space, but with continuous time, is presented. This has some novel features, namely a topological lower bound on the energy of a kink and an explicit static kink which saturates this bound. Kink dynamics in this model is studied using a geodesic approximation which, on comparison with numerical simulations, is found to work well for moderately low kink speeds. At higher speeds the dynamics becomes significantly dissipative, and the approximation fails. Some of the dissipative phenomena observed are explained by means of a dispersion relation for phonons on the spatial lattice. Chapter 3 goes on to quantize the kink sector of this model. A quantum induced potential called the kink Casimir energy is computed numerically in the weak coupling approximation by quantizing the lattice phonons. The effect of this potential on classical kink dynamics is discussed. Chapter 4 presents a study of the low-energy dynamics of a CP(^1) lump on the two-sphere in the geodesic approximation. By considering the isometry group inherited from globalsymmetries of the model, the structure of the induced metric on the unit-charge moduli space is so restricted that the metric can be calculated explicitly. Some totally geodesic submanifolds are found, and the qualitative features of motion on these described. The moduli space is found to be geodesically incomplete. Finally, chapter 5 contains an analysis of long range intervortex forces in the abelian Higgs model, a massive field theory, extending a point source. approximation previously only used in massless theories. The static intervortex potential is rederived from a new viewpoint and used to model type II vortex scattering. Velocity dependent forces are then calculated, providing a model of critical vortex scattering, and leading to a conjecture for the analytic asymptotic form of the metric on the two-vortex moduli space.

This thesis is a study in the field of partial differential equations on differentiable manifolds. In particular non-linear evolution equations with solution solutions are studied by means of differential geometric tools and methods. Differential geometric prolongation technique is applied to the A.K.N.S. system as a unifying system for known 2-dimension solutions. Solution properties are studied in this differential geometric set up. The results are used to obtain a possible model for n-dimensional solutions.

In this thesis, we investigate properties of manifolds with Riemannian metrics which satisfy conditions more general than those of Einstein metrics, including the latter as special cases. The Einstein condition is well known for being the Euler- Lagrange equation of a variational problem. There is not a great deal of difference between such metrics and metrics with Ricci tensor parallel for the latter are locally Riemannian products of the former. More general classes of metrics considered include Ricci- Codazzi and Ricci cyclic parallel. Both of these are of constant scalar curvature. Our study is divided into three parts. We begin with certain metrics in 4-dimensions and conclude our results with three theorems, the first of which is equivalent to a result of Kasner [Kal] while the second and part of the third is known to Derdzinski [Del.2].Next we construct the metrics mentioned above on spheres of odd dimension. The construction is similar to Jensen's [Jel] but more direct and is due essentially to Gray and Vanhecke [GV]. In this way we obtain .beside the standard metric, the second Einstein metric of Jensen. As for the Ricci- Codazzi metrics, they are essentially Einstein, but the Ricci cyclic parallel metrics seem to form a larger class. Finally, we consider subalgebras of the exceptional Lie algebra g2. Making use of computer programmes in 'reduce' we compute all the corresponding metrics on the quotient spaces associated with G2.

The work in this thesis is concerned with the study of stability and scattering of solitons in planar models ie where spacetime is (2+l)-dimensional. We consider both integrable models, where exact solutions can be written in closed form, and non-integrable models, where approximations and numerical methods must be employed. In chapter III we use a 'collective coordinate' approximation to study the scattering of solitons in a model motivated by elementary particle physics. In chapter IV we discuss a method to obtain approximate soliton configurations which can then be used to investigate soliton dynamics. In chapter V we perform a test of the 'collective coordinate' approximation by applying it to the study of classical and quantum soliton scattering in an integrable model, where exact results are known. Chapters VI and VII are concerned with an integrable chiral model. First we construct exact solutions using twistor methods and then we go on to study soliton stability using numerical techniques. Through computer simulations we find that there exist solitons which scatter in a way unlike any previously found in integrable models. Furthermore, this soliton scattering resembles very closely that found in certain non-integrable models, thereby providing a link between the two classes. Finally, chapter VIII is an outlook on current and future research.

This thesis is concerned with various soliton solutions to some of the affine Toda field theories. These are field theories in 1+1 dimensions that possess a rich underlying Lie algebraic structure and they are known to be integrable. The soliton solutions occur as a result of the multi-vacua that appear in the field theory when the coupling constant is taken to be purely imaginary. In chapter one a review of the affine Toda field theories is undertaken. This is meant to be a relatively complete and exhaustive survey of the literature that has appeared on the subject in recent years. A brief introduction to the theory of solitons and the methods of obtaining such solutions in field theory is given in chapter two, resulting in the construction of the relevant machinery for the Toda theories. In chapter three, Hi rota's method is used to construct single and double soliton solutions to these theories. As a consequence of these explicit formulae the fusing structure of the solitons may be investigated and shown to be equivalent to that found in the classical particle regime, supplemented by further 'annihilations' of 'soliton-antisoliton'. The calculations of the double soliton solutions are claimed to be original in this context. The fusing has also been examined by Olive, Turok and Underwood(^16) through an abstract group-theoretical approach to the affine Toda field theories, however very few explicit formulae are given by them, and hence all the solutions given here are important in their own right. An algebra-independent analysis of such phenomena is undertaken in chapter four where a vertex operator construction is given for the relevant interaction functions. Some properties of these functions are noted; (some of these facts correspond with those in [16] concerning the fusing structure of the solitons).

This thesis is concerned with Hopf hypersurfaces of Kähler and nearly Kähler manifolds and gives special emphasis to the cases of hypersurfaces of complex projective spaces and of the 6-sphere endowed with its nearly Kähler almost complex structure. Although there is already a wealth of investigations done in the case of complex space forms and the 6-sphere, a full classification of these hypersurfaces in the former spaces was done under assumption of constancy of the rank of its focal map. Here, the classification is revisited and this assumption is removed although a complete classification is still not obtained. The characterization of the Hopf hypersurfaces of the 6-sphere as tubular hypersurfaces around almost complex curves is used to determine among these hypersurfaces special examples which have constant mean curvature or are Einstein hypersurfaces. The invariants needed to decide when a pair of hypersurfaces of S(^6) and CP(^n) are respectively G(_2)-congruent and holomorphically congruent are determined and this result is applied to characterize the hypersurfaces of these spaces whose Hopf vector fields are also Killing field. Finally, the linearly full almost complex 2-spheres of S(^6) with at most two singularities are determined up to G(^C)(_2)-congruence of their directrix curves and this is used to determine the space of linearly full almost complex 2-spheres of S(^6) with suitably small induced area.

Let Г be a metacyclic group of order pq with p and q prime. We shall show that the Г-cohomology and character of a Г-lattice determine its genus. Let N/L be a Galois extension with group Г, then U(_N), the torsion-free units of N, is a Г-lattice and the isomorphism Q o U(_N) = Q o ɅS(_oo) gives its character. In certain cases we can determine its cohomology and thus its genus; in particular, when = h(_N) = 1 and L = Q we show that the genus of U(_N) depends only on the number of non-split, ramified primes in N/L. We shall also investigate U(_N) in the factorizability defect Grothendieck group.