In this thesis, we prove that irreducible residual Fontaine-La aille representations of the absolute Galois group of an unrami ed extension of Qp have smooth representable crystalline framed deformation problems, provided that the Hodge-Tate weights lie in the Fontaine-La aille range. We then extend this result to the agged lifting problem associated to any Fontaine-La aille upper triangular representation whose ag is of maximal length. We calculate the relative dimension of these various crystalline lifting functors in terms of the underlying Hodge-Tate weight structures, and also apply these results to give an alternative proof of the fact that every such residual representation admits a so-called \universally twistable lift". Finally we give some brief indications as to the various directions in which these results might be generalised.

For any group G and generating set X we shall be primarily concerned with three sets of words over X: the word problem, the reduced word problem, and the irreducible word problem. We explain the relationships between these three sets of words and give necessary and sufficient conditions for a language to be the word problem (or the reduced word problem) of a group. We prove that the groups which have context-free reduced word problem with respect to some finite monoid generating set are exactly the context-free groups, thus proving a conjecture of Haring-Smith. We also show that, if a group G has finite irreducible word problem with respect to a monoid generating set X, then the reduced word problem of G with respect to X is simple. In addition, we show that the reduced word problem is recursive (or recursively enumerable) precisely when the word problem is recursive. The irreducible word problem corresponds to the set of words on the left hand side of a special rewriting system which is confluent on the equivalence class containing the identity. We show that the class of groups which have monoid presentations by means of finite special []-confluent string-rewriting systems strictly contains the class of plain groups (the groups which are free products of a finitely generated free group and finitely many finite groups), and that any group which has an infinite cyclic central subgroup can be presented by such a string-rewriting system if and only if it is the direct product of an infinite cyclic group and a finite cyclic group.

Torsion is a property of space-time which is not incorporated into the standard formulation of general relativity but which appears as a consequence of unification schemes for fundamental forces. It is, therefore, important to understand its physical consequence. This thesis begins with an introduction to a non-propagating version of torsion theory as an extension to general relativity. The theory can be described in terms of a pair coupled field equations with torsion algebraically linked to elementary particle spin. In order to develop the theory it is necessary to postulate a form for the energy-momentum tensor of spinning matter which is not prescribed in the classical domain. The two main candidates that have been proposed for a spinning fluid are considered. Chapter two contains an independent reworking of Zerilli's [1] perturbation calculation of a particle falling into a Schwarzschild black hole. The perturbation equations are found and the resulting wave equations are derived. The special case of a particle falling radially is considered in detail. Chapter three contains new work which employs the method of Zerilli in torsion theory to consider a particle with spin falling radially into a black hole. The changes to the black hole are found for each of the two energy-momentum tensors of Chapter one. This enables us to discount one of these as unphysical. The differential equations describing the gravitational radiation released by this system are derived. Finally in Chapter four these equations are solved to find the gravitational radiation from a spinning particle falling radially. These may be significant for observational assessments of torsion theory.

This thesis makes manifest the roles of the normed division algebras R,C,H and O in various supergravity theories. Of particular importance are the octonions O, which frequently occur in connection with maximal supersymmetry, and hence also in the context of string and M-theory. Studying the symmetries of M-theory is perhaps the most straightforward route towards understanding its nature, and the division algebras provide useful tools for such study via their deep relationship with Lie groups. After reviews of supergravity and the definitions and properties of R,C,H and O, a division-algebraic formulation of pure super Yang-Mills theories is developed. In any spacetime dimension a Yang-Mills theory with Q real supercharge components is written over the division algebra with dimension Q/2. In particular then, maximal Q = 16 super Yang-Mills theories are written over the octonions, since O is eight-dimensional. In such maximally supersymmetric theories, the failure of the supersymmetry algebra to close off-shell (using the conventional auxiliary field formalism) is shown to correspond to the non-associativity of the octonions. Making contact with the idea of 'gravity as the square of gauge theory', these division-algebraic Yang-Mills multiplets are then tensored together in each spacetime dimension to produce a pyramid of supergravity theories, with the Type II theories at the apex in ten dimensions. The supergravities at the base of the pyramid have global symmetry groups that fill out the famous Freudenthal-Rosenfeld-Tits magic square. This magic square algebra is generalised to a 'magic pyramid algebra', which describes the global symmetries of each Yang-Mills-squared theory in the pyramid. Finally, a formulation of eleven-dimensional supergravity over the octonions is presented. Toroidally compactifying this version of the theory to four or three spacetime dimensions leads to an interpretation of the dilaton vectors (which organise the coupling of the seven or eight dilatons to the other bosonic fields) as the octavian integers - the octonionic analogue of the integers.

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The problem of classifying the reduced phase spaces of the natural torus action on a generic coadjoint orbit of SU(n) is considered. The concept of a stratified fibre bundle is defined. It is proved that the orbit map of an equivariant fibre bundle is a stratified fibre bundle. This result is then used to give an iterative description of the reduced phase spaces of the torus action on a generic coadjoint orbit of SU(n). The theory is illustrated with a detailed examination of the n = 3 case, that of the two torus action on a coadjoint orbit of SU(3).

One concern in the homological theory of Banach algebras is the identification of projective algebras and projective closed ideals of algebras. Besides being of independent interest, this question is closely connected to the continuous Hochschild cohomology. In this thesis we give necessary and sufficient conditions for the left projectivity and biprojectivity of Banach algebras defined by locally trivial continuous fields of Banach algebras. We identify projective C*-algebras A defined by locally trivial continuous fields U = fW, (At)t2W,Qg such that each C*-algebra At has a strictly positive element. We also identify projective Banach algebras A defined by locally trivial continuous fields U = fW, (K(Et))t2W,Qg such that each Banach space Et has an extended unconditional basis. In particular, for a left projective Banach algebra A defined by locally trivial continuous fields U = fW, (At)t2W,Qg we prove that W is paracompact. We also show that the biprojectivity of A implies that W is discrete. In the case U is a continuous field of elementary C*-algebras satisfying Fell’s condition (not nessecarily a locally trivial field) we show that the left projectivity of A defined by U, under some additional conditions on U, implies paracompactness of W. For the above Banach algebras A we give applications to the second continuous Hochschild cohomology group H2(A, X) of A and to the strong splittability of singular extensions of A.

The objects of study in this thesis are partially commutative and differential graded algebraic structures. In fact my thesis is in two parts. The first on partially commutative algebraic structures is concerned with automorphism groups of partially commutative groups and their finite presentations. The second on differential graded algebraic structures is concerned with differential graded modules.

We use the classification of finite simple groups to verify a conjecture of Robinson for finite groups G where G/Op(G) has trivial intersection Sylow p-subgroups. Groups of this type are said to have p-local rank one, and it is hoped that this invariant will eventually form the basis for inductive arguments, providing reductions for the conjecture, or even a proof using the results presented here as a base. A positive outcome for Robinson's conjecture would imply Alperin's weight conjecture. It is shown that in proving Robinson's conjecture it suffices to demonstrate only that it holds for finite groups in which Op(G) is both cyclic and central. Part of the proof of the former result is used to complete the verification of Dade's inductive conjecture for the Ree groups of type G2.

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