In this publication, we investigate the groups H(m, n, p, q |k) defined by the presentation < x, y|x^m, y^n, (xy)^p, (xy^k)^q > and determine finiteness and infiniteness for these parameters as far as possible using geometric arguments, principally through pictures and curvature. We state and subsequently prove theorems relating to infiniteness, and also discuss troublesome cases where spherical pictures may arise. We also provide a list of known finite groups and unresolved cases in Appendix A.

In this thesis we investigate the properties of various Banach function algebras and uniform algebras. We are particularly interested in regularity of Banach function algebras and extensions of uniform algebras. The first chapter contains the background in normed algebras, Banach function algebras, and uniform algebras which will be required throughout the thesis. In the second chapter we investigate the classicalisation of certain compact subsets of the complex plane obtained by deleting a sequence of open disks from a closed disk. Sets obtained in this manner are called Swiss cheese sets. We give a new topological proof of the Feinstein-Heath classicalisation theorem along with similar results. We conclude the chapter with an application of the classicalisation results. The results in this chapter are joint with H. Yang. In the third chapter we study Banach function algebras of functions satisfying a generalised notion of differentiability. These algebras were first investigated by Bland and Feinstein as a way to describe the completion of certain normed algebras of complex-differentiable functions. We prove a new version of chain rule in this setting, generalising a result of Chaobankoh, and use this chain rule to give a new proof of the quotient rule. We also investigate naturality and homomorphisms between these algebras. In the fourth chapter we continue the study of the notion of differentiability from the third chapter. We investigate a new notion of quasianalyticity in this setting and prove an analogue of the classical Denjoy-Carleman theorem. We describe those functions which satisfy a notion of analyticity, and give an application of these results. In the fifth chapter we investigate various methods for constructing extensions of uniform algebras. We study the structure of Cole extensions, introduced by Cole and later investigated by Dawson, relative to certain projections. We also discuss a larger class of extensions, which we call generalised Cole extensions, originally introduced by Cole and Feinstein. In the final chapter we investigate extensions of derivations from uniform algebras. We prove that there exists a non-trivial uniform algebra such that every derivation extends with the same norm to every generalised Cole extension of that algebra. A non-trivial, weakly amenable uniform algebra satisfies this property. We also investigate a sequence of extensions of a derivation from the disk algebra.

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In this thesis, we study the modular representations of the special linear group of degree two over a finite field in defining characteristic. In particular, we study the automorphisms of derived category of representations. We have been able to obtain a new type of autoequivalence. This autoequivalence has some uncommon features. It is more conveniently conceived and proved using the representation theory of its Brauer subgroup but at the same time it can be very neatly described, using a type of derived equivalence called perverse equivalence, in global settings.

In this thesis we introduce nuclear dimension and compare it with a stronger form of the completely positive approximation property. We show that the approximations forming this stronger characterisation of the completely positive approximation property witness finite nuclear dimension if and only if the underlying C*-algebra is approximately finite dimensional. We also extend this result to nuclear dimension at most omega. We review interactions between separably acting injective von Neumann algebras and separable nuclear C*-algebras. In particular, we discuss aspects of Connes' work and how some of his strategies have been used by C^*-algebraist to estimate the nuclear dimension of certain classes of C*-algebras. We introduce a notion of coloured isomorphisms between separable unital C*-algebras. Under these coloured isomorphisms ideal lattices, trace spaces, commutativity, nuclearity, finite nuclear dimension and weakly pure infiniteness are preserved. We show that these coloured isomorphisms induce isomorphisms on the classes of finite dimensional and commutative C*-algebras. We prove that any pair of Kirchberg algebras are 2-coloured isomorphic and any pair of separable, simple, unital, finite, nuclear and Z-stable C*-algebras with unique trace which satisfy the UCT are also 2-coloured isomorphic.

Let G be a finite group and F a field. The group algebra FG decomposes as a direct sum of two-sided ideals, called the blocks of FG. In this thesis the structure of the centre of a block for various groups is investigated. Studying these subalgebras yields information about the relationship between two block algebras and, in certain cases, forms a vital tool in establishing the non-existence of an important equivalence in the context of modular representation theory. In particular, the focus lies on determining the Loewy structure for the centre of a block, which so far has not been studied in detail but is fundamental in gaining a better understanding of the block itself. For finite groups G with non-abelian, trivial intersection Sylow p-subgroups, the analysis of the Loewy structure of the centre of a block allows us to deduce that a stable equivalence of Morita type does not induce an algebra isomorphism between the centre of the principal block of G and its Sylow normaliser. This was already known for the Suzuki groups; the techniques will be generalised to extend the result to cover the Ree groups of type ^2G_2(q).In addition, the three sporadic simple groups with the trivial intersection property, M_11, McL and J_4, together with some small projective special unitary groups are studied with respect to showing the non-existence of an isomorphism between the centre of the principal block and the centre of its Brauer correspondent. Finally, the Loewy structure of centres of various principal blocks are calculated. In particular, some small sporadic simple groups and groups with normal, elementary abelian Sylow p-subgroups are considered. For the latter, some specific formulae for the Loewy length are derived, which generalises recent results on groups with cyclic Sylow p-subgroups.

Using the higher tame symbol and Kawada and Satake’s Witt vector method, A.N. Parshin developed class field theory for positive characteristic higher local fields, defining reciprocity maps separately for the tamely ramified and wildly ramified cases. We prove reciprocity laws for these symbols using techniques of Morrow for the Witt symbol and Romo for the higher tame symbol. We then extend this method of defining a reciprocity map to the case of positive characteristic local- global fields associated to points and curves on an algebraic surface over a finite field.

The main goal of this thesis is the description of the section ring of a surface R(S,L) = O∞n=0 H0(S,nL) where L is an ample base point free divisor defining a covering map φL: S -> P2 such that φ*OS = OP2 O Ω1P2 O Ω1P2 O Op2(-3). This is an abelian surface with a polarization of type (1,3) which was studied before in [BL94, Cas99, Cas12]. Given a covering map φ: X -> Y, following the methods introduced by Miranda for general d covers, in chapter 3 we will define a cover homomorphism that will induce a commutative and associative multiplication in φ*OX. Chapter 4 focuses in the OP2-modules Hom (S2Ω1P2,Ω1P2) that will be used to define a commutative multiplication for our surface. Chapter 5 is about the associative condition. It is a computational method based on the paper [Rei90]. In the last chapter we use the ring R(S,L) to prove that the moduli space of abelian surfaces with a polarization of type (1,3) and canonical level structure is rational. We will also show how to use the same method to find models for covering maps such that φ*OS = OP2 O Ω1P2(-m1) O Ω1P2(-m2) O OP2(-m1-m2-3). The last section contains new problems whose goal is to construct and study algebraic varieties given by the vanishing of a high codimensional Gorenstein ideal.

This thesis contains a series of studies about 2-adic integral Galois representations unramified outside a finite set of primes. There are two main focuses of research: the study of 2-adic integral Galois representations and the study on how to compare two 2-adic integral Galois representations. Firstly, when studying a representation, we develop methods to determine whether the residual image is reducible or irreducible: in the irreducible case the residual image is completely determined. On the other hand, when the residual image is reducible we are able to make a choice of a stable lattice to completely determine the residual image. Lastly, from the choice of lattice, we are able to extend our methods to determine whether the representation is trivial modulo 2k+1 assuming that is trivial modulo 2k. Secondly, when comparing two 2-adic integral Galois representations, we are able to determine whether the representations are isogenous that is, after conjugation if necessary, their residual representations are the same. In some cases, this process follows the approach given in [11] by Ron Livné. Finally, the idea behind these studies was the notion of what we call a "Black Box representation", i.e., a system that will provide the characteristic polynomial of the representation for any prime not in the set of primes.

In this thesis we first investigate PBW deformations of Koszul, Calabi-Yau algebras, and we then study moduli spaces of representations of algebras defined by tilting bundles. These classes of algebras are generalisations of the skew group algebras appearing as noncommutative resolutions in the McKay correspondence, and the results we prove are motivated by corresponding results for skew group algebras. Koszul, Calabi-Yau algebras are Morita equivalent to path algebras with relations defined by superpotentials, and we classify which PBW deformations of these algebras still have relations defined by a superpotential and show that these deformations also retain the Calabi-Yau property. As an application of these results we show that symplectic reflection algebras are Calabi-Yau and can be interpreted as path algebras with relations defined by a superpotential. We then investigate when a variety with a tilting bundle can be produced as a moduli space of representations of an algebra defined by the tilting bundle. We find a set of conditions ensuring a variety and tilting bundle can be reconstructed in such a manner, and we show that these conditions hold in a large class of examples, which includes situations arising in the minimal model Program where the variety may be singular. As an example we show that the minimal resolution of a rational surface singularity can be produced as a moduli space for a noncommutative algebra such that the tautological bundle is a tilting bundle defining the noncommutative algebra.