We introduce the class of (D,A)-stacked algebras, which generalise the classes of Koszul algebras, d-Koszul algebras and (D,A)-stacked monomial algebras. We show that the Ext algebra of a (D,A)-stacked algebra is finitely generated in degrees 0, 1, 2 and 3. After investigating some general properties of E(Ʌ) for this class of algebras, we look at a regrading of E(Ʌ) and give examples for which the regraded Ext algebra is a Koszul algebra. Following this we give a general construction of a (D,A)-stacked algebra ~Ʌ from a d-Koszul algebra Ʌ, setting D = dA, with A ≥ 1. From this construction we relate the homological properties of ~Ʌ and Ʌ, including the projective resolutions and the structure of the Ext algebra.

The seemingly distinct areas of group theory, formal language theory and complexity theory interact in an important way when one considers decision problems in groups, such as the question of whether a word in the generators of the group represents the identity or not. In general, these problems are known to be undecidable. Much work has been done on the solvability of these problems in certain groups, however less has been done on the resource bounds needed to solve them, in particular with regard to space considerations. The focus of the work presented here is that of groups with (deterministic) context-sensitive decision problems, that is those that have such problems decidable in (deterministic) linear space. A classification of such groups (similarly to the way that the groups with, for example, regular or context-free word problem, have been classified) seems untenable at present. However, we present a series of interesting results with regard to such groups, with the intentions that this will lead to a better understanding of this area. Amongst these results, we emphasise the difficulty of the conjugacy problem by showing that a group may have unsolvable conjugacy problem, even if it has a subgroup of finite index with context-sensitive conjugacy problem. Our main result eliminates the previously-considered possibility that the groups with context-sensitive word problem could be classified as the set of groups which are subgroups of automatic groups, by constructing a group with context-sensitive word problem which is not a subgroup of an automatic group. We also consider a range of other issues in this area, in an attempt to increase the understanding of the sort of groups under consideration.

The concept of a groupoid was first introduced in 1926 by H. Brandt in his fundamental paper [7]. The idea behind it is a small category in which every arrow is invertible. This notion of groupoid can be thought of as a generalisation of the notion of a group. Namely, a group is a groupoid with only one object. After the introduction of topological and differentiable groupoids by Ehresmann in 1950 in his paper on connections [19], the concept has been widely studied by many mathematicians in many areas of topology, geometry and physics. In this thesis, we deal with topological groupoids as the main object of study. We first develop the main concepts of homotopy theory of topological groupoids. Also, we study general versions of Morita equivalence between topological groupoids, which lead us to discuss topological stacks. The main objective of this thesis is then to develop and analyse a notion of Lusternik-Schnirelmann category for general topological groupoids and topological stacks, generalising the classical work by Lusternik and Schnirelmann for topological spaces and manifolds [30] and for orbifolds and Lie groupoids as introduced by Colman [11]. Fundamental in the classical definition of the LS-category of a smooth manifold or topological space is the concept of a categorical set. A subset of a space is said to be categorical if it is contractible in the space. The Lusternik-Schnirelmann category cat(X) of a topological space X is defined to be the least number of categorical open sets required to cover X, if that number is finite. Otherwise the category cat(X) is said to be infinite. Here using a generalised notion of categorical subgroupoid and substack, we generalise the notion of the Lusternik-Schnirelmann category to topological groupoids and topological stacks with the intention of providing a new useful tool and invariant to study homotopy types of topological groupoids and topological stacks, which will be important also to understand the geometry and Morse theory of Lie groupoids and differentiable stacks from a purely homotopical viewpoint.

The model theory of finite and pseudofinite fields as well as the model theory of finite and pseudofinite groups have been and are thoroughly studied. A close relation has been found between algebraic and model theoretic properties of pseudofinite fields and psedudofinite groups. In this thesis we present results contributing to the beginning of the study of model theory of finite and pseudofinite rings. In particular we classify the theory of ultraproducts of finite residue rings in the context of generalised stability theory. We give sufficient and necessary conditions for the theory of such ultraproducts to be NIP, simple, NTP2 but not simple nor NIP, or TP2 . Further, we show that for any fixed positive l in N the class of finite residue rings {Zp=p^l Zp : p in P} forms an l-dimensional asymptotic class. We discuss related classes of finite residue rings in the context of R-multidimensional asymptotic classes. Finally we present a classification of simple and semisimple (in the algebraic sense) pseudofinite rings, we study NTP2 classes of J-semisimple rings and we discuss NIP classes of finite rings and ultraproducts of these NIP classes.

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We investigate a class of noncommutative algebras, which we call connected quantized Weyl algebras, with a simple description in terms of generators and relations. We already knew of two families, both of which arise from cluster mutation in mutationperiodic quivers, and we show that for generic values of a scalar parameter q these are the only examples. We then investigate the ring-theoretic properties of these two families, determining their prime spectra, automorphism groups and some results on their Krull and global dimensions. The theory of ambiskew polynomial rings and generalised Weyl algebras is useful here and we obtain a description of the height 1 prime ideals in certain generalised Weyl algebras, along with some results on the dimension theory of these rings. We also investigate the semiclassical limit Poisson algebras of the connected quantized Weyl algebras, and compare the prime spectra and Poisson prime spectra of the corresponding rings. We also show that the quantum cluster algebra without coefficients for an acyclic quiver is simple, and extend this result to find a simple localisation in the case where there are coefficients. Finally, we investigate quantum cluster algebra structures related to the connected quantized Weyl algebras discussed earlier, and use these to illustrate the previous result.

For an odd prime p, we aim to do some calculations of connective K-theory of elementary abelian groups V(r), where V(r) denotes an elementary abelian p-group of rank r. The methods involve a combination of Adams spectral sequence (ASS) calculations together with local cohomology calculations. The overall plan builds on and takes its inspiration from work of J. Greenlees and R. Bruner. As a step towards the Gromov-Lawson-Rosenberg (GLR) conjecture for V(r), the thesis calculates the complex connective K-cohomology, ku^*(BV(r)), for r ≤ 3, and the complex connective K-homology, ku_*(BV (r)) for p = 3 and r ≤ 2.

This research belongs to the field of Representation Theory and tries to solve questions through homological algebraic methods. This project deals with the study of symmetries of the plane and aims at measuring how much a mathematical object of importance for that study fails to satisfy the property of not needing bracketing when multiplying three elements together, which is called associativity. More precisely, we study the rational representations of GL2(Fp), the general linear group of order 2 over an algebraically closed field of prime characteristic p. Representations are a means to understand group or algebra elements as linear transformations on a vector space of a given dimension, and it is possible to 'build' representations from smaller ones, e.g. the set of so-called standard representations. The way to glue these building blocks together is governed by the algebra of extensions between standard representations. In a series of papers culminating with [MT13], Miemietz and Turner described precisely the algebra structure of that extension algebra. It is the homology of a differential-graded algebra and this project aims at estimating how non-associative it is by computing its A1-algebra structure. For any p, we give the quiver of that extension algebra, and for p = 2, we show that there exists a subalgebra of the extension algebra which admits a trivial A1-algebra structure, and what's more, in a somewhat peculiar way. We also give its quiver and discuss some of its properties.

The relationship between a Banach space X and its Banach algebra of bounded operators B(X) is rich and complex; this is especially so for non-classical Banach spaces. In this thesis we consider questions of the following form: does there exist a Banach space X such that B(X) has a particular (Banach algebra) property? If not, is there a quotient of B(X) with the property? The first of these is the uniqueness-of-norm problem for Calkin algebras: does there exist a Banach space whose Calkin algebra lacks a unique complete norm? We show that there does indeed exist such a space, answering a classical open question [101]. Secondly, we turn our attention to splittings of extensions of Banach algebras. Work of Bade, Dales and Lykova [12] inspired the problem of whether there exist a Banach space X and an extension of B(X) which splits algebraically but not strongly; this asks for a special type of discontinuous homomorphism from B(X). Using the categorical notion of a pullback we obtain, jointly with N. J. Laustsen [71], new general results about extensions and prove that such a space exists. The same space is used to answer our third question, which goes back to Helemskii, in the positive: is there a Banach space X such that B(X) has homological bidimension at least two? The proof uses techniques developed (with N. J. Laustsen [71]) during the solution to the second question. We use two main Banach spaces to answer our questions. One is due to Read [90], the other to Argyros and Motakis [8]; the former plays a much more prominent role. Together with Laustsen [72], we prove a major original result about Read’s space which allows for the new applications. The conclusion of the thesis examines a class of operators on Banach spaces which have previously received little attention; these are a weak analogue of inessential operators.

Let N1 denote the restricted nullcone of the Lie algebra g of a simple algebraic group in characteristic p>0, i.e. the set of x∈g such that x|p| = 0. For representatives e1,...,en of the nilpotent orbits of g we find the irreducible components of gei∩N1 for g = G2 and F4 in good characteristic p. We do the same for g = E6 with the exception of three nilpotent orbits. We use this information to determine the irreducible components of the restricted nilpotent commuting variety C1nil(g)= {(x,y) ∈ N1×N1 : [x,y] = 0} for g = G2 and F4. We do the same for g = E6 with the exception of when p=7 where we describe C1nil(g) as the union of an irreducible set of dimension 78 and one of dimension 76 which may or may not be an irreducible component.