When analyzing symmetric problems and the methods for solving them, multigrid and algebraic multigrid in particular, one of the primary tools at the analyst's disposal is the energy norm associated with the problem. The lack of this tool is one of the many reasons analysis of nonsymmetric problems and methods for solving them is substantially more difficult than in the symmetric case. We show that there is an analog to the energy norm for a nonsymmetric matrix A, associated with a new absolute value we term the "form" absolute value. This new absolute value can be described as a symmetric positive definite solution to the matrix equation A*|A|-1A = |A|; it exists and is unique in particular whenever A has positive symmetric part. We then develop a novel convergence theory for a general two-level multigrid iteration for any such A, making use of the form absolute value. In particular, we derive a convergence bound in terms of a smoothing property and separate approximation properties for the interpolation and restriction (a novel feature). Finally, we present new algebraic multigrid heuristics designed specifically targeting this new theory, which we evaluate with numerical tests.

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Over the last 20 years, cluster algebras have been widely studied, with numerous links to different areas of mathematics and physics. These algebras have a cluster structure given by successively mutating seeds, which can be thought of as living on some graph or tree. In this way one can use various combinatorial tools to discover more about these cluster structures and the cluster algebras themselves. This thesis considers some of the combinatorics at play here. Mutation-finite quivers have been classified, with links to triangulations of surfaces and semi-simple Lie algebras, while comparatively little is known about mutation-infinite quivers. We introduce a classification of the minimal types of these mutation-infinite quivers before studying their properties. We show that these minimal mutation-infinite quivers admit a maximal green sequence and that the cluster algebras which they generate are equal to their related upper cluster algebras. Automorphisms of skew-symmetric cluster algebras are known to be linked to automorphisms of their exchange graphs. In the final chapter we discuss how this idea can be extended to skew-symmetrizable cluster algebras by using the symmetrizing weights to add markings to the exchange graphs. This opens possible opportunities to study orbifold mapping class groups using combinatoric graph theory.

This thesis is on the representations of connected reductive groups over finite quotients of a complete discrete valuation ring. Several aspects of higher Deligne–Lusztig representations are studied. First we discuss some properties analogous to the finite field case; for example, we show that the higher Deligne–Lusztig inductions are compatible with the Harish-Chandra inductions. We then introduce certain subvarieties of higher Deligne–Lusztig varieties, by taking pre-images of lower level groups along reduction maps; their constructions are motivated by efforts on computing the representation dimensions. In special cases we show that their cohomologies are closely related to the higher Deligne–Lusztig representations. Then we turn to our main results. We show that, at even levels the higher Deligne–Lusztig representations of general linear groups coincide with certain explicitly induced representations; thus in this case we solved a problem raised by Lusztig. The generalisation of this result for a general reductive group is completed jointly with Stasinski; we also present this generalisation. Some discussions on the relations between this result and the invariant characters of finite Lie algebras are also presented. In the even level case, we give a construction of generic character sheaves on reductive groups over rings, which are certain complexes whose associated functions are higher Deligne–Lusztig characters; they are accompanied with induction and restriction functors. By assuming some properties concerning perverse sheaves, we show that the induction and restriction functors are transitive and admit a Frobenius reciprocity.

In 2002, Fomin and Zelevinsky introduced a cluster algebra; a dynamical system that has already proved to be ubiquitous within mathematics. In particular, it was shown by Fomin, Shapiro and Thurston that some cluster algebras arise from orientable surfaces. Subsequently, Dupont and Palesi extended this construction to non-orientable surfaces, giving birth to quasi-cluster algebras. The finite type cluster algebras possess the remarkable property of their exchanges graphs being polytopal. We discover that the finite type quasi-cluster algebras enjoy a closely related property, namely, their exchange graphs are spherical. Revealing yet more connections we unify these two frameworks via Lam and Pylyavskyy's Laurent phenomenon algebras, showing that both orientable and non-orientable marked surfaces have an associated LP-algebra. The integration of these structures is attempted in two ways. Firstly we show that the quasi-cluster algebras of unpunctured surfaces have LP structures. Secondly, to obtain a connection for all marked surfaces, we consider laminations, forging the notion of the laminated quasi-cluster algebra. We show that each marked surface exhibits a lamination which supplies the laminated quasi-cluster algebra with an LP structure.

This thesis combines the fields of functional analysis and topology. $C^\ast$-algebras are analytic objects used in non-commutative geometry and in particular we consider an invariant of them, namely $E$-theory. $E$-theory is a sequence of abelian groups defined in terms of homotopy classes of morphisms of $C^\ast$-algebras. It is a bivariant functor from the category where objects are $C^\ast$-algebras and arrows are $\ast$-homomorphisms to the category where objects are abelian groups and arrows are group homomorphisms. In particular, $E$-theory is a cohomology theory in its first variable and a homology theory in its second variable. We prove in the case of real graded $C^\ast$-algebras that $E$-theory has $8$-fold periodicity. Further we create a spectrum for $E$-theory. More precisely, we use the notion of quasi-topological spaces and form a quasi-spectrum, that is a sequence of based quasi-topological spaces with specific structure maps. We consider actions of the orthogonal group and we obtain a orthogonal quasi-spectrum which we prove has a smash product structure using the categorical framework. Then we obtain stable homotopy groups which give us $E$-theory. Finally, we combine these ideas and a relation between $E$-theory and $K$-theory to obtain connections of the $E$-theory orthogonal quasi-spectrum to $K$-theory and $K$-homology orthogonal quasi-spectra.

Higher amalgamation is a model theoretic property. It was also studied under the name generalised independence theorem. This property is defined in stable, or more generally simple or rosy theories. In this thesis we study how higher amalgamation behaves under expansion by finite covers and algebraic covers. We first show that finite and algebraic covers are mild expansions, in the sense that they preserve many model theoretic properties and behave well when imaginaries are added to them. Then we show that in pregeometric theories higher amalgamation over ; implies higher amalgamation over parameters. We also show that in general this is not true. In fact, for any stable theory with an algebraic closed set which is not a model we construct a finite cover which fails 4-Amalgamation. With some additional assumption we can also preserve higher amalgamation over the empty set. We apply this result to abelian groups and show that (Z/4Z)ω satisfies these assumptions. Then we take the opposite direction: rather then investigating covers which have malicious properties towards amalgamation, we construct covers which will make higher amalgamation become true. First we give a new proof for the fact that there exists an algebraic cover of any stable Teq acleq(Ø) with higher amalgamation over Ø. A proof sketch of this was given by Hrushovski and a full proof appeared in an unpublished work by D. Evans. The new proof uses the notion of symmetric witness which was introduced by Goodrick, Kim and Kolesnikov. We also show with a similar approach that there exists an algebraic cover of any stable, omega-categorical theory with higher amalgamation over parameters.

Introduced in 2008 by Khovanov and Lauda, and independently by Rouquier, the quiver Hecke algebras are a family of infinite dimensional graded algebras which categorify the negative part of the quantum group associated to a graph. Infinite types these algebras are known to have nice homological properties, in particular they are affine quasi-hereditary. In this thesis we utilise the affine quasi-hereditary structure to create finite dimensional quotients which preserve some of the homological structure of the original algebra.

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The thesis is concerned with groups acting on three dimensional spaces. The groups are assumed to have a compact simply-connected fundamental region. The action of the group is given partially by its action on the boundary of the fundamental region. This boundary is naturally split up by its intersection with transforms of the fundamental region. We assume that each such intersection is a single proper face of the boundary unless the transforming element is of order two, in which case there can be two faces. We also assume that any point of the boundary has only a finite number of transforms under the group which lie on the boundary. This enables one to give generators and defining relations for the group. The generators correspond to faces of the boundary inequivalent under the group, and defining relations to inequivalent lines. In these circumstances two questions arise: Is the three-dimensional space a manifold? Is the group finite? If the space is not a manifold, then the group cannot be finite. So an answer to the first question gives some information about the second. Another theorem which is a corollary of the methods used in proving the first theorem is:- Given a group acting on a three-dimensional space with a fundamental region satisfying the conditions above then the group has a soluble word problem.