The major results of this work concern perfect ideals of ordered vector spaces, and a representation theory for ordered vector spaces. Perfect ideals are characterized by the property that their annihilators in the order dual are ideals. We obtain a number of conditions for an ordered vector space which are equivalent to the intersection of the set of perfect maximal ideals being 0. We also obtain conditions which permit an ordered vector space to be represented as a subspace of the sections of a vector bundle. This generalizes the representation theory for odered vector spaces with unit. / Thesis / Doctor of Philosophy (PhD)

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Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / Let G be a connected and simply connected semisimple algebraic group over an algebraically closed field of positive prime characteristic. Let L([lambda]) and [upside-down triangle]([lambda]) be the simple and induced finite dimensional rational G-modules with p-singular dominant highest weight [lambda]. In this thesis, the concept of Weyl filtration dimension of a finite dimensional rational G-module is studied for some highest weight modules with p-singular highest weights inside the p2-alcove when G is of type B[subscript]2. In chapter 4, intertwining morphisms, a diagonal G-module morphism and tilting modules are used to compute the Weyl filtration dimension of L([lambda]) with [lambda] p-singular and inside the p[superscript]2-alcove. It is shown that the Weyl filtration dimension of L([lambda]) coincides with the Weyl filtration dimension of [upside-down triangle]([lambda]) for almost all (all but one of the 6 facet types) p-singular weights inside the p[superscript]2-alcove. In chapter 5 we study some submodule structures of Weyl (and there translations), Vogan, and tilting modules with both p-regular and p-singular highest weights. Most results are for the p[superscript]2 -alcove only although some concepts used are alcove independent.

Anscombe (1964) presents influential arguments that 'before' and 'after' cannot denote converse relations, despite intuitions to the contrary. These arguments, I claim, rely on ambiguity of certain 'before'- and 'after'-sentences, ambiguity that arises from the interaction of tense and aspect with the temporal ordering relations denoted by 'before' and 'after'. To account for this ambiguity, I adopt a Discourse Representation Theory-based analysis of tense and aspect (Kamp & Reyle 2011) and apply it to a set of examples that exhibit the variety of readings available for 'before'- and 'after'-sentences. I argue that certain readings of stative 'after'-sentences support the existence of an inceptive coercion operator, equivalent in effect to the aspectual verb 'begin'. This operator has much in common with 'earliest', an operator proposed by Beaver & Condoravdi (2003), but it is motivated by independent aspectual considerations. I conclude with a discussion of areas for future research. / text

In this thesis, we investigate two expressions for symmetric group characters: Kerov?s universal character polynomials and Stanley?s character polynomials. We give a new explicit form for Kerov?s polynomials, which exactly evaluate the characters of the symmetric group scaled by degree and a constant. We use this explicit expression to obtain specific information about Kerov polynomials, including partial answers to positivity questions. We then use the expression obtained for Kerov?s polynomials to obtain results about Stanley?s character polynomials.

The present dissertation proposes integrating Discourse Representation Theory (DRT), information structure (IS) and Combinatory Categorial Grammar (CCG) into a single framework. It achieves this by making two new contributions to computational treatment of information structure. First, it presents an uncomplicated approach to incorporating information structure in DRT. Second, it shows how the new DRT representation can be integrated into a unification-based grammar framework in a straightforward manner. We foresee the main application of the new formalism to be in spoken language systems: the approach presented here has the potential to considerably facilitate spoken language systems benefiting from insights derived from information structure. The DRT representation with information structure which is proposed in this dissertation is simpler than the previous attempts to include information structure in DRT. We believe that the simplicity of the Information-Structure-marked Discourse Representation Structure (IS-DRS) is precisely what makes it attractive and easy to use for practical tasks like determining the intonation in spoken language applications. The IS component in ISDRS covers a range of aspects of information structural semantics. A further advantage of IS-DRS is that in its case a single semantic representation is suitable for both the generation of context-appropriate prosody and automatic reasoning. A semantic representation on its own is useful for describing and analysing a language. However, it is of even greater utility if it is accompanied by a mechanism that allows one to directly infer the semantic representation from a natural language expression. We incorporated the IS-DRS into the Categorial Grammar (CG) framework, developing a unification based realisation of Combinatory Categorial Grammar, which we call Unification-based Combinatory Categorial Grammar (UCCG). UCCG inherits elements from Combinatory Categorial Grammar and Unification Categorial Grammar. The UCCG framework is developed gradually throughout the dissertation. The information structural component is included as the final step. The IS-DRSs for linguistic expressions are built up compositionally from the IS-DRSs of their sub-expressions. Feature unification is the driving force in this process. The formalism is illustrated by numerous examples which are characterised by different levels of syntactic complexity and diverse information structure. We believe that the main assets of both the IS-DRSs as well as the Unification-based Combinatory Categorial Grammar framework are their simplicity, transparency, and inherent suitability for computational implementation. This makes them an appealing choice for use in practical applications like spoken language systems.

Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / Let A[subscript]n(S) be a coefficient free commutative cluster algebra over a field K. A cluster automorphism is an element of Aut.[subscript]KK(t[subscript]1,[dot, dot, dot],t[subscript]n) which leaves the set of all cluster variables, [chi][subscript]s invariant. In Chapter 2, the group of all such automorphisms is studied in terms of the orbits of the symmetric group action on the set of all seeds of the field K(t[subscript]1,[dot,dot, dot],t[subscript]n). In Chapter 3, we set up for a new class of non-commutative algebras that carry a non-commutative cluster structure. This structure is related naturally to some hyperbolic algebras such as, Weyl Algebras, classical and quantized universal enveloping algebras of sl[subscript]2 and the quantum coordinate algebra of SL(2). The cluster structure gives rise to some combinatorial data, called cluster strings, which are used to introduce a class of representations of Weyl algebras. Irreducible and indecomposable representations are also introduced from the same data. The last section of Chapter 3 is devoted to introduce a class of categories that carry a hyperbolic cluster structure. Examples of these categories are the categories of representations of certain algebras such as Weyl algebras, the coordinate algebra of the Lie algebra sl[subscript]2, and the quantum coordinate algebra of SL(2).

Let G be a semisimple linear algebraic group over an algebraically closed field of prime characteristic. In this thesis we outline the theory of such groups and their cohomology. We then concentrate on algebraic groups in rank 1 and 2, and prove some new results in their bounding cohomology.

Let R(G) denote the category of smooth representations of a p-adic group. Bernstein has constructed an indexing set B(G) such that R(G) decomposes into a direct sum over s ∈ B(G) of full subcategories Rs(G) known as Bernstein subcategories. Bushnell and Kutzko have developed a method to study the representations contained in a given subcategory. One attempts to associate to that subcategory a smooth irreducible representation (τ,W) of a compact open subgroup J < G. If the functor V ↦ HomJ(W,V) is an equivalence of categories from Rs(G) → H(G,τ)mod we call (J,τ) a type. Given a Levi subgroup L < G and a type (JL, τL) for a subcategory of representations on L, Bushnell and Kutzko further show that one can construct a type on G that “lies over” (JL, τL) by constructing an object known as a cover. In particular, a cover implements induction of H(L,τL)-modules in a manner compatible with parabolic induction of L-representations. In this thesis I construct a cover for certain representations of the Siegel Levi subgroup of Sp(2k) over an archimedean local field of characteristic zero. In partic- ular, the representations I consider are twisted by highly ramified characters. This compliments work of Bushnell, Goldberg, and Stevens on covers in the self-dual case. My construction is quite concrete, and I also show that the cover I construct has a useful property known as splitness. In fact, I prove a fairly general theorem characterizing when covers are split.

The goal of this thesis is to first give an overview of the modern approach, using the paper of A. Vershik and A. Okounkov, to inductively parametrizing all irreducible representations of the symmetric groups. This theory is then used to answer questions concerning to central projections in the group algebra. We index units first by partitions, and then by so called standard tableaux. We also present a new result and discuss future research exploring the connections between this theory and Quantum Information.