The Cuntz semigroup is an isomorphism invariant for C*-algebras consisting of a semigroup with a compatible (though not algebraic) ordering. Its construction is similar to that of the Murray-von Neumann semigroup (from which the ordered K_0-group arises by the Grothendieck construction), but using positive elements in place of projections. Both rich in structure and sensitive to subtleties of the C*-algebra, the Cuntz semigroup promises to be a useful tool in the classification program for nuclear C*-algebras. It has already delivered on this promise, particularly in the study of regularity properties and the classification of nonsimple C*-algebras. The first part of this thesis introduces the Cuntz semigroup, highlights structural properties, and outlines some applications. The main result of this thesis, however, contributes to the understanding of what the Cuntz semigroup looks like for particular examples of (nonsimple) C*-algebras. We consider separable C*-algebras given as the tensor product of a commutative C*-algebra C_0(X) with a simple, approximately subhomogeneous algebra A, under the regularity hypothesis that A is Z-stable. (The Z-stability hypothesis is needed even to describe of the Cuntz semigroup of A.) For these algebras, the Cuntz semigroup is described in terms of the Cuntz semigroup of A and the Murray-von Neumann semigroups of C(K,A) for compact subsets K of X. This result is a marginal improvement over one proven by the author in [Tikuisis, A. "The Cuntz semigroup of continuous functions into certain simple C*-algebras." Internat. J. Math., to appear] (there, A is assumed to be unital), although improvements have been made to the techniques used. The second part of this thesis provides the basic theory of approximately subhomogeneous algebras, including the important computational concept of recursive subhomogeneous algebras. Theory to handle nonunital approximately subhomogeneous algebras is novel here. In the third part of this thesis lies the main result. The Cuntz semigroup computation is achieved by defining a Cuntz-equivalence invariant I(.) on the positive elements of the C*-algebra, picking out certain data from a positive element which obviously contribute to determining its Cuntz class. The proof of the main result has two parts: showing that the invariant I(.) is (order-)complete, and describing its range.

We prove in this thesis three main results, involving the distribution of values of $L'/L(\sigma,\chi_D)$,$D$ being a fundamental discriminant, and $\chi_D$ the real character attached to it. We prove two Omega theorems for $L'/L(1,\chi_D)$, compute the moments of $L'/L(1,\chi_D)$, and construct under GRH, for each $\sigma>1/2$,a density function ${\cal Q}_\sigma$ such that \[\#\{D ~~\text{fundamental discriminants, such that}~~ |D|\leq Y,~~ \text{and}~~ \alpha \leq L'/L(\sigma,\chi_D)\leq \beta \} \]\[ \sim \frac{6}{\pi^2\sqrt{2\pi}} Y \int_\alpha^\beta {\cal Q}_\sigma(x)dx . \]

In this thesis we use a diffeo-geometric framework based on manifolds hat are locally modeled on ``convenient'' vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold M, we construct a weak symplectic structure on each leaf I_w of a foliation of the space of compact oriented isotropic submanifolds in M equipped with top degree forms of total measure 1. These forms are called weightings and such manifolds are said to be weighted. We show that this symplectic structure on the particular leaves consisting of weighted Lagrangians is equivalent to a heuristic weak symplectic structure of Weinstein. When the weightings are positive, these symplectic spaces are symplectomorphic to reductions of a weak symplectic structure of Donaldson on the space of embeddings of a fixed compact oriented manifold into M. When M is compact, by generalizing a moment map of Weinstein we construct a symplectomorphism of each leaf I_w consisting of positive weighted isotropics onto a coadjoint orbit of the group of Hamiltonian symplectomorphisms of M equipped with the Kirillov-Kostant-Souriau symplectic structure. After defining notions of Poisson algebras and Poisson manifolds, we prove that each space I_w can also be identified with a symplectic leaf of a Poisson structure. Finally, we discuss a kinematic description of spaces of weighted submanifolds.

This thesis studies the pre-quantization of quasi-Hamiltonian group actions from a cohomological viewpoint. The compatibility of pre-quantization with symplectic reduction and the fusion product are established, and are used to understand the necessary and sufficient conditions for the pre-quantization of M(G,S), the moduli space of at flat G-bundles over a closed surface S. For a simply connected, compact, simple Lie group G, M(G,S) is known to be pre-quantizable at integer levels. For non-simply connected G, however, integrality of the level is not sufficient for pre-quantization, and this thesis determines the obstruction, namely a certain 3-dimensional cohomology class, that places further restrictions on the underlying level. The levels that admit a pre-quantization of the moduli space are determined explicitly for all non-simply connected, compact, simple Lie groups G. Partial results are obtained for the case of a surface S with marked points. Also, it is shown that via the bijective correspondence between quasi-Hamiltonian group actions and Hamiltonian loop group actions, the corresponding notions of prequantization coincide.

The Cuntz semigroup is an isomorphism invariant for C*-algebras consisting of a semigroup with a compatible (though not algebraic) ordering. Its construction is similar to that of the Murray-von Neumann semigroup (from which the ordered K_0-group arises by the Grothendieck construction), but using positive elements in place of projections. Both rich in structure and sensitive to subtleties of the C*-algebra, the Cuntz semigroup promises to be a useful tool in the classification program for nuclear C*-algebras. It has already delivered on this promise, particularly in the study of regularity properties and the classification of nonsimple C*-algebras. The first part of this thesis introduces the Cuntz semigroup, highlights structural properties, and outlines some applications. The main result of this thesis, however, contributes to the understanding of what the Cuntz semigroup looks like for particular examples of (nonsimple) C*-algebras. We consider separable C*-algebras given as the tensor product of a commutative C*-algebra C_0(X) with a simple, approximately subhomogeneous algebra A, under the regularity hypothesis that A is Z-stable. (The Z-stability hypothesis is needed even to describe of the Cuntz semigroup of A.) For these algebras, the Cuntz semigroup is described in terms of the Cuntz semigroup of A and the Murray-von Neumann semigroups of C(K,A) for compact subsets K of X. This result is a marginal improvement over one proven by the author in [Tikuisis, A. "The Cuntz semigroup of continuous functions into certain simple C*-algebras." Internat. J. Math., to appear] (there, A is assumed to be unital), although improvements have been made to the techniques used. The second part of this thesis provides the basic theory of approximately subhomogeneous algebras, including the important computational concept of recursive subhomogeneous algebras. Theory to handle nonunital approximately subhomogeneous algebras is novel here. In the third part of this thesis lies the main result. The Cuntz semigroup computation is achieved by defining a Cuntz-equivalence invariant I(.) on the positive elements of the C*-algebra, picking out certain data from a positive element which obviously contribute to determining its Cuntz class. The proof of the main result has two parts: showing that the invariant I(.) is (order-)complete, and describing its range.

Let G be a simple algebraic group. Labelled trivalent graphs called webs can be used to produce invariants in tensor products of minuscule representations. For each web, a configuration space of points in the affine Grassmannian is constructed. This configuration space gives a natural way of calculating the invariant vectors coming from webs. In the case of G = SL_3, non-elliptic webs yield a basis for the invariant spaces. The non-elliptic condition, which is equivalent to the condition that the dual diskoid of the web is CAT(0), is explained by the fact that affine buildings are CAT(0). In the case of G = SL_n, a sufficient condition for a set of webs to yield a basis is given. Using this condition and a generalization of a technique by Westbury, a basis is constructed for SL_n. Due to the geometric Satake correspondence there exists another natural basis of invariants, the Satake basis. This basis arises from the underlying geometry of the affine Grassmannian. There is an upper unitriangular change of basis from the basis constructed above to the Satake basis. An example is constructed showing that the Satake, web and dual canonical basis of the invariant space are all different. The natural action of rotation on tensor factors sends invariant space to invariant space. Since the rotation of web is still a web, the set of vectors coming from webs is fixed by this action. The Satake basis is also fixed, and an explicit geometric and combinatorial description of this action is developed.

Let $F$ be a non-Archimedean local field and $G$ be the general linear group $\mathrm{GL}_n$ over $F$. Bushnell and Henniart described the essentially tame local Langlands correspondence of $G(F)$ using rectifiers, which are certain characters defined on tamely ramified elliptic maximal tori of $G(F)$. They obtained such result by studying the automorphic induction character identity. We relate this formula to the spectral transfer character identity, based on the theory of twisted endoscopy of Kottwitz, Langlands, and Shelstad. In this article, we establish the following two main results. (i) To show that the automorphic induction character identity is equal to the spectral transfer character identity when both are normalized by the same Whittaker data. (ii) To express the essentially tame local Langlands correspondence using admissible embeddings constructed by Langlands-Shelstad $\chi$-data and to relate Bushnell-Henniart's rectifiers to certain transfer factors.

We prove in this thesis three main results, involving the distribution of values of $L'/L(\sigma,\chi_D)$,$D$ being a fundamental discriminant, and $\chi_D$ the real character attached to it. We prove two Omega theorems for $L'/L(1,\chi_D)$, compute the moments of $L'/L(1,\chi_D)$, and construct under GRH, for each $\sigma>1/2$,a density function ${\cal Q}_\sigma$ such that \[\#\{D ~~\text{fundamental discriminants, such that}~~ |D|\leq Y,~~ \text{and}~~ \alpha \leq L'/L(\sigma,\chi_D)\leq \beta \} \]\[ \sim \frac{6}{\pi^2\sqrt{2\pi}} Y \int_\alpha^\beta {\cal Q}_\sigma(x)dx . \]

John Napier, born at Merchiston in 1550, published The Whole Revelation of St. John in 1594; and he appears to have regarded that theological polemic as his most important achievement. Napier's invention of logarithms (with greatly advanced spherical trigonometry) was published in 1614 as Descriptio Canonis Logarithmorum; whereupon the mathematicians of Europe instantly acclaimed Napier as the greatest of them all. In 1617 he published Rabdologiae, which explained several devices for aiding calculation: (1) numbering rods to aid multiplication (known as 'Napier's bones'); (2) other rods to aid evaluation of square and cube roots; (3) the first publication of binary arithmetic, as far as square root extraction; and (4) the Promptuary for multiplication of numbers (up to 10 digits each), which has a strong claim to be regarded as the first calculating machine. Napier's explanation of the construction of his logarithms was published posthumously in 1619 as Constructio Canonis Logarithmorum, in which he developed much of the differential calculus in order to define his logarithms as the solution of a differential equation and then constructed strict upper and lower bounds for the solution. His incomplete manuscript on arithmetic and algebra (written in the early 1590s) was published in 1839 as De Arte Logistica. This thesis provides the first English translations of De Arte Logistica and of Rabdologiae, and it reprints Edward Wright's English translation (1616) of the Descriptio and W. R. Macdonald's English translation (1889) of the Constructio. Extensive commentaries are given on Napier's work on arithmetic, algebra, trigonometry and logarithms. The history of trigonometry is traced from ancient Babylonia and Greece through mediaeval Islam to Renaissance Europe. Napier's logarithms (and spherical trigonometry) resulted in an explosion of logarithms over most of the world, with European ships using logarithms for navigation as far as Japan by 1640. / Subscription resource available via Digital Dissertations only.

John Napier, born at Merchiston in 1550, published The Whole Revelation of St. John in 1594; and he appears to have regarded that theological polemic as his most important achievement. Napier's invention of logarithms (with greatly advanced spherical trigonometry) was published in 1614 as Descriptio Canonis Logarithmorum; whereupon the mathematicians of Europe instantly acclaimed Napier as the greatest of them all. In 1617 he published Rabdologiae, which explained several devices for aiding calculation: (1) numbering rods to aid multiplication (known as 'Napier's bones'); (2) other rods to aid evaluation of square and cube roots; (3) the first publication of binary arithmetic, as far as square root extraction; and (4) the Promptuary for multiplication of numbers (up to 10 digits each), which has a strong claim to be regarded as the first calculating machine. Napier's explanation of the construction of his logarithms was published posthumously in 1619 as Constructio Canonis Logarithmorum, in which he developed much of the differential calculus in order to define his logarithms as the solution of a differential equation and then constructed strict upper and lower bounds for the solution. His incomplete manuscript on arithmetic and algebra (written in the early 1590s) was published in 1839 as De Arte Logistica. This thesis provides the first English translations of De Arte Logistica and of Rabdologiae, and it reprints Edward Wright's English translation (1616) of the Descriptio and W. R. Macdonald's English translation (1889) of the Constructio. Extensive commentaries are given on Napier's work on arithmetic, algebra, trigonometry and logarithms. The history of trigonometry is traced from ancient Babylonia and Greece through mediaeval Islam to Renaissance Europe. Napier's logarithms (and spherical trigonometry) resulted in an explosion of logarithms over most of the world, with European ships using logarithms for navigation as far as Japan by 1640. / Subscription resource available via Digital Dissertations only.