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This thesis is comprised of three logically independent parts. As the title suggests, each part is related to vector bundles on abelian varieties. We first use Brill-Noether theory to study the geometry of a general curve in its canonical embedding. We prove that there is no $g$ for which the canonical embedding of a general curve of genus $g$ lies on the Segre embedding of any product of three or more projective spaces. We then consider non-degenerate line bundles on abelian varieties. Central to our work is Mumford's index theorem. We give an interpretation of this theorem, and then prove that non-degenerate line bundles, with nonzero index, exhibit positivity analogous to ample line bundles. As an application, we determine the asymptotic behaviour of families of cup-product maps. Using this result, we prove that vector bundles, which are associated to these families, are asymptotically globally generated. To illustrate our results, we consider explicit examples. We also prove that simple abelian varieties, for which our results apply in all possible instances, exist. This is achieved by considering a particular class of abelian varieties with real multiplication. The final part of this thesis concerns the theory of theta and adelic theta groups. We extend and refine work of Mumford, Umemura, and Mukai. For example, we determine the structure and representation theory of theta groups associated to a class of vector bundles which we call simple semi-homogeneous vector bundles of separable type. We also construct, and clarify functorial properties enjoyed by, adelic theta groups associated to line bundles. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2013-06-24 17:14:21.687

Yu, Hok Pun. / Thesis submitted in: October 2008. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 55-56). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Complex Tori --- p.8 / Chapter 2.1 --- Homomorphisms of complex tori --- p.9 / Chapter 2.2 --- Cohomology of Complex Tori --- p.10 / Chapter 3 --- Line bundles on complex tori --- p.11 / Chapter 3.1 --- First Chern classes --- p.11 / Chapter 3.2 --- Semicharacters on line bundles --- p.12 / Chapter 3.3 --- Theorem of the Square --- p.14 / Chapter 4 --- Principally polarized abelian varieties --- p.16 / Chapter 4.1 --- Riemann Relations --- p.17 / Chapter 4.2 --- Characteristics of line bundles --- p.20 / Chapter 4.3 --- Theta Functions --- p.21 / Chapter 4.4 --- The Ox(l) bundle --- p.22 / Chapter 4.5 --- Metric on Ox(l) --- p.23 / Chapter 4.6 --- Abelian Varieties and Elliptic Curves --- p.24 / Chapter 5 --- Isogeny of Abelian Varieties --- p.26 / Chapter 5.1 --- Symmetric Line Bundles --- p.27 / Chapter 5.2 --- Theta Relations --- p.28 / Chapter 5.3 --- Theta Divisors --- p.30 / Chapter 6 --- Jacobians --- p.32 / Chapter 6.1 --- Jacobian as an abelian variety --- p.33 / Chapter 6.2 --- Abel-Jacobi Theorem --- p.36 / Chapter 6.3 --- Torelli´ةs theorem --- p.42 / Chapter 7 --- The Heisenberg Group --- p.43 / Chapter 8 --- Balanced Embedding into the Projective Space --- p.50

Given a function (more generally, a measure) on a locally compact Abelian group, one can define the Toeplitz operators as certain integral transforms of functions on the dual group, where the kernel is the Fourier transform of the original function or measure. In the case of the unit circle, this corresponds to forming a matrix out of the Fourier coefficients in a particular way. We will study the asymptotic eigenvalue distributions of these Toeplitz operators.

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1894. / Vita.

Thesis (Ph.D.)--Kent State University, 2008. / Title from PDF t.p. (viewed Sept. 17, 2009). Advisor: Donald White. Keywords: Huppert's Conjecture; character degrees; nonabelian finite simple groups Includes bibliographical references (p. 103-105).

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This thesis focuses on the subject of varieties for modules for elementary abelian p-groups. Given a homogeneous polynomial over an algebraically closed field of char- acteristic 2 we will give constructions for modules of small dimension having that polynomial as variety. This is similar to an earlier construction given by Jon Carlson but our modules will in general be of considerably smaller dimension. We also investigate the connection between the variety of a module and its Loewy length. We show that working over an algebraically closed field of characteristic 2 with modules of Loewy length 2 allows us to find modules with any hypersurface as their variety. On the other hand we also demonstrate that in odd characteristic p, with modules of Loewy length p, the only possible varieties are finite unions of linear hypersurfaces.