Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 70-73). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2000. / Typescript. Vita. Includes bibliographical references (leaves 106-111). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2000. / Typescript. Vita. Includes bibliographical references (leaves 106-111). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 70-73). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2000. / Typescript. Vita. Includes bibliographical references (leaves 71-78). Also available on the Internet.

Thesis (M. Phil.)--University of Hong Kong, 2002. / Includes bibliographical references (leaves 115-118).

This dissertation provides the first known examples of finite co-volume Kleinian groups which have co- final towers of finite index subgroups with positive rank gradient. We prove that if the fundamental group of an orientable finite volume hyperbolic 3-manifold has fi nite index in the reflection group of a right-angled ideal polyhedron in H^3 then it has a co-fi nal tower of fi nite sheeted covers with positive rank gradient. The manifolds we provide are also known to have co- final towers of covers with zero rank gradient. We also prove that the reflection groups of compact right-angled hyperbolic polyhedra satisfying mild conditions have co-fi nal towers of fi nite sheeted covers with positive rank gradient. / text

Orientifolds of type II string theory offer a promising toolkit for model builders, especially when one includes not only the usual fluxes from NSNS and RR field strengths, but also fluxes that are T-dual to the NSNS three-form flux. These additional ingredients can help stabilize moduli and lead to D-term contributions to the effective scalar potential. We describe in general how these fluxes appear as parameters of an effective N = 1 supergravity theory in four dimensions for type IIA and type IIB string theory. We also show how these fluxes arise from compactifications on six-dimensional spaces that can be described by toroidal fibers twisted over a toroidal base. This approach leads us to a more subtle treatment of the quantization of the general NSNS fluxes. We illustrate these phenomena with examples of certain orientifolds of T⁶/Z₄. / text

In this dissertation, we present some analysis of Ricci flow on complete noncompact manifolds. The first half of the dissertation concerns the formation of Type-II singularity in Ricci flow on [mathematical equation]. For each [mathematical equation] , we construct complete solutions to Ricci flow on [mathematical equation] which encounter global singularities at a finite time T such that the singularities are forming arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate [mathematical equation]. Near the origin, blow-ups of such a solution converge uniformly to the Bryant soliton. Near spatial infinity, blow-ups of such a solution converge uniformly to the shrinking cylinder soliton. As an application of this result, we prove that there exist standard solutions of Ricci flow on [mathematical equation] whose blow-ups near the origin converge uniformly to the Bryant soliton. In the second half of the dissertation, we fully analyze the structure of the Lichnerowicz Laplacian of a Bergman metric g[subscript B] on a complex hyperbolic space [mathematical equation] and establish the linear stability of the curvature-normalized Ricci flow at such a geometry in complex dimension [mathematical equation]. We then apply the maximal regularity theory for quasilinear parabolic systems to prove a dynamical stability result of Bergman metric on the complete noncompact CH[superscript m] under the curvature-normalized Ricci flow in complex dimension [mathematical equation]. We also prove a similar dynamical stability result on a smooth closed quotient manifold of [mathematical symbols]. In order to apply the maximal regularity theory, we define suitably weighted little Hölder spaces on a complete noncompact manifold and establish their interpolation properties. / text

Bott-Samelson varieties associated to reductive algebraic groups are much studied in representation theory and algebraic geometry. They not only provide resolutions of singularities for Schubert varieties but also have interesting geometric properties of their own. A distinguished feature of Bott-Samelson varieties is that they admit natural affine coordinate charts, which allow explicit computations of geometric quantities in coordinates. Poisson geometry dates back to 19th century mechanics, and the more recent theory of quantum groups provides a large class of Poisson structures associated to reductive algebraic groups. A holomorphic Poisson structure Π on Bott-Samelson varieties associated to complex semisimple Lie groups, referred to as the standard Poisson structure on Bott-Samelson varieties in this thesis, was introduced and studied by J. H. Lu. In particular, it was shown by Lu that the Poisson structure Π was algebraic and gave rise to an iterated Poisson polynomial algebra associated to each affine chart of the Bott-Samelson variety. The formula by Lu, however, was in terms of certain holomorphic vector fields on the Bott-Samelson variety, and it is much desirable to have explicit formulas for these vector fields in coordinates. In this thesis, the holomorphic vector fields in Lu’s formula for the Poisson structure Π were computed explicitly in coordinates in every affine chart of the Bott-Samelson variety, resulting in an explicit formula for the Poisson structure Π in coordinates. The formula revealed the explicit relations between the Poisson structure and the root system and the structure constants of the underlying Lie algebra in any basis. Using a Chevalley basis, it was shown that the Poisson structure restricted to every affine chart of the Bott-Samelson variety was defined over the integers. Consequently, one obtained a large class of iterated Poisson polynomial algebras over any field, and in particular, over fields of positive characteristic. Concrete examples were given at the end of the thesis. / published_or_final_version / Mathematics / Master / Master of Philosophy