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We introduce the naive notion of a relative open book decomposition for contact 3-manifolds with torus boundary. We then use this to construct nice, minimal genus open book decompositions compatible with all of the universally tight contact structures (as well as a few others) on torus-bundles over S¹, following Honda's classification. In an accurate sense, we find Stein fillings of 'half' of the torus bundles. In addition, these give the first examples of open books compatible with the universally tight contact structures on circle bundles over higher genus surfaces, as well, following a pattern introduced by a branched covering of B⁴. Some interesting examples of open books without positive monodromy are emphasized.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

Thesis (Dr. rer. nat.)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2001. / Pt. 1. with Gero Kleineidam. Includes bibliographical references.

Honors Project--Smith College, Northampton, Mass., 2010. / Includes bibliographical references (p. 64-65).

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

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

In 1984, T. Kobayashi gave a classification of the genus two 3-manifolds with a nontrivial torus decomposition. The intent of this study is to extend this classification to the genus two, torally bounded 3-manifolds with a separating non-trivial torus decomposition. These 3-manifolds are also known as the tunnel-1 generalized satellite knot exteriors. The main result of the study is a full decomposition of the exterior of a tunnel-1 satellite knot in an arbitrary 3-manifold. Several corollaries are drawn from this classification. First, Schubert's 1953 results regarding the existence and uniqueness of a core component for satellite knots in the 3-sphere is extended to tunnel-1 satellite knots in arbitrary 3-manifolds. Second, Morimoto and Sakuma's 1991 classification of tunnel-1 satellite knots in the 3-sphere is extended to a classification of the tunnel-1 satellite knots in lens spaces. Finally, for these knot exteriors, a result of Eudave-Muñoz in 1994 regarding the relative position of tunnels and decomposing tori is recovered.

Realistic four-dimensional model building from string theory has been a focus of the string theory community ever since its inception. Toroidal orientifold constructions have emerged as a technically simple class of candidate models. Novel ingredients, such as background fluxes, have been discovered and intensely studied over the past few years. They allow for a (partial) solution of several long standing problems associated with model building in this framework. In this thesis, I summarize progress that has been made in toroidal orientifold constructions in type IIA string theory.This includes a detailed discussion of moduli stabilization and (non-) supersymmetric AdS and Minkowski vacua. Furthermore I commence a systematic study of generalized NSNS, i.e., metric and non-geometric, fluxes. The emergence of novel D-terms is presented in detail. While most of the discussion applies to generic orientifolds of T⁶, most features are exemplified by and studied in terms of a certain orientifold of T⁶/ℤ₄ owing to its somewhat richer structure compared to simpler models studied before. It is also briefly reported on efforts of finding de Sitter vacua and inflation in this class of models. / text

Let M = H³/Γ be a complete, non-compact, oriented geometrically finite hyperbolic 3-manifold without cusps. By constructing a conformal compactification of M x S¹ we functorially associate to M an oriented, conformally flat, compact 4-manifold X (without boundary) with an S¹-action. X determines M as a hyperbolic manifold. Using our functor and the differential geometry of conformally flat 4-manifolds we prove that any Γ as above with a limit set of Hausdorff dimension ≤ 1 is Schottky, Fuchsian or extended Fuchsian. Furthermore, the Hodge theory for H² (X;R) carries over to H¹(M, δM;R) and H²(M;R) which correspond to the spaces of harmonic L²-forms of degree 1 and 2 on M. Comparison of lattices through the Hodge star gives an invariant h(M) ε GL(H²(M;R)/GL(H²(M;Z)) of the hyperbolic structure. Secondly we pay attention to magnetic monopoles on M which correspond to S¹invariant solutions of the anti-self-duality equations on X. The basic result is that we associate to M an infinite collection of moduli spaces of monopoles , labelled by boundary conditions. We prove that the moduli spaces are not empty (under reasonable conditions), compute their dimension , prove orientability , the existence of a compactification and smoothness for generic S¹-invariant conformal structures on X. For these results one doesn't need a hyperbolic structure on M , the existence of a conformal compactification X suffices. A twistor description for monopoles on a hyperbolic M can be given through the twistor space of X , and monopoles turn out to correspond to invariant holomorphic bundles on twistor space. We analyse these bundles. Explicit formulas for monopoles can be found on handlebodies M , and for M = surface x R we describe the moduli spaces in some detail.