Thesis (M.Ed.)--University of Hong Kong, 2003. / Also available in print.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1970. / Typescript. Vita. Includes bibliographical references.

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This thesis is concerned with the theory of invariant bilinear differential pairings on parabolic geometries. It introduces the concept formally with the help of the jet bundle formalism and provides a detailed analysis. More precisely, after introducing the most important notations and definitions, we first of all give an algebraic description for pairings on homogeneous spaces and obtain a first existence theorem. Next, a classification of first order invariant bilinear differential pairings is given under exclusion of certain degenerate cases that are related to the existence of invariant linear differential operators. Furthermore, a concrete formula for a large class of invariant bilinear differential pairings of arbitrary order is given and many examples are computed. The general theory of higher order invariant bilinear differential pairings turns out to be much more intricate and a general construction is only possible under exclusion of finitely many degenerate cases whose significance in general remains elusive (although a result for projective geometry is included). The construction relies on so-called splitting operators examples of which are described for projective geometry, conformal geometry and CR geometry in the last chapter. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1339548 / Thesis (Ph.D.) - University of Adelaide, School of Mathematical Sciences, 2008

We study a number of local and global classification problems in generalized complex geometry. Generalized complex geometry is a relatively new type of geometry which has applications to string theory and mirror symmetry. Symplectic and complex geometry are special cases. In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a complex point arises from a holomorphic Poisson structure. In the proof we use a smoothed Newton’s method along the lines of Nash, Moser and Conn. In the second topic, we consider whether a given regular Poisson structure and transverse complex structure come from a generalized complex structure. We give cohomological criteria, and we find some counterexamples and some unexpected examples, including a compact, regular generalized complex manifold for which nearby symplectic leaves are not symplectomorphic. In the third topic, we consider generalized complex structures with nondegenerate type change; we describe a generalized Calabi-Yau structure induced on the type change locus, and prove a local normal form theorem near this locus. Finally, in the fourth topic, we give a classification of generalized complex principal bundles satisfying a certain transversality condition; in this case, there is a generalized flat connection, and the classification involves a monodromy map to the Courant automorphism group.

We study a number of local and global classification problems in generalized complex geometry. Generalized complex geometry is a relatively new type of geometry which has applications to string theory and mirror symmetry. Symplectic and complex geometry are special cases. In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a complex point arises from a holomorphic Poisson structure. In the proof we use a smoothed Newton’s method along the lines of Nash, Moser and Conn. In the second topic, we consider whether a given regular Poisson structure and transverse complex structure come from a generalized complex structure. We give cohomological criteria, and we find some counterexamples and some unexpected examples, including a compact, regular generalized complex manifold for which nearby symplectic leaves are not symplectomorphic. In the third topic, we consider generalized complex structures with nondegenerate type change; we describe a generalized Calabi-Yau structure induced on the type change locus, and prove a local normal form theorem near this locus. Finally, in the fourth topic, we give a classification of generalized complex principal bundles satisfying a certain transversality condition; in this case, there is a generalized flat connection, and the classification involves a monodromy map to the Courant automorphism group.

Noncommutative Riemannian geometry is an area that has seen intense activity over the past 25 years. Despite this, noncommutative complex geometry is only now beginning to receive serious attention. The theory of quantum groups provides a large family of very interesting potential examples, namely the quantum flag manifolds. Thus far, only the irreducible quantum flag manifolds have been investigated as noncommutative complex spaces. In a series of papers, Heckenberger and Kolb showed that for each of these spaces, there exists a q-deformed Dolbeault double complex. In this thesis a comprehensive framework for noncommutative complex geometry on quantum homogeneous spaces is introduced. The main ingredients used are covariant differential calculi and Takeuchi's categorical equivalence for faithfully at quantum homogeneous spaces. A number of basic results are established, producing a simple set of necessary and sufficient conditions for noncommutative complex structures to exist. It is shown that when applied to the quantum projective spaces, this theory reproduces the q-Dolbeault double complexes of Heckenberger and Kolb. Furthermore, the framework is used to q-deform results from Borel{Bott{ Weil theory, and to produce the beginnings of a theory of noncommutative Kahler geometry.

This thesis takes a Kleinian approach to hyperbolic geometry in order to illustrate the importance of discrete subgroups and their fundamental domains (fundamental regions). A brief history of Euclids Parallel Postulate and its relation to the discovery of hyperbolic geometry be given first. We will explore two models of hyperbolic $n$-space: $U^n$ and $B^n$. Points, lines, distances, and spheres of these two models will be defined and examples in $U^2$, $U^3$, and $B^2$ will be given. We will then discuss the isometries of $U^n$ and $B^n$. These isometries, known as M\"obius transformations, have special properties and turn out to be linear fractional transformations when in $U^2$ and $B^2$. We will then study a bit of topology, specifically the topological groups relevant to the group of isometries of hyperbolic $n$-space, $I(H^n)$. Finally we will combine what we know about hyperbolic $n$-space and topological groups in order to study fundamental regions, fundamental domains, Dirichlet domains, and quotient spaces. Using examples in $U^2$, we will then illustrate how useful fundamental domains are when it comes to visualizing the geometry of quotient spaces.

In this thesis we construct an analogue in tropical geometry for a class of Schubert varieties from classical geometry. In particular, we look at the collection of tropical lines contained in the fan tropical plane. We call these tropical spaces "tropical Schubert prevarieties", and develop them after creating a tropical analogue for flag varieties that we call the "flag Dressian". Having constructed this tropical analogue of Schubert varieties we then determine that the 2-skeleton of these tropical Schubert prevarieties is realizable. In fact, as long as the lift of the fan tropical plane is in general position, only the 2-skeleton of the tropical Schubert prevariety is realizable. / text

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy