The following is my M.Sc. thesis on moduli space techniques in algebraic and symplectic geometry. It is divided into the following two parts: the rst part is devoted to presenting moduli problems in algebraic geometry using a modern perspective, via the language of stacks and the second part is devoted to studying moduli problems from the perspective of symplectic geometry. The key motivation to the rst part is to present the theorem of Keel and Mori [20] which answers the classical question of under what circumstances a quotient exists for the action of an algebraic group G acting on a scheme X. Part two of the thesis is a more elaborate description of the topics found in Chapter 8 of [28].

The following is my M.Sc. thesis on moduli space techniques in algebraic and symplectic geometry. It is divided into the following two parts: the rst part is devoted to presenting moduli problems in algebraic geometry using a modern perspective, via the language of stacks and the second part is devoted to studying moduli problems from the perspective of symplectic geometry. The key motivation to the rst part is to present the theorem of Keel and Mori [20] which answers the classical question of under what circumstances a quotient exists for the action of an algebraic group G acting on a scheme X. Part two of the thesis is a more elaborate description of the topics found in Chapter 8 of [28].

This thesis is devoted to subriemannian optimal transportation problems. In the first part of the thesis, we consider cost functions arising from very general optimal control costs. We prove the existence and uniqueness of an optimal map between two given measures under certain regularity and growth assumptions on the Lagrangian, absolute continuity of the measures with respect to the Lebesgue class, and, most importantly, the absence of sharp abnormal minimizers. In particular, this result is applicable in the case where the cost function is square of the subriemannian distance on a subriemannian manifold with a 2-generating distribution. This unifies and generalizes the corresponding Riemannian and subriemannian results of Brenier, McCann, Ambrosio-Rigot and Bernard-Buffoni. We also establish various properties of the optimal plan when abnormal minimizers are present. The second part of the thesis is devoted to the infinite-dimensional geometry of optimal transportation on a subriemannian manifold. We start by proving the following nonholonomic version of the classical Moser theorem: given a bracket-generating distribution on a connected compact manifold (possibly with boundary), two volume forms of equal total volume can be isotoped by the flow of a vector field tangent to this distribution. Next, we describe formal solutions of the corresponding subriemannian optimal transportation problem and present the Hamiltonian framework for both the Otto calculus and its subriemannian counterpart as infinite-dimensional Hamiltonian reductions on diffeomorphism groups. Finally, we define a subriemannian analog of the Wasserstein metric on the space of densities and prove that the subriemannian heat equation defines a gradient flow on the subriemannian Wasserstein space with the potential given by the Boltzmann relative entropy functional. Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In the third part of the thesis, we discuss when a three dimensional contact subriemannian manifold satisfies such property.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1882. / Vita.

Thesis (M. Phil.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 56-59).

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 177-183). Available also in a digital version from Dissertation Abstracts.

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Thesis (Ph.D.)--University of Adelaide, School of Mathematical Sciences, Discipline of Pure Mathematics, 2004. / "Submitted September 2002 ... Amended September 2004." Bibliography: p. 152-157.

Thesis (Ph. D.)--University of Hong Kong, 2008. / Includes bibliographical references (leaf 31-35) Also available in print.

Thesis (Ph.D.) - Carleton University, 2002. / Includes bibliographical references (p. 133-145). Also available in electronic format on the Internet.