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In this thesis we extend Lerman’s cutting construction to spin-c structures. Every spin-c structure on an even-dimensional Riemannian manifold gives rise to a Dirac operator D+ acting on sections of the associated spinor bundle. The spin-c quantization of a spin-c manifold is defined to be ker(D+)−coker(D+). It is a virtual vector space, and in the presence of a Lie group action, it is a virtual representation. In 2004, Guillemin et al defined signature quantization and showed that it is additive under cutting. We prove that the spin-c quantization of an S^1-manifold is also additive under cutting. Our proof uses the method of localization, i.e., we express the spin-c quantization of a manifold in terms of local data near connected components of the fixed point set. For a symplectic manifold (M,ω), a spin-c prequantization is a spin-c structure together with a connection compatible with ω. We explain how one can cut a spin-c prequantization and show that the choice of a spin-c structure on the complex plane (which is part of the cutting process) must be compatible with the moment level set along which the cutting is performed. Finally, we prove that the spin-c and metaplectic-c groups satisfy a universal property: Every structure that makes the construction of a spinor bundle possible must factor uniquely through a spin-c structure in the Riemannian case, or through a metaplectic-c structure in the symplectic case.

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.

A Lagrange--d'Alembert integrator is a geometric numerical method for finding numerical solutions to the Lagrange--d'Alembert equations for mechanical systems with nonholonomic constraints that are linear in the velocities. The integrator is developed from geometry and principles that are analogues of the continuous theory.<p>Using discrete analogues of the symplectic form and momentum map, the resulting methods are symplectic and momentum preserving whenever the continuous system is symplectic and momentum preserving. In addition, it is possible to, in principle, generate Lagrange--d'Alembert integrators of any method order.

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].

In this thesis we extend Lerman’s cutting construction to spin-c structures. Every spin-c structure on an even-dimensional Riemannian manifold gives rise to a Dirac operator D+ acting on sections of the associated spinor bundle. The spin-c quantization of a spin-c manifold is defined to be ker(D+)−coker(D+). It is a virtual vector space, and in the presence of a Lie group action, it is a virtual representation. In 2004, Guillemin et al defined signature quantization and showed that it is additive under cutting. We prove that the spin-c quantization of an S^1-manifold is also additive under cutting. Our proof uses the method of localization, i.e., we express the spin-c quantization of a manifold in terms of local data near connected components of the fixed point set. For a symplectic manifold (M,ω), a spin-c prequantization is a spin-c structure together with a connection compatible with ω. We explain how one can cut a spin-c prequantization and show that the choice of a spin-c structure on the complex plane (which is part of the cutting process) must be compatible with the moment level set along which the cutting is performed. Finally, we prove that the spin-c and metaplectic-c groups satisfy a universal property: Every structure that makes the construction of a spinor bundle possible must factor uniquely through a spin-c structure in the Riemannian case, or through a metaplectic-c structure in the symplectic case.

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.

<html> <head> <meta http-equiv="Content-Type" content="text/html;charset=iso-8859-1"> </head> Let &Sigma;<em>_g</em> be a closed Riemann surface of genus <em>g</em>. Generalizing Ivan Smith's construction, for each <em>g</em> &ge; 1 and <em>h</em> &ge; 0 we construct an infinite set of infinite families of homotopic but pairwise non-isotopic symplectic surfaces inside the product symplectic manifold &Sigma;<em>_g</em> ×&Sigma;<em>_h</em>. In particular, we achieve all positive genera from these families, providing first examples of infinite families of homotopic but pairwise non-isotopic symplectic surfaces of even genera inside &Sigma;<em>_g</em> ×&Sigma;<em>_h</em>.