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.

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.

Made available in DSpace on 2014-12-02T11:16:50Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-12-05Bitstream added on 2014-12-02T11:21:26Z : No. of bitstreams: 1 000793711.pdf: 1081771 bytes, checksum: 9ad4a08d3ec9d6accf66ef005a138f0a (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Nosso objetivo neste trabalho é demonstrar o Teorema da Densidade Geral que é um resultado análogo ao Teorema de Kupka-Smale para campos de vetores hamiltonianos. O Teorema da Densidade Geral afirma que o conjuntos dos campos hamiltonianos em uma variedade simplética M que possuem a propriedade H2-N é residual em Xk H(M). Começamos estabelecendo as teorias simpléticas linear e não-linear básicas e depois estudamos suas conexões com os sistemas hamiltonianos, provando os principais resultados da teoria e alguns resultados relacionados. Recebem destaque o estudo das curvas genéricas de matrizes simpléticas, a noção de funções geradoras de difeomorfismos simpléticos e sua aplicação na questão da estabilidade dos pontos fixos elípticos de campos hamiltonianos, a qual é respondida parcialmente através da Forma Normal de Birkhoff. Depois de estabelecer os resultados necessários, passamos a estudar a dinâmica hamiltoniana do ponto de vista das famílias a um parâmetro de difeomorfismos simpléticos. Provamos um resultado devido a Pugh e consideramos a questão da estabilidade estrutural de certas famílias de difeomorfismos simpléticos. Finalmente, provamos o Teorema da Densidade Geral usando a noção de pseudotransversalidade dada no Apêndice C. Este trabalho é baseado nas notas de aula Lectures on Hamiltonian Systems do professor R. Clark Robinson / In this work our goal is to prove the General Density Theorem which is an analogous result for hamiltonian vector fields of the Kupka-Smale Theorem. The General Density Theorem states that the set of hamiltonian vector fields on a symplectic manifold M that has the property H2-N is a residual subset of Xk H(M). We begin by stating the basic linear and nonlinear symplectic theory and then we study its connections with hamiltonian systems, proving some of the main theorems of the theory and other related results. Here we give special attention to topics like generic curves of symplectic matrices, generating functions of symplectic diffeomorphisms and their applications in the problem of the stability of eliptic fixed points of hamiltonian systems, which is partially solved using the Birkhoff Normal Form. After stating the necessary results, we begin to study some hamiltonian dynamics using one-parameter families of symplectic diffeomorphisms. We prove a result stated by Pugh and consider the problem of structural stability of a certain type of one-parameter family. Finally we prove the General Density Theorem using the notion of pseudotransversality given in Appendix C. This work is based on the lecture notes Lectures on Hamiltonian Systems of professor R. Clark Robinson

Orientador: Vanderlei Minori Horita / Banca: Thiago Aparecido Catalan / Banca: Claudio Aguinaldo Buzzi / Resumo: Nosso objetivo neste trabalho é demonstrar o Teorema da Densidade Geral que é um resultado análogo ao Teorema de Kupka-Smale para campos de vetores hamiltonianos. O Teorema da Densidade Geral afirma que o conjuntos dos campos hamiltonianos em uma variedade simplética M que possuem a propriedade H2-N é residual em Xk H(M). Começamos estabelecendo as teorias simpléticas linear e não-linear básicas e depois estudamos suas conexões com os sistemas hamiltonianos, provando os principais resultados da teoria e alguns resultados relacionados. Recebem destaque o estudo das curvas genéricas de matrizes simpléticas, a noção de funções geradoras de difeomorfismos simpléticos e sua aplicação na questão da estabilidade dos pontos fixos elípticos de campos hamiltonianos, a qual é respondida parcialmente através da Forma Normal de Birkhoff. Depois de estabelecer os resultados necessários, passamos a estudar a dinâmica hamiltoniana do ponto de vista das famílias a um parâmetro de difeomorfismos simpléticos. Provamos um resultado devido a Pugh e consideramos a questão da estabilidade estrutural de certas famílias de difeomorfismos simpléticos. Finalmente, provamos o Teorema da Densidade Geral usando a noção de pseudotransversalidade dada no Apêndice C. Este trabalho é baseado nas notas de aula Lectures on Hamiltonian Systems do professor R. Clark Robinson / Abstract: In this work our goal is to prove the General Density Theorem which is an analogous result for hamiltonian vector fields of the Kupka-Smale Theorem. The General Density Theorem states that the set of hamiltonian vector fields on a symplectic manifold M that has the property H2-N is a residual subset of Xk H(M). We begin by stating the basic linear and nonlinear symplectic theory and then we study its connections with hamiltonian systems, proving some of the main theorems of the theory and other related results. Here we give special attention to topics like generic curves of symplectic matrices, generating functions of symplectic diffeomorphisms and their applications in the problem of the stability of eliptic fixed points of hamiltonian systems, which is partially solved using the Birkhoff Normal Form. After stating the necessary results, we begin to study some hamiltonian dynamics using one-parameter families of symplectic diffeomorphisms. We prove a result stated by Pugh and consider the problem of structural stability of a certain type of one-parameter family. Finally we prove the General Density Theorem using the notion of pseudotransversality given in Appendix C. This work is based on the lecture notes Lectures on Hamiltonian Systems of professor R. Clark Robinson / Mestre

This work investigates the dynamics of magnetic flows on closed orientable Riemannian surfaces. These flows are determined by triples (M, g, σ), where M is the surface, g is the metric and σ is a 2-form on M . Such dynamical systems are described by the Hamiltonian equations of a function E on the tangent bundle TM endowed with a symplectic form ω_σ, where E is the kinetic energy. Our main goal is to prove existence results for a) periodic orbits, and b) Poincare sections for motions on a fixed energy level Σ_m := {E = m^2/2} ⊂ T M . We tackle this problem by studying the contact geometry of the level set Σ_m . This will allow us to a) count periodic orbits using algebraic invariants such as the Symplectic Cohomology SH of the sublevels ({E ≤ m^2/2}, ω_σ ); b) find Poincare sections starting from pseudo-holomorphic foliations, using the techniques developed by Hofer, Wysocki and Zehnder in 1998. In Chapter 3 we give a proof of the invariance of SH under deformation in an abstract setting, suitable for the applications. In Chapter 4 we present some new results on the energy values of contact type. First, we give explicit examples of exact magnetic systems on T^2 which are of contact type at the strict critical value. Then, we analyse the case of non-exact systems on M different from T^2 and prove that, for large m and for small m with symplectic σ, Σ_m is of contact type. Finally, we compute SH in all cases where Σ_m is convex. On the other hand, we are also interested in non-exact examples where the contact property fails. While for surfaces of genus at least two, there is always a level not of contact type for topological reasons, this is not true anymore for S^2 . In Chapter 5, after developing the theory of magnetic flows on surfaces of revolution, we exhibit the first example on S^2 of an energy level not of contact type. We also give a numerical algorithm to check the contact property when the level has positive magnetic curvature. In Chapter 7 we restrict the attention to low energy levels on S^2 with a symplectic σ and we show that these levels are of dynamically convex contact type. Hence, we prove that, in the non-degenerate case, there exists a Poincare section of disc-type and at least an elliptic periodic orbit. In the general case, we show that there are either 2 or infinitely many periodic orbits on Σ_m and that we can divide the periodic orbits in two distinguished classes, short and long, depending on their period. Then, we look at the case of surfaces of revolution, where we give a sufficient condition for the existence of infinitely many periodic orbits. Finally, we discuss a generalisation of dynamical convexity introduced recently by Abreu and Macarini, which applies also to surfaces with genus at least two.

In this thesis we discuss various classical problems in enumerative geometry. We are focused on ideas and methods which can be used explicitly for practical computations. Our approach is based on studying the limits of elliptic stable envelopes with shifted equivariant or Kahler variables from elliptic cohomology to K-theory. We prove that for a variety X we can obtain K-theoretic stable envelopes for the variety of the G-fixed points of X, where G is a cyclic group acting on X preserving the symplectic form. We formalize the notion of symplectic duality, also known as 3-dimensional mirror symmetry. We obtain a factorization theorem about the limit of elliptic stable envelopes to a wall, which generalizes the result of M. Aganagic and A. Okounkov. This approach allows us to extend the action of quantum groups, quantum Weyl groups, R-matrices etc., to actions on the K-theory of the symplectic dual variety. In the case of X = Hilb, our results imply the conjectures of E. Gorsky and A. Negut. We propose a new approach to K-theoretic quantum difference equations.