This thesis explores aspects of generalized geometry, a geometric framework introduced by Hitchin and Gualtieri in the early 2000s. In the first part, we introduce a new class of submanifolds in stable generalized complex manifolds, so-called Lagrangian branes with boundary. We establish a correspondence between stable generalized complex geometry and log symplectic geometry, which allows us to prove results on local neighbourhoods and small deformations of this new type of submanifold. We further investigate Lefschetz thimbles in stable generalized complex Lefschetz fibrations and show that Lagrangian branes with boundary arise in this context. Stable generalized complex geometry provides the simplest examples of generalized complex manifolds which are neither complex nor symplectic, but it is sufficiently similar to symplectic geometry for a multitude of symplectic results to generalize. Our results on Lefschetz thimbles in stable generalized complex geometry indicate that Lagrangian branes with boundary are part of a potential generalisation of the Wrapped Fukaya category to stable generalized complex manifolds. The work presented in this thesis should be seen as a first step towards the extension of Floer theory techniques to stable generalized complex geometry, which we hope to develop in future work. The second part of this thesis studies Dorfman brackets, a generalisation of the Courant- Dorfman bracket, within the framework of double vector bundles. We prove that every Dorfman bracket can be viewed as a restriction of the Courant-Dorfman bracket on the standard VB-Courant algebroid, which is in this sense universal. Dorfman brackets have previously not been considered in this context, but the results presented here are reminiscent of similar results on Lie and Dull algebroids: All three structures seem to fit into a more general duality between subspaces of sections of the standard VB-Courant algebroid and brackets on vector bundles of the form T M ⊕ E ∗ , E → M a vector bundle. We establish a correspondence between certain properties of the brackets on one, and the subspaces on the other side.

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The rigid body has been one of the most noteworthy applications of Newtonian mechanics. Applying the principles of classical mechanics to the rigid body is by no means routine. The equations of motion, though discovered two hundred and fifty years ago by Euler, have remained quite elusive since their introduction. Understanding the rigid body has required the applications of concepts from integrable systems, algebraic geometry, Lie groups, representation theory, and symplectic geometry to name a few. Moreover, several important developments in these fields have in fact originated with the study of the rigid body and subsequently have grown into general theories with much wider applications.In this work, we study the stability of equilibria of non-degenerate orbits of the generalized rigid body. The energy-Casimir method introduced by V.I. Arnold in 1966 allows us to prove stability of certain non-degenerate equilibria of systems on Lie groups. Applied to the three dimensional rigid body, it recovers the classical Euler stability theorem [12]: rotations around the longest and shortest principal moments of inertia are stable equilibria. This method has not been applied to the analysis of rigid body dynamics beyond dimension n = 3. Furthermore, no conditions for the stability of equilibria are known at all beyond n = 4, in which case the conditions are not of the elegant longest/shortest type [10].Utilizing the rich geometric structures of the symmetry group G = SO(2n), we obtain stability results for generic equilibria of the even dimensional free rigid body. After obtaining a general expression for the generic equilibria, we apply the energy-Casimir method and find that indeed the classical longest/shortest conditions on the entries of the inertia matrix are suffcient to prove stability of generic equilibria for the generalized rigid body in even dimensions.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21541号 / 理博第4448号 / 新制||理||1639(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 小野 薫, 教授 向井 茂, 教授 望月 拓郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

This thesis is mainly concerned with the study of hypersymplectic structures in gauge theory. These structures arise via applications of the hypersymplectic quotient construction to the action of the gauge group on certain spaces of connections and Higgs fields. Motivated by Kobayashi-Hitchin correspondences in the case of hyperkähler moduli spaces, we first study the relationship between hypersymplectic, complex and paracomplex quotients in the spirit of Kirwan's work relating Kähler quotients to GIT quotients. We then study dimensional reductions of the ASD equations on $mathbb R^{2,2}$. We discuss a version of twistor theory for hypersymplectic manifolds, which we use to put the ASD equations into Lax form. Next, we study Schmid's equations from the viewpoint of hypersymplectic quotients and examine the local product structure of the moduli space. Then we turn towards the integrability aspects of this system. We deduce various properties of the spectral curve associated to a solution and provide explicit solutions with cyclic symmetry. Hitchin's harmonic map equations are the split signature analogue of the self-duality equations on a Riemann surface, in which case it is known that there is a smooth hyperkähler moduli space. In the case at hand, we cannot expect to obtain a globally well-behaved moduli space. However, we are able to construct a smooth open set of solutions with small Higgs field, on which we then analyse the hypersymplectic geometry. In particular, we exhibit the local product structures and the family of complex structures. This is done by interpreting the equations as describing certain geodesics on the moduli space of unitary connections. Using this picture we relate the degeneracy locus to geodesics with conjugate endpoints. Finally, we present a split signature version of the ADHM construction for so-called split signature instantons on $S^2 imes S^2$, which can be given an interpretation as a hypersymplectic quotient.

Esta tese estuda a geometria de problemas variacionais através da linearização e projetivização das suas equações de Euler - Lagrange. O processo de linearização fornece a passagem das equações de Euler - Lagrange para as equações de Jacobi; a minimalidade (local) de extremais está determinada pelo conceito de ponto conjugado, que tem natureza projetiva. Propriedades de minimalidade local são transformadas em propriedades de auto-interseção de uma curva na variedade de Grassmann adequada. Desenvolvemos este processo em duas aplicações: 1) O estudo da minimalidade local de extremais de problemas variacionais de ordem superior. Neste caso, encontramos uma curva não degenerada de planos isotrópicos num espaço vetorial simplético, que, após prolongamento por derivadas, fornece uma curva degenerada de planos Lagrangeanos cujas auto-interseções determinam a minimalidade. 2) No caso mais clássico de problemas de ordem um, estudamos a versão linear - projetiva do problema inverso: dada uma equação diferencial de ordem dois, quando ela é a equação de Euler - Lagrange de um problema variacional? Veremos que as condições do problema inverso linear - projetivo fornecem informações sobre os possíveis Lagrangianos, por exemplo a assinatura. / In this work we study the geometry of high order calculus of variations through the linearization and projectivization of their Euler Lagrange equations. The linearization process provides the passage from the Euler Lagrange equations to the Jacobi equations; the (local) minimality properties of the extremal is determined by conjugate points, which is a projective concept. Minimaltiy properties of the extremals are transformed into self-intersection propertie of curves in the appropriate Grassmann manifold. We develop this process in two instances: 1) The study of minimality properties of extremals of higher-order variational problems. In this case, we find a non-degenerate curve of isotropic subspaces, that, after prolongation by derivatives, gives a degenerate curve of Lagrangian planes whose self-intersections determine minimality. 2) In the classical case of order one variational problems, we study a projective-linear version of the inverse problem: given a second order differential equation, when is it the Euler-Lagrange equation of a variational problem? We will see that the conditions given by the linear projective inverse problem provides information about the possible Lagrangians, for example, its signature.

Mon travail de thèse se situe dans le cadre de l'analyse semi-classique. Il se divise en trois parties. Dans la première, j'ai étudié l'opérateur de Klein-Gordon semi-classique en dimension un. Dans la zone où le potentiel reste sous le niveau d'énergie, il existe pour cet opérateur des constructions de solutions WKB, similaires à celles développées pour l'opérateur de Schrödinger. Sous certaines hypothèses, on a prolongé ces solutions hors de cette zone, grâce aux méthodes utilisées près des points tournants pour l'opérateur de Schrödinger. On a ensuite étudié un exemple pour lequel on peut faire des calculs explicites. Enfin, en dimension quelconque, on a obtenu une nouvelle majoration des fonctions propres, lorsque la distance d'Agmon associée à cet opérateur a un gradient lipschitzien. La deuxième partie concerne l'opérateur de Schrödinger et l'étude des résonances en dimension un. Lorsque le potentiel présente deux puits et une mer pour les niveaux d'énergies considérés, on a obtenu des conditions de non croisement des résonances ainsi que leur graphe, grâce à la construction de modes. En présence d'un nombre quelconque de puits, cela permet également de calculer une estimation de la partie imaginaire des résonances dans le cas d'une interaction simple. Enfin, dans la troisième partie, on considère un opérateur de Schrödinger dont le potentiel présente un maximum non dégénéré. On a étudié les résonances générées par une courbe homocline qui passe par ce maximum. En dimension un, on a obtenu une condition de quantification, et par suite les résonances recherchées. En dimension quelconque, on a construit une solution asymptotique sortante le long de cette courbe, en adaptant la méthode de B. Helffer et J. Sjöstrand pour le fond de puits non résonnant. Une transformation FBI permet ensuite de conjecturer un premier niveau de résonances.

Le sujet de cette thèse est la géométrie de contact, en particulier l'étude des orbites périodiques du champ de Reeb. Colin et Honda ont conjecturé que sur une variété hyperbolique munie d'une structure de contact universellement tendue, le nombre d'orbites périodiques de Reeb croit exponentiellement avec la période. Dans les cas non hyperboliques, ils prédisent un comportement polynomial de l'homologie de contact. On montre dans ce texte qu'une variété possédant une composante hyperbolique qui fibre sur le cercle porte une infinité de structures de contact non isomorphes pour lesquelles le nombre d'orbites périodiques de tout champ de Reeb non dégénéré croit exponentiellement avec la période. Ce résultat s'obtient grâce à un résultat de croissance de l'homologie de contact. De plus, on calcule l'homologie de contact et sa croissance dans un cas non hyperbolique : celui des structures universellement tendues non transversales aux fibres sur un fibré en cercles. Enfin, on étudie l'effet d'un recollement de rocade sur les orbites périodiques de Reeb. Cette opération décrit une modification élémentaire de la structure de contact. Elle consiste en l'attachement d'un demi-disque vrillé le long d'un arc legendrien contenu dans le bord de la variété. On montre que les orbites de Reeb créées s'expriment comme mots en les cordes de Reeb de l'arc d'attachement. On calcule l'homologie de contact d'un voisinage produit d'une surface convexe après recollement de rocade ainsi que de certaines structures sur le tore plein.

L'ensemble des opérateurs de Cauchy-Riemann réels sur un fibré vectoriel complexe N muni d'une structure réelle cN au-dessus d'une courbe réelle est un espace affine de dimension infinie. L'union des déterminants de ces opérateurs est un fibré en droites réelles au-dessus de cet espace. L'objet de cette thèse est l'étude de l'action des automorphismes du fibré (N, cN) sur les orientations de ce fibré déterminant ainsi que de ses conséquences sur l'orientabilité des espaces de modules de courbes réelles dans une variété symplectique réelle. Nous commençons par interpréter l'action des automorphismes qui induisent l'identité sur le fibré en droites complexes det(N) en termes d'action sur les structures Pin± de la partie réelle de N. Nous remarquons ensuite qu'un automorphisme au-dessus de l'identité agit sur les classes de bordisme de structures Spin réelles de la courbe et nous utilisons cette action afin d'obtenir une description en termes topologiques de l'action sur les orientations du fibré déterminant. Enfin, pour comprendre l'action des automorphismes de (N, cN) qui ne relèvent pas l'identité, nous introduisons la notion de relevé d'un difféomorphisme de la courbe associé à un diviseur compatible avec (N, cN) et nous calculons le signe de l'action d'un tel relevé sur les orientations du fibré déterminant. Dans une dernière partie, nous appliquons les résultats obtenus à l'étude de l'orientabilité des espaces de modules de courbes réelles dans des variétés symplectiques réelles. Nous calculons en particulier la première classe de Stiefel-Whitney de l'espace de modules des courbes réelles dans l'espace projectif complexe de dimension trois.

This thesis studies U(1) Chern-Simons theory and its relation to the results of Chris Beasley and Edward Witten (2005). Using the partition function formalism, we are led to compare U(1) Chern-Simons theory as constructed by Manoliu (1998) to the results of Beasley and Witten (2005). This leads to an explicit calculation of the U(1) Chern Simons partition function on a closed Sasakian three-manifold and opens the door to studying rigorous extensions of this theory to more general gauge groups and three-manifold geometries.