Thèses sur le sujet « Symplectic and Poisson geometry »

This thesis is concerned principally with tangential geometry and the applications of these concepts to tangentially symplectic foliations. The subject of tangential geometry is still at an elementary stage. The author here systematises current concepts and results and extends them, leading to the definition of vertical connections and vertical G-structures. Tangentially symplectic foliations are then characterised in terms of vertical symplectic forms. Some significant particular cases are discussed.

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.

Orientador: Paulo Regis Caron Ruffino
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: O objetivo desse trabalho é estudar as equações de Hamilton no contexto estocástico. Sendo necessário para tal um pouco de conhecimento a cerca dos seguintes assuntos: cálculo estocástico, geometria de segunda ordem, estruturas simpléticas e de Poisson. Abordamos importantes resultados, dentre eles o teorema de Darboux (coordenadas locais) em variedades simpléticas, teorema de Lie-Weinstein que de certa forma generaliza o teorema de Darboux em variedades de Poisson. Veremos que apesar de o ambiente natural para se estudar sistemas hamiltonianos ser variedades simpléticas, no caso estocástico esses sistemas se adaptam bem em variedades de Poisson. Além disso, para atingir a nossa meta, estudaremos equações diferenciais estocásticas em variedades de dimensão finita usando o operador de Stratonovich
Abstract: This dissertation deals with transfering Hamilton's equations in stochastic context. This requires some knowledge about the following: stochastic calculus, second order geometry and Poisson and simplectic structures. Important results that will be discussed in this theory are Darboux's theorem (local coordinates) for simplectic manifolds, and Lie-Weintein's theorem that is in a certain way of Darboux's theorem on Poisson manifolds. We will see that although the natural environment for studying hamiltonian systems is symplectic manifolds, if we have a Poisson structure we will still be able to study them. Moreover, to achieve our goal, we will study stochastic differential equations on finite dimensional manifolds using the Stratonovich operator
Mestrado
Geometria Estocastica
Mestre em Matemática

In this thesis we develop a mathematically rigorous framework of the so-called ''classical limit'' of quantum systems and their semi-classical properties. Our methods are based on the theory of strict, also called C*- algebraic deformation quantization. Since this C*-algebraic approach encapsulates both quantum as classical theory in one single framework, it provides, in particular, an excellent setting for studying natural emergent phenomena like spontaneous symmetry breaking (SSB) and phase transitions typically showing up in the classical limit of quantum theories. To this end, several techniques from functional analysis and operator algebras have been exploited and specialised to the context of Schrödinger operators and quantum spin systems. Their semi-classical properties including the possible occurrence of SSB have been investigated and illustrated with various physical models. Furthermore, it has been shown that the application of perturbation theory sheds new light on symmetry breaking in Nature, i.e. in real, hence finite materials. A large number of physically relevant results have been obtained and presented by means of diverse research papers.

We introduce some basic concepts from symplectic geometry, classical mechanics and integrable systems. We use this theory to show that the rational and the trigonometric Calogero-Moser systems, that is the Hamiltonian systems with Hamiltonian  and  respectively are integrable systems. We do this using symplectic reduction on .

This essay is a detailed survey of an article from 1996 published by Franc Forstneric, where he studies symplectic automorphisms of C2n. The vision is to introduce the density property for holomorphic symplectic manifolds. Our idea is that of Dror Varolin when he in 2001 introduced the concept of density property for Stein manifolds. The main result here is the introduction of symplectic shears on C2n equipped with a holomorphic symplectic form and to show that the group generated by finite compositions of symplectic shears is dense in the group of symplectic automorphisms of C2n in the compact-open topology. We give a complete background of the tools from the theory of ordinary differential equations, smooth manifolds, and complex and symplectic geometry that is needed in order to prove this result.
Den här uppsatsen är en detaljerad undersökning av en artikel från 1996 publicerad av Franc Forstneric där han studerar symplektiska automorfismer av C2n. Visionen är att introducera täthetsegenskapen för holomorfa symplektiska mångfalder. Våran idé är som den av Dror Varolin när han 2001 introducerade täthetsegenskapen för Stein mångfalder. Huvudresultatet här är införandet av symplektiska skjuvningar på C2n med en holomorfisk symplektisk form och att visa att gruppen som genereras av ändliga sammansättningar av symplektiska skjuvningar är tät i gruppen av symplektiska automorfismer av C2n i den kompakt-öppna topologin. Vi ger en fullständig bakgrund av de verktyg från teorin om ordinära differentialekvationer, släta mångfalder och komplex och symplektisk geometri som behövs för att visa detta.

Motivé par la dualité de Howe dans la théorie des représentations de groupes de Lie, on cherche une construction analogue en géométrie symplectique, c'est-à-dire on souhaite que sa quantification géométrique décomposé de manière Howe-duale. On trouve que dans le contexte symplectique, le cadre correct est donné par deux groupes de Lie agissant sur la même variété symplectique si ces actions commutent et satisfont la condition de Howe symplectique, i. e., ces actions sont hamiltoniennes et leurs fonctions collectives sont leurs centralisateurs mutuelles dans l'algèbre de Poisson des fonctions lisses sur la variété symplectique. Une fois cette condition est remplie, nous pouvons décrire la structure d'orbites en détail. En particulier, il y a une bijection entre les orbites coadjointes dans une image d'application moment et celles dans l'image de l'autre application moment – or, il est cette bijection que nous appelerons la correspondance d’orbites coadjointes. On poursuit l'étude de la correspondance d’orbites coadjointes et on montre que, si les groupes de Lie qui agissent sont compacts et la variété symplectique est préquantifiable, l'intégralité est préservée par la correspondance. Ainsi, il est possible d'associer en même temps des représentations irréductibles aux deux orbites de la correspondance. Donc, nous avons une bijection entre certaines parties des duaux unitaires des deux groupes de Lie qui agissent sur la variété symplectique. En appliquant des résultats connus qui assurent que la quantification et la réduction commutent, nous constatons que la quantification d’une variété kählerienne (vue comme une représentation du produit des deux groupes qui agissent sur la variété) admet une décomposition en somme direct sans multiplicités de produits tensoriels des représentations irréductibles des deux groupes, les paires étant données par la bijection obtenue précédemment –parfaitement en accord avec la dualité de Howe. Ce résultat principal est accompagné par l’étude de la structure locale d’une variété avec deux actions hamiltoniennes qui commutent, ce qui donne une version locale de la correspondance d'orbites, ainsi que par des réflexions sur la relation entre la correspondance d'orbites coadjointes et la correspondance de feuilles symplectiques généralisées dans des paires duales singulières
Motivated by the representation-theoretic notion of Howe duality, we seek an analogous construction in symplectic geometry in the sense that its geometric quantization decomposes in a Howe dual fashion. We find that in the symplectic context, the correct setting is given by two Lie groups acting on a symplectic manifold when these two actions commute and satisfy the symplectic Howe ondition, i. e., these actions are Hamiltonian and their collective functions are their mutual centralizers in the Poisson algebra of smooth functions on the symplectic manifold. Once this condition is satisfied, we can describe the orbit structure in detail. In particular, there is a bijection between the coadjoint orbits in one moment image and those in the other moment image – this bijection is what we call the coadjoint orbit correspondence. We study the coadjoint orbit correspondence further and show, if the acting Lie groups are compact and the symplectic manifold is prequantizable, that it preserves integrality of the coadjoint orbits, so to both coadjoint orbits in the correspondence an irreducible representation can be associated. We thus have a bijection between certain parts of the unitary duals of both Lie groups acting on the symplectic manifold. Applying known results about the interchangeability of quantization and reduction, we see that for a Kähler manifold, its quantization (as a representation of the product of both groups acting on the manifold) decomposes into a multiplicity-free direct sum of tensor products of irreducibles of the individual groups, the pairs being given by the bijection obtained before – as one would expect according to Howe duality. This main result is accompanied by a study of the local structure of a manifold carrying two commuting Hamiltonian action which proves a local version of the orbit correspondence and by a discussion about the relation of the coadjoint orbit correspondence to the generalized symplectic leaf correspondence in singular dual pairs

Le but de cette thèse est d'étudier les structures symplectiques dans la catégorie des variétés linéaires par morceaux (PL). La question centrale est de déterminer si toute variété symplectique lisse $(M,omega)$ peut être triangulée de manière symplectique, au sens où il existe une variété linéaire par morceaux $K$ et une triangulation $h :K -> M$ telle que $h^*omega$ est une forme symplectique constante par morceaux. Nous étudions d'abord un problème plus simple, qui consiste à trianguler les formes volumes lisses. Étant donnée une variété lisse $M$ munie d'une forme volume $Omega$, nous montrons qu'il existe une triangulation lisse $h :K -> M$ telle que $h^*Omega$ est une forme volume constante par morceaux. En particulier, les variétés symplectiques lisses de dimension 2 admettent donc des triangulations symplectiques. Étant donnée une variété symplectique fermée $(M,omega)$, nous montrons ensuite que pour certaines triangulations lisses $h :K -> M$, on peut, par une modification arbitrairement petite du complexe $K$, supposer que la forme $h^*omega$ est de rang maximal le long de tous les simplexes de $K$. Ce résultat permet d'approximer arbitrairement bien toute variété symplectique fermée par une variété symplectique PL. Nous nous intéressons finalement au cas d'une sous-variété symplectique $M$ d'un espace ambiant qui admet lui-même une triangulation symplectique. Nous montrons qu'il est possible de construire un cobordisme entre la sous-variété $M$ considérée et une approximation lisse par morceaux de celle-ci, triangulée par un complexe symplectique.
In this thesis, we study symplectic structures in a piecewise linear (PL) setting. The central question is to determine whether a smooth symplectic manifold can be triangulated symplectically, in the sense that there exists a triangulation $h :K -> M$ such that $h^*omega$ is a piecewise constant symplectic form on $K$. We first focus on a simpler related problem, and show that any smooth volume form $Omega$ on $M$ can be triangulated. This means that there always exists a triangulation $h :K -> M$ such that $h^*Omega$ is a piecewise constant volume form. In particular, symplectic surfaces admit symplectic triangulations. Given a closed symplectic manifold $(M,omega)$, we then prove that there exists triangulations $h :K -> M$ for which the piecewise smooth form $h^*omega$ has maximal rank along all the simplices of $K$. This result allows to approximate arbitrarily closely any closed symplectic manifold by a PL one. Finally, we investigate the case of a symplectic submanifold $M$ of an ambient space which is itself symplectically triangulated, and give the construction of a cobordism between $M$ and a piecewise smooth approximation of $M$, triangulated by a symplectic complex.
Doctorat en Sciences
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La tesi si concentra sullo studio degli automorfismi di varietà olomorfe simplettiche irriducibili di tipo K3^[n], ovvero varietà equivalenti per deformazione allo schema di Hilbert di n punti su una superficie K3, per n > 1. Negli ultimi anni, molti teoremi classici riguardanti la classificazione degli automorfismi non-simplettici di superfici K3 sono stati estesi alle varietà di tipo K3^[2]. Siamo quindi interessati a comprendere se tali risultati possono essere ulteriormente generalizzati anche al caso di varietà di tipo K3^[n], per n > 2. Nella prima parte della tesi descriviamo il gruppo degli automorfismi dello schema di Hilbert di n punti su una superficie K3 proiettiva generica, il cui reticolo di Picard è generato da un singolo fibrato ampio. Mostriamo che, se il gruppo non è triviale, esso è generato da una involuzione non-simplettica, la cui esistenza è determinata da condizioni aritmetiche che coinvolgono il numero n di punti e la polarizzazione della superficie. In aggiunta a tale caratterizzazione numerica, individuiamo anche delle condizioni necessarie e sufficienti per l'esistenza dell'involuzione riguardanti la struttura del reticolo di Picard dello schema di Hilbert. La seconda parte della tesi è dedicata allo studio degli automorfismi non-simplettici di ordine primo su varietà di tipo K3^[n]. Dopo aver investigato le proprietà del reticolo invariante dell'automorfismo e del suo complemento ortogonale all'interno del secondo reticolo di coomologia della varietà, forniamo una classificazione per le loro classi di isometria. Affrontiamo quindi il problema di individuare varietà di tipo K3^[n] dotate di automorfismi non-simplettici che inducano ognuna delle possibili azioni in coomologia presenti nella nostra classificazione. Nel caso delle involuzioni, e degli automorfismi di ordine primo dispari per n=3, 4, siamo in grado di realizzare tutti i casi ammissibili, presentando una costruzione esplicita della varietà o almeno dimostrandone l'esistenza. Tra i numerosi esempi esibiti, è di particolare rilievo un nuovo automorfismo di ordine tre su una famiglia di dimensione dieci di varietà di Lehn-Lehn-Sorger-van Straten di tipo K3^[4]. Infine, per n < 6 descriviamo le famiglie di deformazione massimali di varietà di tipo K3^[n] dotate di una involuzione non-simplettica.
We study automorphisms of irreducible holomorphic symplectic manifolds of type K3^[n], i.e. manifolds which are deformation equivalent to the Hilbert scheme of n points on a K3 surface, for some n > 1. In the first part of the thesis we describe the automorphism group of the Hilbert scheme of n points on a generic projective K3 surface, i.e. a K3 surface whose Picard lattice is generated by a single ample line bundle. We show that, if it is not trivial, the automorphism group is generated by a non-symplectic involution, whose existence depends on some arithmetic conditions involving the number of points n and the polarization of the surface. We also determine necessary and sufficient conditions on the Picard lattice of the Hilbert scheme for the existence of the involution. In the second part of the thesis we study non-symplectic automorphisms of prime order on manifolds of type K3^[n]. We investigate the properties of the invariant lattice and its orthogonal complement inside the second cohomology lattice of the manifold, providing a classification of their isometry classes. We then approach the problem of constructing examples (or at least proving the existence) of manifolds of type K3^[n] with a non-symplectic automorphism inducing on cohomology each specific action in our classification. In the case of involutions, and of automorphisms of odd prime order for n=3,4, we are able to realize all possible cases. In order to do so, we present a new non-symplectic automorphism of order three on a ten-dimensional family of Lehn-Lehn-Sorger-van Straten eightfolds of type K3^[4]. Finally, for n < 6 we describe deformation families of large dimension of manifolds of type K3^[n] equipped with a non-symplectic involution.
Nous allons étudier les automorphismes des variétés symplectiques holomorphes irréductibles de type K3^[n], c'est-à-dire des variétés équivalentes par déformation au schéma de Hilbert de n points sur une surface K3, pour n > 1. Dans la première partie de la thèse, nous classifions les automorphismes du schéma de Hilbert de n points sur une surface K3 projective générique, dont le réseau de Picard est engendré par un fibré ample. Nous montrons que le groupe des automorphismes est soit trivial soit engendré par une involution non-symplectique et nous déterminons des conditions numériques et géométriques pour l’existence de l’involution. Dans la deuxième partie, nous étudions les automorphismes non-symplectiques d’ordre premier des variétés de type K3^[n]. Nous déterminons les propriétés du réseau invariant de l'automorphisme et de son complément orthogonal dans le deuxième réseau de cohomologie de la variété et nous classifions leurs classes d’isométrie. Dans le cas des involutions, e des automorphismes d’ordre premier impair pour n = 3, 4, nous montrons que toutes les actions en cohomologie dans notre classification sont réalisées par un automorphism non-symplectique sur une variété de type K3^[n]. Nous construisons explicitement l’immense majorité de ces automorphismes et, en particulier, nous présentons la construction d’un nouvel automorphisme d’ordre trois sur une famille de dimension dix de variétés de Lehn-Lehn-Sorger-van Straten de type K3^[4]. Pour n < 6, nous étudions aussi les espaces de modules de dimension maximal des variétés de type K3^[n] munies d’une involution non-symplectique.

This thesis presents a series of results obtained in [13, 18, 19, 23{25, 87]. In [19], we prove a Darboux theorem for derived schemes with symplectic forms of degree k < 0, in the sense of [142]. We use this to show that the classical scheme X = t₀(X) has the structure of an algebraic d-critical locus, in the sense of Joyce [87]. Then, if (X, s) is an oriented d-critical locus, we prove in [18] that there is a natural perverse sheaf P·_X,s on X, and in [25], we construct a natural motive MF_X,s, in a certain quotient ring M^μ_X of the μ-equivariant motivic Grothendieck ring M^μ_X, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants [102]. In [13], we obtain similar results for k-shifted symplectic derived Artin stacks. We apply this theory to categorifying Donaldson-Thomas invariants of Calabi-Yau 3-folds, and to categorifying Lagrangian intersections in a complex symplectic manifold using perverse sheaves, and to prove the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with 'orientation data', as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory [102], and on intersections L??M of oriented Lagrangians L,M in an algebraic symplectic manifold (S,ω). In [23] we show that if (S,ω) is a complex symplectic manifold, and L,M are complex Lagrangians in S, then the intersection X= L??M, as a complex analytic subspace of S, extends naturally to a complex analytic d-critical locus (X, s) in the sense of Joyce [87]. If the canonical bundles K_L,K_M have square roots K^1/2_L, K^1/2_M then (X, s) is oriented, and we provide a direct construction of a perverse sheaf P·_L,M on X, which coincides with the one constructed in [18]. In [24] we have a more in depth investigation in generalized Donaldson-Thomas invariants DT^α(τ) defined by Joyce and Song [85]. We propose a new algebraic method to extend the theory to algebraically closed fields K of characteristic zero, rather than K = C, and we conjecture the extension of generalized Donaldson-Thomas theory to compactly supported coherent sheaves on noncompact quasi-projective Calabi-Yau 3-folds, and to complexes of coherent sheaves on Calabi-Yau 3-folds.

This research seeks to understand the Poisson Geometry of the Ablowitz-Ladik equations (AL), an integrable discretization of the Non-linear Schrodinger equation (NLS) first proposed by Ablowitz and Ladik in the 70's. More specifically, to argue that the AL hierarchy (an integrable hierarchy of equations which comprises AL) can be explicitly viewed as a hierarchy of commuting flows which: (1) are Hamiltonian with respect to both a (known) Poisson operator J, and a (new) non-local, skew, almost Poisson operator K, on the appropriate space; (2) can be recursively generated from an operator R = KJ⁻¹. This thesis also clarifies the geometric framework that underlies a certain class of evolving geodesic linkages related to the AL hierarchy via the evolution for their "discrete" geodesic curvature. In this regard, our results include a geometric interpretation of a compatibility condition associated to a Lax pair for AL and, consequently, a bijective correspondence between AL flows and linkage flows.

In this thesis we study the classi cation problem of boundary singularities of functions in symplectic and volume-preserving geometry. In particular we generalise several well known theorems concerning the classi cation of isolated singularities of functions and volume forms in the presence of a \boundary", i.e. a germ of a xed smooth hypersurface. The results depend in turn on a generalisation of the relative de Rham cohomology and the corresponding Gauss-Manin theory to the case of isolated boundary singularities and in particular, on a relative version of the so called Brieskorn-Deligne-Sebastiani theorem, concerning the niteness and freeness of certain cohomology modules.

Symplectic geometry can be traced back to Lagrange and his work on celestial mechanics and has since then been a very active field in mathematics, partly because of the applications it offers but also because of the beauty of the objects it deals with. I this thesis we begin by the simplest fact of symplectic geometry. We give the definition of a symplectic space and of the symplectic group, Sp(n). A symplectic space is the data of an even-dimensional space and of a form which satisfies a number of properties. Having done this we give a definition of the Lagrangian Grassmannian Lag(n) which consists of all n-dimensional subspaces of the symplectic space on which the symplectic form vanishes. We carefully study the topology of these spaces and their universal coverings. It is of great interest to know how the elements of the Lagrangian Grassmannian intersect each other. A lot of efforts have therefore been made to construct intersection indices for elements of Lag(n). They have gone under many names but have had a sole purpose, namely to give us a way to determine how these elements intersect. We show how these elements are constructed and extend the definition to paths of elements of Lag(n) and Sp(n). We end this thesis by extending the definition of an index defined by Conley and Zehnder bu using the properties of the Leray index. Their index plays a significant role in the theory of periodic Hamiltonian orbit.

Dans cette thèse, on définit et on étudie les notions de structure de Poisson et coïsotrope sur un champ dérivé, dans le contexte de la géométrie algébrique dérivée. On considère deux présentations différentes de structure de Poisson : la première est purement algébrique, alors que la deuxième est plus géométrique. On montre que les deux approches sont en fait équivalentes. On introduit aussi la notion de structure coïsotrope sur un morphisme de champs dérivés, encore une fois en présentant deux définitions équivalentes : la première est basée sur une généralisation appropriée de l'opérade Swiss-Cheese de Voronov, tandis que la deuxième est formulée en termes de champs de multivecteurs rélatifs. En particulier, on montre que le morphisme identité admet une unique structure coïsotrope ; cela produit une application d'oubli des structures de Poisson n-décalées aux structures de Poisson (n-1)-décalées. On montre aussi que l'intersection de deux morphismes coïsotropes dans un champ de Poisson n-décalée est naturellement equipée d'une structure de Poisson (n-1)-décalée canonique. En outre, on fournit une équivalence entre l'espace de structures coïsotropes non-dégénérées et l'espace des structures Lagrangiennes en géométrie dérivée, introduites dans les travaux de Pantev-Toën-Vaquié-Vezzosi
In this thesis, we define and study Poisson and coisotropic structures on derived stacks in the framework of derived algebraic geometry. We consider two possible presentations of Poisson structures of different flavour: the first one is purely algebraic, while the second is more geometric. We show that the two approaches are in fact equivalent. We also introduce the notion of coisotropic structure on a morphism between derived stacks, once again presenting two equivalent definitions: one of them involves an appropriate generalization of the Swiss Cheese operad of Voronov, while the other is expressed in terms of relative polyvector fields. In particular, we show that the identity morphism carries a unique coisotropic structure; in turn, this gives rise to a non-trivial forgetful map from n-shifted Poisson structures to (n-1)-shifted Poisson structures. We also prove that the intersection of two coisotropic morphisms inside a n-shifted Poisson stack is naturally equipped with a canonical (n-1)-shifted Poisson structure. Moreover, we provide an equivalence between the space of non-degenerate coisotropic structures and the space of Lagrangian structures in derived geometry, as introduced in the work of Pantev-Toën-Vaquié-Vezzosi

Résumé de la thèse :ce travail porte sur la notion d'espace symétrique symplectique extrinsèque. Ces espaces sont des espaces symétriques symplectiques dont la structure est induite par le plongement dans variété symplectique ambiante munie d'une connexion.

Par analogie à la théorie standard des espaces symétriques, nous démontrons un théorème d'équivalence entre les espaces symétriques symplectiques extrinsèques d'une variété qui est elle-même un espace symétrique symplectique.

La définition d'un espace symétrique symplectique extrinsèque fait intervenir l'existence d'affinités globales de la variété ambiante, les ``symétries extrinsèques', qui induisent la structure symétrique de la sous-variété ;ceci mène à poser une question du type :quelles sont les variétés possédant ``beaucoup' de ces affinités~? Une question précise ainsi qu'une réponse sont fournies dans un contexte où la variété ambiante est seulement supposée munie d'une structure

symplectique et d'une connexion symplectiques. Nous considérons également le cas où ces symétries commutent avec un champ $K$ d'endomorphismes symplectiques fixé de la variété, de carré $pmId$. Nous définissons une notion de courbure sectionnelle pour plans $K$-stables et montrons que les espaces à $K$-courbure sectionnelle constantes sont localement symétriques de type Ricci.

Par suite nous étudions les espaces symétriques symplectiques extrinsèques dans un espace vectoriel symplectique. Nous montrons par exemple qu'un tel espace, s'ils est de dimension deux, est forcément intrinsèquement plat (c.-à-d. à courbure intrinsèque nulle), mais que son image n'est pas forcément un plan affin de l'espace vectoriel ambiant. Nous décrivons en fait explicitement tous les espaces

symétriques symplectiques extrinsèques, dans un espace vectoriel, dont la courbure intrinsèque s'annule identiquement. Nous décrivons également une famille d'exemples d'espaces extrinsèques, dont nous montrons qu'elle fournit la totalité des espaces extrinsèques de codimension $2$, dans un espace vectoriel.

Enfin, nous décrivons quelques exemples d'espaces symétriques symplectiques extrinsèques qui sont totalement géodésiques, dans un espace de type Ricci particulier.

Doctorat en Sciences
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Nous allons étudier les automorphismes des variétés symplectiques holomorphes irréductibles de type K3^[n], c'est-à-dire des variétés équivalentes par déformation au schéma de Hilbert de n points sur une surface K3, pour n > 1.Dans la première partie de la thèse, nous classifions les automorphismes du schéma de Hilbert de n points sur une surface K3 projective générique, dont le réseau de Picard est engendré par un fibré ample. Nous montrons que le groupe des automorphismes est soit trivial soit engendré par une involution non-symplectique et nous déterminons des conditions numériques et géométriques pour l’existence de l’involution.Dans la deuxième partie, nous étudions les automorphismes non-symplectiques d’ordre premier des variétés de type K3^[n]. Nous déterminons les propriétés du réseau invariant de l'automorphisme et de son complément orthogonal dans le deuxième réseau de cohomologie de la variété et nous classifions leurs classes d’isométrie. Dans le cas des involutions, e des automorphismes d’ordre premier impair pour n = 3, 4, nous montrons que toutes les actions en cohomologie dans notre classification sont réalisées par un automorphism non-symplectique sur une variété de type K3^[n]. Nous construisons explicitement l’immense majorité de ces automorphismes et, en particulier, nous présentons la construction d’un nouvel automorphisme d’ordre trois sur une famille de dimension dix de variétés de Lehn-Lehn-Sorger-van Straten de type K3^[4]. Pour n < 6, nous étudions aussi les espaces de modules de dimension maximal des variétés de type K3^[n] munies d’une involution non-symplectique
We study automorphisms of irreducible holomorphic symplectic manifolds of type K3^[n], i.e. manifolds which are deformation equivalent to the Hilbert scheme of n points on a K3 surface, for some n > 1. In the first part of the thesis we describe the automorphism group of the Hilbert scheme of n points on a generic projective K3 surface, i.e. a K3 surface whose Picard lattice is generated by a single ample line bundle. We show that, if it is not trivial, the automorphism group is generated by a non-symplectic involution, whose existence depends on some arithmetic conditions involving the number of points n and the polarization of the surface. We also determine necessary and sufficient conditions on the Picard lattice of the Hilbert scheme for the existence of the involution.In the second part of the thesis we study non-symplectic automorphisms of prime order on manifolds of type K3^[n]. We investigate the properties of the invariant lattice and its orthogonal complement inside the second cohomology lattice of the manifold, providing a classification of their isometry classes. We then approach the problem of constructing examples (or at least proving the existence) of manifolds of type K3^[n] with a non-symplectic automorphism inducing on cohomology each specific action in our classification. In the case of involutions, and of automorphisms of odd prime order for n=3,4, we are able to realize all possible cases. In order to do so, we present a new non-symplectic automorphism of order three on a ten-dimensional family of Lehn-Lehn-Sorger-van Straten eightfolds of type K3^[4]. Finally, for n < 6 we describe deformation families of large dimension of manifolds of type K3^[n] equipped with a non-symplectic involution

Contents: Part I: Symplectic Geometry Chapter 1: Symplectic Spaces and Lagrangian Planes Chapter 2: The Symplectic Group Chapter 3: Multi-Oriented Symplectic Geometry Chapter 4: Intersection Indices in Lag(n) and Sp(n) Part II: Heisenberg Group, Weyl Calculus, and Metaplectic Representation Chapter 5: Lagrangian Manifolds and Quantization Chapter 6: Heisenberg Group and Weyl Operators Chapter 7: The Metaplectic Group Part III: Quantum Mechanics in Phase Space Chapter 8: The Uncertainty Principle Chapter 9: The Density Operator Chapter 10: A Phase Space Weyl Calculus

I investigate some algebras and calculi naturally associated with the symplectic and metric Clifford algebras. In particular, I reformulate the well known Lepage decomposition for the symplectic exterior algebra in geometrical form and present some new results relating to the simple subspaces of the decomposition. I then present an analogous decomposition for the symmetric exterior algebra with a metric. Finally, I extend this symmetric exterior algebra into a new calculus for the symmetric differential forms on a pseudo-Riemannian manifold. The importance of this calculus lies in its potential for the description of bosonic systems in Quantum Theory.

In this project we introduce the general idea of geometric quantization and demonstratehow to apply the process on a few examples. We discuss how to construct a line bundleover the symplectic manifold with Dirac’s quantization conditions and how to determine if we are able to quantize a system with the help of Weil’s integrability condition. To reducethe prequantum line bundle we employ real polarization such that the system does notbreak Heisenberg’s uncertainty principle anymore. From the prequantum bundle and thepolarization we construct the sought after Hilbert space.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002.
Includes bibliographical references (p. 239-240).
Enumerative geometry of algebraic varieties is a fascinating field of mathematics that dates back to the nineteenth century. We introduce new computational tools into this field that are motivated by recent progress in symplectic topology and its influence on enumerative geometry. The most straightforward applications of the methods developed are to enumeration of rational curves with a cusp of specified nature in projective spaces. A general approach for counting positive-genus curves with a fixed complex structure is also presented. The applications described include enumeration of rational curves with a (3,4)-cusp, genus-two and genus-three curves with a fixed complex structure in the two-dimensional complex projective space, and genus-two curves with a fixed complex structure in the three-dimensional complex projective space. Our constructions may be applicable to problems in symplectic topology as well.
by Aleksey Zinger.
Ph.D.

Let X be a simply connected compact Riemannian symmetric space, let U be the universal covering group of the identity component of the isometry group of X, and let g denote the complexification of the Lie algebra of U, g=u^C. Each u-compatible triangular decomposition g= n_- + h + n_+ determines a Poisson Lie group structure pi_U on U. The Evens-Lu construction produces a (U, pi_U)-homogeneous Poisson structure on X. By choosing the basepoint in X appropriately, X is presented as U/K where K is the fixed point set of an involution which stabilizes the triangular decomposition of g. With this presentation, a connection is established between the symplectic foliation of the Evens-Lu Poisson structure and the Birkhoff decomposition of U/K. This is done through reinterpretation of results of Pickrell. Each symplectic leaf admits a natural torus action. It is shown that these actions are Hamiltonian and the momentum maps are computed using triangular factorization. Finally, local formulas for the Evens-Lu Poisson structure are displayed in several examples.

Symplectic fillings of standard tight contact structures on lens spaces are understood and classified. The situation is different if one considers non-standard tight structures (i.e. those that are virtually overtwisted), for which a classification scheme is still missing. In this work we use different approaches and employ various techniques to improve our knowledge of symplectic fillings of virtually overtwisted contact structures. We study curves configurations on surfaces to solve the problem in the case of a specific family of lens spaces. Then we give general constraints on the topology of Stein fillings of any lens space by looking at algebraic properties of integer lattices and at geometric slicing of solid tori. Furthermore, we try to place these manifolds in the context of algebraic geometry, in order to determine whether Stein fillings can be realized as Milnor fibers of hypersurfce singularities, finding a series of necessary conditions for this to happen. In the concluding part of the thesis, we focus on the connections between planar contact 3-manifolds and the theory of Artin presentations.

Mirror symmetry is the phenomenon, originally discovered by physicists, that Calabi-Yau manifolds come in dual pairs, with each member of the pair producing the same physics. Mathematicians studying enumerative geometry became interested in mirror symmetry around 1990, and since then, mirror symmetry has become a major research topic in pure mathematics. One important problem in mirror symmetry is that there may be several ways to construct a mirror dual for a Calabi-Yau manifold. Hence it is a natural question to ask: when two different mirror symmetry constructions apply, do they agree?We specifically consider two mirror symmetry constructions for K3 surfaces known as BHK and LPK3 mirror symmetry. BHK mirror symmetry was inspired by the LandauGinzburg/Calabi-Yau correspondence, while LPK3 mirror symmetry is more classical. In particular, for algebraic K3 surfaces with a purely non-symplectic automorphism of order n, we ask if these two constructions agree. Results of Artebani Boissi`ere-Sarti originally showed that they agree when n = 2, and more recently Comparin-Lyon-Priddis-Suggs showed that they agree when n is prime. However, the n being composite case required more sophisticated methods. Whenever n is not divisible by four (or n = 16), this problem was solved by Comparin and Priddis by studying the associated lattice theory more carefully. In this thesis, we complete the remaining case of the problem when n is divisible by four by finding new isomorphisms and deformations of the K3 surfaces in question, develop new computational methods, and use these results to complete the investigation, thereby showing that the BHK and LPK3 mirror symmetry constructions also agree when n is composite.

In questa tesi studiamo alcuni sistemi lineari completi associati a divisori di schemi di Hilbert di 2 punti su una superficie K3 proiettiva complessa con gruppo di Picard di rango 1, e le mappe razionali indotte. Queste varietà sono chiamate quadrati di Hilbert su superfici K3 generiche, e sono esempi di varietà irriducibili olomorfe simplettiche (varietà IHS). Nella prima parte della tesi, usando la teoria dei reticoli, gli operatori di Nakajima e il modello di Lehn–Sorger, diamo una base per il sottospazio vettoriale dell’anello di coomologia singolare a coefficienti razionali generato dalle classi di Hodge razionali di tipo (2, 2) sul quadrato di Hilbert di una qualsiasi superficie K3 proiettiva. In seguito sfruttiamo un teorema di Qin e Wang insieme a un risultato di Ellingsrud, Göttsche e Lehn per ottenere una base del reticolo delle classi di Hodge integrali di tipo (2, 2) sul quadrato di Hilbert di una qualsiasi superficie K3 proiettiva. Nella seconda parte della tesi studiamo il problema seguente: se X è il quadrato di Hilbert di una superficie K3 generica che ammette un divisore ampio D con q(D) = 2, dove q è la forma quadratica di Beauville-Bogomolov-Fujiki, descrivere geometricamente la mappa razionale indotta dal sistema lineare completo |D|. Il risultato principale della tesi mostra che tale X, tranne nel caso del quadrato di Hilbert di una superficie quartica generica di P^3, è una doppia EPW sestica, cioè il ricoprimento doppio di una EPW sestica, una ipersuperficie normale di P^5, ramificato nel suo luogo singolare. Inoltre la mappa razionale indotta da |D| coincide proprio con tale ricoprimento doppio. Gli strumenti principali per ottenere questo risultato sono la descrizione del reticolo delle classi integrali di Hodge di tipo (2, 2) della prima parte della tesi e l’esistenza di un’involuzione anti-simplettica su tali varietà per un teorema di Boissière, Cattaneo, Nieper-Wißkirchen e Sarti.
In this thesis we study some complete linear systems associated to divisors of Hilbert schemes of 2 points on complex projective K3 surfaces with Picard group of rank 1, together with the rational maps induced. We call these varieties Hilbert squares of generic K3 surfaces, and they are examples of irreducible holomorphic symplectic (IHS) manifold. In the first part of the thesis, using lattice theory, Nakajima operators and the model of Lehn–Sorger, we give a basis for the subvector space of the singular cohomology ring with rational coefficients generated by rational Hodge classes of type (2, 2) on the Hilbert square of any projective K3 surface. We then exploit a theorem by Qin and Wang together with a result by Ellingsrud, Göttsche and Lehn to obtain a basis of the lattice of integral Hodge classes of type (2, 2) on the Hilbert square of any projective K3 surface. In the second part of the thesis we study the following problem: if X is the Hilbert square of a generic K3 surface admitting an ample divisor D with q(D)=2, where q is the Beauville–Bogomolov–Fujiki form, describe geometrically the rational map induced by the complete linear system |D|. The main result of the thesis shows that such an X, except on the case of the Hilbert square of a generic quartic surface of P^3, is a double EPW sextic, i.e., the double cover of an EPW sextic, a normal hypersurface of P^5, ramified over its singular locus. Moreover, the rational map induced by |D| is a morphism and coincides exactly with this double covering. The main tools to obtain this result are the description of integral Hodge classes of type (2, 2) of the first part of the thesis and the existence of an anti-symplectic involution on such varieties due to a theorem by Boissière, Cattaneo, Nieper-Wißkirchen and Sarti.
Dans cette thèse, nous étudions certains systèmes linéaires complets associés aux diviseurs des schémas de Hilbert de 2 points sur des surfaces K3 projectives complexes avec groupe de Picard de rang 1, et les fonctions rationnelles induites. Ces variétés sont appelées carrés de Hilbert sur des surfaces K3 génériques, et sont un exemple de variété symplectique holomorphe irréductible (variété IHS). Dans la première partie de la thèse, en utilisant la théorie des réseaux, les opérateurs de Nakajima et le modèle de Lehn–Sorger, nous donnons une base pour le sous-espace vectoriel de l’anneau de cohomologie singulière à coefficients rationnels engendré par les classes de Hodge rationnels de type (2, 2) sur le carré de Hilbert de toute surface K3 projective. Nous exploitons ensuite un théorème de Qin et Wang ainsi qu’un résultat de Ellingsrud, Göttsche et Lehn pour obtenir une base du réseau des classes de Hodge intégraux de type (2, 2) sur le carré de Hilbert d’une surface K3 projective quelconque. Dans la deuxième partie de la thèse, nous étudions le problème suivant : si X est le carré de Hilbert d’une surface K3 générique tel que X admet un diviseur ample D avec q(D) = 2, où q est la forme quadratique de Beauville–Bogomolov–Fujiki, on veut décrire géométriquement la fonction rationnelle induite par le système linéaire complet |D|. Le résultat principal de la thèse montre qu’une telle X, sauf dans le cas du carré de Hilbert d’une surface quartique générique de P^3, est une double sextique EPW, c’est-à-dire le revêtement double d’une sextique EPW, une hypersurface normale de P^5, ramifié sur son lieu singulier. En plus la fonction rationnelle induite par |D| est exactement ce revêtement double. Les outils principaux pour obtenir ce résultat sont la description des classes de Hodge intégraux de type (2, 2) de la première partie de la thèse et l’existence d’une involution anti-symplectique sur de telles variétés par un théorème de Boissière, Cattaneo, Nieper-Wißkirchen et Sarti.

The mathematical modeling and performance analysis of cellular networks have seen a major paradigm shift with the application of stochastic geometry. The main purpose of stochastic geometry is to endow probability distributions on the locations of the base stations (BSs) and users in a network, which, in turn, provides an analytical handle on the performance evaluation of cellular networks. To preserve the tractability of analysis, the common practice is to assume complete spatial randomness} of the network topology. In other words, the locations of users and BSs are modeled as independent homogeneous Poisson point processes (PPPs). Despite its usefulness, the PPP-based network models fail to capture any spatial coupling between the users and BSs which is dominant in a multi-tier cellular network (also known as the heterogeneous cellular networks (HetNets)) consisting of macro and small cells. For instance, the users tend to form hotspots or clusters at certain locations and the small cell BSs (SBSs) are deployed at higher densities at these locations of the hotspots in order to cater to the high data demand. Such user-centric deployments naturally couple the locations of the users and SBSs. On the other hand, these spatial couplings are at the heart of the spatial models used in industry for the system-level simulations and standardization purposes. This dissertation proposes fundamentally new spatial models based on stochastic geometry which closely emulate these spatial couplings and are conductive for a more realistic and fine-tuned performance analysis, optimization, and design of cellular networks. First, this dissertation proposes a new class of spatial models for HetNets where the locations of the BSs and users are assumed to be distributed as Poisson cluster process (PCP). From the modeling perspective, the proposed models can capture different spatial couplings in a network topology such as the user hotspots and user BS coupling occurring due to the user-centric deployment of the SBSs. The PCP-based model is a generalization of the state-of-the-art PPP-based HetNet model. This is because the model reduces to the PPP-based model once all spatial couplings in the network are ignored. From the stochastic geometry perspective, we have made contributions in deriving the fundamental distribution properties of PCP, such as the distance distributions and sum-product functionals, which are instrumental for the performance characterization of the HetNets, such as coverage and rate. The focus on more refined spatial models for small cells and users brings to the second direction of the dissertation, which is modeling and analysis of HetNets with millimeter wave (mm-wave) integrated access and backhaul (IAB), an emerging design concept of the fifth generation (5G) cellular networks. While the concepts of network densification with small cells have emerged in the fourth generation (4G) era, the small cells can be realistically deployed with IAB since it solves the problem of high capacity wired backhaul of SBSs by replacing the last-mile fibers with mm-wave links. We have proposed new stochastic geometry-based models for the performance analysis of IAB-enabled HetNets. Our analysis reveals some interesting system-design insights: (1) the IAB HetNets can support a maximum number of users beyond which the data rate drops below the rate of a single-tier macro-only network, and (2) there exists a saturation point of SBS density beyond which no rate gain is observed with the addition of more SBSs. The third and final direction of this dissertation is the combination of machine learning and stochastic geometry to construct a new class of data driven network models which can be used in the performance optimization and design of a network. As a concrete example, we investigate the classical problem of wireless link scheduling where the objective is to choose an optimal subset of simultaneously active transmitters (Tx-s) from a ground set of Tx-s which will maximize the network-wide sum-rate. Since the optimization problem is NP-hard, we replace the computationally expensive heuristic by inferring the point patterns of the active Tx-s in the optimal subset after training a determinantal point process (DPP). Our investigations demonstrate that the DPP is able to learn the spatial interactions of the Tx-s in the optimal subset and gives a reasonably accurate estimate of the optimal subset for any new ground set of Tx-s.
Doctor of Philosophy
The high speed global cellular communication network is one of the most important technologies, and it continues to evolve rapidly with every new generation. This evolution greatly depends on observing performance-trends of the emerging technologies on the network models through extensive system-level simulations. Since these simulation models are extremely time-consuming and error prone, the complementary analytical models of cellular networks have been an area of active research for a long time. These analytical models are intended to provide crisp insights on the network behavior such as the dependence of network performance metrics (such as coverage or rate) on key system-level parameters (such as transmission powers, base station (BS) density) which serve as the prior knowledge for more fine-tuned simulations. Over the last decade, the analytical modeling of the cellular networks has been driven by stochastic geometry. The main purpose of stochastic geometry is to endow the locations of the base stations (BSs) and users with probability distributions and then leverage the properties of these distributions to average out the spatial randomness. This process of spatial averaging allows us to derive the analytical expressions of the system-level performance metrics despite the presence of a large number of random variables (such as BS and user locations, channel gains) under some reasonable assumptions. The simplest stochastic geometry based model of cellular networks, which is also the most tractable, is the so-called Poisson point process (PPP) based network model. In this model, users and BSs are assumed to be distributed as independent homogeneous PPPs. This is equivalent to saying that the users and BSs independently and uniformly at random over a plane. The PPP-based model turned out to be a reasonably accurate representation of the yesteryear’s cellular networks which consisted of a single tier of macro BSs (MBSs) intended to provide a uniform coverage blanket over the region. However, as the data-hungry devices like smart-phones, tablets, and application like online gaming continue to flood the consumer market, the network configuration is rapidly deviating from this baseline setup with different spatial interactions between BSs and users (also termed spatial coupling) becoming dominant. For instance, the user locations are far from being homogeneous as they are concentrated in specific areas like residential and commercial zones (also known as hotspots). Further, the network, previously consisting of a single tier of macro BSs (MBSs), is becoming increasingly heterogeneous with the deployment of small cell BSs (SBSs) with small coverage footprints and targeted to serve the user hotspots. It is not difficult to see that the network topology with these spatial couplings is quite far from complete spatial randomness which is the basis of the PPP-based models. The key contribution of this dissertation is to enrich the stochastic geometry-based mathematical models so that they can capture the fine-grained spatial couplings between the BSs and users. More specifically, this dissertation contributes in the following three research directions. Direction-I: Modeling Spatial Clustering. We model the locations of users and SBSs forming hotspots as Poisson cluster processes (PCPs). A PCP is a collection of offspring points which are located around the parent points which belong to a PPP. The coupling between the locations of users and SBSs (due to their user-centric deployment) can be introduced by assuming that the user and SBS PCPs share the same parent PPP. The key contribution in this direction is the construction of a general HetNet model with a mixture of PPP and PCP-distributed BSs and user distributions. Note that the baseline PPP-based HetNet model appears as one of the many configurations supported by this general model. For this general model, we derive the analytical expressions of the performance metrics like coverage probability, BS load, and rate as functions of the coupling parameters (e.g. BS and user cluster size). Direction-II: Modeling Coupling in Wireless Backhaul Networks. While the deployment of SBSs clearly enhances the network performance in terms of coverage, one might wonder: how long network densification with tens of thousands of SBSs can meet the everincreasing data demand? It turns out that in the current network setting, where the backhaul links (i.e. the links between the BSs and core network) are still wired, it is not feasible to densify the network beyond some limit. This backhaul bottleneck can be overcome if the backhaul links also become wireless and the backhaul and access links (link between user and BS) are jointly managed by an integrated access and backhaul (IAB) network. In this direction, we develop the analytical models of IAB-enabled HetNets where the key challenge is to tackle new types of couplings which exist between the rates on the wireless access and backhaul links. Such couplings exist due to the spatial correlation of the signal qualities of the two links and the number of users served by different BSs. Two fundamental insights obtained from this work are as follows: (1) the IAB HetNets can support a maximum number of users beyond which the network performance drops below that of a single-tier macro-only network, and (2) there exists a saturation point of SBS density beyond which no performance gain is observed with the addition of more SBSs. Direction-III: Modeling Repulsion. In this direction, we focus on modeling another aspect of spatial coupling imposed by the intra-point repulsion. Consider a device-to-device (D2D) communication scenario, where some users are transmitting some on-demand content locally cached in their devices using a common channel. Any reasonable multiple access scheme will ensure that two nearly users are never simultaneously active as they will cause severe mutual interference and thereby reducing the network-wide sum rate. Thus the active users in the network will have some spatial repulsion. The locations of these users can be modeled as determinantal point processes (DPPs). The key property of DPP is that it forms a bridge between stochastic geometry and machine learning, two otherwise non-overlapping paradigms for wireless network modeling and design. The main focus in this direction is to explore the learning framework of DPP and bring together advantages of stochastic geometry and machine learning to construct a new class of data-driven analytical network models.

This thesis primarily discusses the results of two papers, [Hu] and [HaHu]. The first is an overview of algebraic-geometric techniques for integrable systems in which the AKS theorem is proven. Under certain conditions, this theorem asserts the commutatvity and (potential) non-triviality of the Hamiltonian flow of Ad*-invariant functions once they're restricted to subalgebras. This theorem is applied to the case of coadjoint orbits on loop algebras, identifying the flow with a spectral curve and a line bundle via the Lax equation. These results play an important role in the discussion of [HaHu], wherein we consider three levels of spaces, each possessing a linear family of Poisson spaces. It is shown that there exist Poisson mappings between these levels. We consider the two cases where the underlying Riemann surface is an elliptic curve, as well as its degeneration to a Riemann sphere with two points identified (the trigonometric case). Background in necessary areas is provided.

Ultra dense networks (UDNs) allow for efficient spatial reuse of the spectrum, giving rise to substantial capacity and power gains. In order to exploit those gains, tractable mathematical models need to be derived, allowing for the analysis and optimization of the network operation. In this course, stochastic geometry has emerged as a powerful tool for large-scale analysis and modeling of wireless cellular networks. In particular, the employment of stochastic geometry has been proven instrumental for the characterization of the network performance and for providing significant insights into network densification. Fundamental issues, however, remain open in order to use stochastic geometry tools for the optimization of wireless networks, with the biggest challenge being the lack of tractable closed form expressions for the derived figures of merit. To this end, the present thesis revisits stochastic geometry and provides a novel stochastic geometry framework with a twofold contribution. The first part of the thesis focuses on the derivation of simple, albeit accurate closed form approximations for the ergodic rate of Poisson cellular networks under a noise limited, an interference limited and a general case scenario. The ergodic rate constitutes the most sensible figure of merit for characterizing the system performance, but due to the inherent intractability of the available stochastic geometry frameworks, had not been formulated in closed form hitherto. To demonstrate the potential of the aforementioned tractable expressions with respect to network optimization, the present thesis proposes a flexible connectivity paradigm and employs part of the developed expressions to optimize the network connectivity. The proposed flexible connectivity paradigm exploits the downlink uplink decoupling (DUDe) configuration, which is a promising framework providing substantial capacity and outage gains in UDNs and introduces the DUDe connectivity gains into the 5G era and beyond. Subsequently, the last part of the thesis provides an analytical formulation of the probability density function (PDF) of the aggregate inter-cell interference in Poisson cellular networks. The introduced PDF is an accurate approximation of the exact PDF that could not be analytically formulated hitherto, even though it constituted a crucial tool for the analysis and optimization of cellular networks. The lack of an analytical expression for the PDF of the interference in Poisson cellular networks had imposed the use of intricate formulas, in order to derive sensible figures of merit by employing only the moment generating function (MGF). Hence, the present thesis introduces an innovative framework able to simplify the analysis of Poisson cellular networks to a great extent, while addressing fundamental issues related to network optimization and design.
Las redes ultra densas (UDNs) permiten una reutilización espacial del espectro, proporcionando ventajas en términos de mejora de capacidad y ahorro de potencia. Para explotar estas ventajas se necesitan modelos matemáticos simples que permitan el análisis y la optimización de la operación de la red. Por esta razón, la geometría estocástica se ha convertido en una potente herramienta para el análisis de redes celulares. En particular, el empleo de la geometría estocástica ha sido fundamental para la caracterización del rendimiento de la red y para proporcionar información importante sobre la densificación de la misma. Sin embargo, hay problemas fundamentales que deben resolverse para utilizar estas herramientas de geometría estocástica, siendo el mayor desafío la falta de expresiones simples de forma cerrada para las funciones objetivo de interés. Por este motivo, la presente tesis examina la geometría estocástica y proporciona un marco novedoso con una doble contribución. La primera parte de la tesis se centra en la derivación de aproximaciones cerradas simples pero ajustadas para la capacidad ergódica de las redes de Poisson en escenarios limitados por ruido, por interferencia y por ambos. La capacidad ergódica constituye la figura de mérito más apropiada para caracterizar el rendimiento del sistema, pero no se ha formulado en forma cerrada debido a la complejidad inherente de las expresiones de geometría estocástica disponibles. Para demostrar el potencial de las expresiones simples propuestas, la presente tesis propone un paradigma de conectividad flexible y utiliza parte de las expresiones desarrolladas para optimizar la conectividad de la red. El paradigma de conectividad flexible propuesto explota la configuración de "Downlink Uplink Decoupling" (DUDe), que es un marco que proporciona ventajas sustanciales en términos de incremento de capacidad y reducción de la probabilidad de bloqueo en UDNs e introduce mejoras de conectividad DUDe en la era de 5G. Más adelante, la última parte de la tesis proporciona una formulación analítica de la función de densidad de probabilidad (PDF) de la interferencia agregada en las redes celulares de Poisson. La PDF desarrollada es una aproximación precisa de la PDF exacta que hasta ahora no se ha podido formular analíticamente, a pesar de que se trata de una herramienta crucial para el análisis y la optimización de las redes celulares. La falta de una expresión analítica para la PDF de la interferencia en las redes celulares de Poisson había impuesto el uso de fórmulas complejas, a fin de derivar funciones objetivas apropiadas empleando solo la función generadora de momentos (MGF). Por lo tanto, la presente tesis presenta un marco innovador capaz de simplificar el análisis de las redes celulares de Poisson y así resolver problemas fundamentales relacionados con la optimización y el diseño de la red.

Ginzburg and Weinstein proved in [GW92] that for a compact, semisimple Lie group K endowed with the Lu-Weinstein Poisson structure, there exists a Poisson diffeomorphism from the dual Poisson Lie group K* to the dual k* of the Lie algebra of K endowed with the Lie-Poisson structure. We investigate the possibility of extending this result to the pseudo-unitary groups SU (p, q ), which are semisimple but not compact. The main results presented here are the following. (1) The Ginzburg-Weinstein proof hinges on the existence of a certain vector field X on k*. We prove that for any p, q, the analogous vector field for the SU (p, q ) case exists on an open subset of k*. (2) Each generic dressing orbit ψ(λ) in the Poisson dual AN can be embedded in the complex flag manifold K/T . We show that for SU (1, 1) and SU (1, 2), the induced Poisson structure π(λ) on ψ(λ) extends smoothly to the entire flag manifold. (3) Finally, we prove the Ginzburg-Weinstein theorem for the SU (1, 1) case in two different ways: first, by constructing the vector field X in coordinates and proving that it satisfies the necessary properties, and second, by adapting the approach of [FR96] to the SU (1, 1) case.

Vehicular communication networks are essential to the development of intelligent navigation systems and improvement of road safety. Unlike most terrestrial networks of today, vehicular networks are characterized by stringent reliability and latency requirements. In order to design efficient networks to meet these requirements, it is important to understand the system-level performance of vehicular networks. Stochastic geometry has recently emerged as a powerful tool for the modeling and analysis of wireless communication networks. However, the canonical spatial models such as the 2D Poisson point process (PPP) does not capture the peculiar spatial layout of vehicular networks, where the locations of vehicular nodes are restricted to roadways. Motivated by this, we consider a doubly stochastic spatial model that captures the spatial coupling between the vehicular nodes and the roads and analyze the performance of vehicular communication networks. We model the spatial layout of roads by a Poisson line process (PLP) and the locations of nodes on each line (road) by a 1D PPP, thereby forming a Cox process driven by a PLP or Poisson line Cox process (PLCP). In this dissertation, we develop the theory of the PLCP and apply it to study key performance metrics such as coverage probability and rate coverage for vehicular networks under different scenarios. First, we compute the signal-to-interference plus noise ratio (SINR)-based success probability of the typical communication link in a vehicular ad hoc network (VANET). Using this result, we also compute the area spectral efficiency (ASE) of the network. Our results show that the optimum transmission probability that maximizes the ASE of the network obtained for the Cox process differs significantly from that of the conventional 1D and 2D PPP models. Second, we calculate the signal-to-interference ratio (SIR)-based downlink coverage probability of the typical receiver in a vehicular network for the cellular network model in which each receiver node connects to its closest transmitting node in the network. The conditioning on the serving node imposes constraints on the spatial configuration of interfering nodes and also the underlying distribution of lines. We carefully handle these constraints using various fundamental distance properties of the PLCP and derive the exact expression for the coverage probability. Third, building further on the above mentioned works, we consider a more complex cellular vehicle-to-everything (C-V2X) communication network in which the vehicular nodes are served by roadside units (RSUs) as well as cellular macro base stations (MBSs). For this setup, we present the downlink coverage analysis of the typical receiver in the presence of shadowing effects. We address the technical challenges induced by the inclusion of shadowing effects by leveraging the asymptotic behavior of the Cox process. These results help us gain useful insights into the behavior of the networks as a function of key network parameters, such as the densities of the nodes and selection bias. Fourth, we characterize the load on the MBSs due to vehicular users, which is defined as the number of vehicular nodes that are served by the MBS. Since the limited network resources are shared by multiple users in the network, the load distribution is a key indicator of the demand of network resources. We first compute the distribution of the load on MBSs due to vehicular users in a single-tier vehicular network. Building on this, we characterize the load on both MBSs and RSUs in a heterogeneous C-V2X network. Using these results, we also compute the rate coverage of the typical receiver in the network. Fifth and last, we explore the applications of the PLCP that extend beyond vehicular communications. We derive the exact distribution of the shortest path distance between the typical point and its nearest neighbor in the sense of path distance in a Manhattan Poisson line Cox process (MPLCP), which is a special variant of the PLCP. The analytical framework developed in this work allows us to answer several important questions pertaining to transportation networks, urban planning, and personnel deployment.
Doctor of Philosophy
Vehicular communication networks are essential to the development of intelligent transportation systems (ITS) and improving road safety. As the in-vehicle sensors can assess only their immediate environment, vehicular nodes exchange information about critical events, such as accidents and sudden braking, with other vehicles, pedestrians, roadside infrastructure, and cellular base stations in order to make critical decisions in a timely manner. Considering the time-sensitive nature of this information, it is of paramount importance to design efficient communication networks that can support the exchange of this information with reliable and high-speed wireless links. Typically, prior to actual deployment, any design of a wireless network is subject to extensive analysis under various operational scenarios using computer simulations. However, it is not viable to rely entirely on simulations for the system design of highly complex systems, such as the vehicular networks. Hence, it is necessary to develop analytical methods that can complement simulators and also serve as a benchmark. One of the approaches that has gained popularity in the recent years for the modeling and analysis of large-scale wireless networks is the use of tools from stochastic geometry. In this approach, we endow the locations of wireless nodes with some distribution and analyze various aspects of the network by leveraging the properties of the distribution. Traditionally, wireless networks have been studied using simple spatial models in which the wireless nodes can lie anywhere on the domain of interest (often a 1D or a 2D plane). However, vehicular networks have a unique spatial geometry because the locations of vehicular nodes are restricted to roadways. Therefore, in order to model the locations of vehicular nodes in the network, we have to first model the underlying road systems. Further, we should also consider the randomness in the locations of vehicles on each road. So, we consider a doubly stochastic model called Poisson line Cox process (PLCP), in which the spatial layout of roads are modeled by random lines and the locations of vehicles on the roads are modeled by random set of points on these lines. As is usually the case in wireless networks, multiple vehicular nodes and roadside units (RSUs) operate at the same frequency due to the limited availability of radio frequency spectrum, which causes interference. Therefore, any receiver in the network obtains a signal that is a mixture of the desired signal from the intended transmitter and the interfering signals from the other transmitters. The ratio of the power of desired signal to the aggregate power of the interfering signals, which is called as the signal-to-interference ratio (SIR), depends on the locations of the transmitters with respect to the receiver. A receiver in the network is said to be in coverage if the SIR measured at the location of the receiver exceeds the required threshold to successfully decode the message. The probability of occurrence of this event is referred to as the coverage probability and it is one of the fundamental metrics that is used to characterize the performance of a wireless network. In our work, we have analytically characterized the coverage probability of the typical vehicular node in the network. This was the first work to present the coverage analysis of a vehicular network using the aforementioned doubly stochastic model. In addition to coverage probability, we have also explored other performance metrics such as data rate, which is the number of bits that can be successfully communicated per unit time, and spectral efficiency. Our analysis has revealed interesting trends in the coverage probability as a function of key system parameters such as the density of roads in a region (total length of roads per unit area), and the density of vehicles on the roads. We have shown that the vehicular nodes in areas with high density of roads have lower coverage than those in areas with sparsely distributed roads. On the other hand, the coverage probability of a vehicular node improves as the density of vehicles on the roads increases. Such insights are quite useful in the design and deployment of network infrastructure. While our research was primarily focused on communication networks, the utility of the spatial models considered in these works extends to other areas of engineering. For a special variant of the PLCP, we have derived the distribution of the shortest path distance between an arbitrary point and its nearest neighbor in the sense of path distance. The analytical framework developed in this work allows us to answer several important questions pertaining to infrastructure planning and personnel deployment.

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.

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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

L’objectif principal de cette thèse est l’analyse des performances des réseaux LTE-A (Long Term Evolution- Advanced) au travers de la géométrie stochastique. L’analyse mathématique des réseaux cellulaires est un problème difficile, pour lesquels ils existent déjà un certain nombre de résultats mais qui demande encore des efforts et des contributions sur le long terme. L’utilisation de la géométrie aléatoire et des processus ponctuels de Poisson (PPP) s’est avérée être une approche permettant une modélisation pertinente des réseaux cellulaires et d’une complexité faible (tractable). Dans cette thèse, nous nous intéressons tout particulièrement à des modèles s’appuyant sur ces processus de Poisson : PPP-based abstraction. Nous développons un cadre mathématique qui permet le calcul de quantités reflétant les performances des réseaux LTE-A, tels que la probabilité d’erreur, la probabilité et le taux de couverture, pour plusieurs scénarios couvrant entre autres le sens montant et descendant. Nous considérons également des transmissions multi-antennes, des déploiements hétérogènes, et des systèmes de commande de puissance de la liaison montante. L’ensemble de ces propositions a été validé par un grand nombre de simulations. Le cadre mathématique développé dans cette thèse se veut général, et doit pouvoir s’appliquer à un nombre d’autres scénarios importants. L’intérêt de l’approche proposée est de permettre une évaluation des performances au travers de l’évaluation des formules, et permettent en conséquences d’éviter des simulations qui peuvent prendre énormément de temps en terme de développement ou d’exécution
The main focus of this thesis is on performance analysis and system optimization of Long Term Evolution - Advanced (LTE-A) cellular networks by using stochastic geometry. Mathematical analysis of cellular networks is a long-lasting difficult problem. Modeling the network elements as points in a Poisson Point Process (PPP) has been proven to be a tractable yet accurate approach to the performance analysis in cellular networks, by leveraging the powerful mathematical tools such as stochastic geometry. In particular, relying on the PPP-based abstraction model, this thesis develops the mathematical frameworks to the computations of important performance measures such as error probability, coverage probability and average rate in several application scenarios in both uplink and downlink of LTE-A cellular networks, for example, multi-antenna transmissions, heterogeneous deployments, uplink power control schemes, etc. The mathematical frameworks developed in this thesis are general enough and the accuracy has been validated against extensive Monte Carlo simulations. Insights on performance trends and system optimization can be done by directly evaluating the formulas to avoid the time-consuming numerical simulations

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.

En esta tesis se estudian problemas relacionados con acciones de grupos finitos en 4-variedades diferenciables y simplécticas. Se prueba que toda 4-variedad diferenciable cerrada X admite una constante C>0 tal que cualquier grupo finito G que actúa en X de manera efectiva y diferenciable tiene un subgrupo H abeliano o nilpotente de clase 2 que satisface [G:H]