Μια από τις πιο επιτυχείς προσεγγίσεις της γεωμετρίας είναι αυτή που πρότεινε ο Γερμανός μαθηματικός Felix Klein στο γνωστό Πρόγραμμα Erlangen. To πρόγραμμα αυτό αποτέλεσε ένα γενικό σχέδιο ταξινόμησης των διάφορων γεωμετριών που εμφανίστηκαν μετά την ανακάλυψη των μη Ευκλείδειων γεωμετριών, με τεράστιες επιπτώσεις όχι μόνο στα μαθηματικά αλλά και στη θεωρητική φυσική. Σύμγωνα με τον Klein, το αντικείμενο της γεωμετρίας είναι μια πολλαπλότητα στην οποία δρα μια ομάδα μετασχηματισμών, η οποία συνήθως είναι μια ομάδα Lie. Στη περίπτωση που η ομάδα δρα μεταβατικά πάνω στην πολλαπλότητα, τότε οδηγούμαστε στην περίτωση των ομογενών χώρων. Κλασικά παραδείγματα τέτοιων χώρων αποτελούν η σφαίρα και ο πραγματικός ή μιγαδικός προβολικός χώρος. Η βασική ιδιότητα των ομογενών χώρων είναι ότι αν γνωρίζουμε την τιμή κάποιου γεωμετρικού μεγέθους (για παράδειγμα της καμπυλότητας) σε ένα σημείο του χώρου τότε χρησιμοποιώντας κατάλληλες απεικονίσεις μεταφοράς μπορούμε να υπολογίσουμε την τιμή του μεγέθους αυτού σε οποιοδήποτε άλλο σημείο του χώρου. Στην εργασία μας περιγράφουμε τη γεωμετρία των χώρων αυτών χρησιμοποιώντας εργαλεία από τη θεωρία των ομάδων Lie. Το δεύτερο σκέλος της εργασίας αφορά τη θεωρία των πολλαπλοτήτων σημαιών, οι οποίες αποτελούν και μια ιδιαίτερη κλάση ομογενών χώρων. Μια πολλαπλότητα σημαιών είναι η τροχιά της συζυγούς αναπαράστασης μιας ημιαπλής ομάδας Lie. Οι χώροι αυτοί δέχονται μια κομψή αλγεβρική περιγραφή χρησιμοποιώντας τη δομική θεωρία των ημιαπλών αλγεβρών Lie και ταξινομούνται από τα χρωματιστά διαγράμματα Dynkin. / One of the most successful approaches to geometry is that suggested by the german mathematician Felix Klein and his famous Erlangen programm. According to Klein, a geometry is the study of all these objects which remain invariant under the action of a transormation group. Usually this group is a Lie group. If the above action is transitive then the space is called Homogeneous space. Classical examples of these spaces are the sphere and the real or complex projective space. The basic property of homogeneous spaces is that if we know the value of a geometrical object (e.g curvature) at a given point, then we can calculate the value of this quantity at any other point by using certain translations maps. In this project we describe the geometry of homogeneous spaces, by using tools of the Lie group theory. The second part of this project has to do with generalized flag manifolds, which are an important class of homogeneous spaces. A flag manifold is the orbit of the adjoint representation of a compact semisimple Lie group. These spaces admit a nice algebraic description by using the structure theory of semisimple Lie algebras and are classified by painted Dynkin diagramms.

Orientador: Luiz Antonio Barrera San Martin / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-10T09:55:18Z (GMT). No. of bitstreams: 1 Santos_LaercioJosedos_D.pdf: 632399 bytes, checksum: 25069afc192f7633f7f8cd6bdce8e96e (MD5) Previous issue date: 2007 / Resumo: Este trabalho divide-se em duas partes. Na primeira parte, obtemos condições necessárias e suficientes para que uma família de classes laterais de um subgrupo de Lie gere um subsemigrupo com interior não vazio. Aplicamos essas condições aos pares simétricos, onde o grupo é semi-simples. Como consequência, mostramos que o subgrupo dos pontos fixos de vários automorfismos involutivos é maximal como semigrupo. Na segunda parte, definimos a função característica de um subsemigrupo de um grupo de Lie semi-simples e, encontramos um subconjunto do domínio de definição dessa função. Fizemos isto usando a teoria geral de semigrupos em grupos semi-simples. Usamos a função característica de um semigrupo, com algumas hipóteses adicionais, para introduzir uma métrica Riemanniana nas órbitas do subgrupo das unidades do semigrupo. Com essa métrica, obtemos uma condição necessária para que um subgrupo possa ser imerso em um semigrupo próprio com interior não vazio / Abstract: This work is made of two parts. In the first one, we gave necessary and sufficient conditions for a family of cosets of a Lie subgroup to generate a subsemigroup with nonempty interior. We apply these conditions to symmetric pairs where the group is semi-simple. As a consequence we prove that for several involutive automorphisms the fixed points subgroup is a maximal semigroup. In the second part, we define a characteristic function of a subsemigroup of a semi- simple Lie group and we find a subset where the function is defined. This is made through general theory of semigroups in semi-simple groups. The characteristic function is used, together with some additional hypothesis, for to create a Riemannian metric in the orbits of the unity subgroup of the semigroup. With this metric we gave a necessary condition for a subgroup be embedded in a proper semigroup with nonempty interior / Doutorado / Teoria de Lie / Doutor em Matemática

Orientador: Caio José Colletti Negreiros / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T12:45:02Z (GMT). No. of bitstreams: 1 Grama_LinoAndersondaSilva_D.pdf: 1119551 bytes, checksum: d2dc2c993629f40f7976e91497c5d219 (MD5) Previous issue date: 2011 / Resumo: O principal objetivo deste trabalho é o estudo de aplicações harmônicas em variedades flag generalizadas. Na primeira parte do trabalho, consideramos aplicações cujo domínio é uma superfície de Riemann. Provamos que toda aplicação holomorfa-horizontal na variedade flag é uma aplicação equiharmônica (ie, harmônica com respeito a cada métrica invariante na variedade flag). Obtemos também as fórmulas de Plucker para curvas holomorfa-horizontais na variedade flag maximal. Na segunda parte do trabalho, consideramos aplicações harmônicas cujo domínio possui dimensão 1 ( ie, geodésicas) na variedade flag. Provamos que toda variedade ag generalizada admite curvas que são geodésicas com respeito a cada métrica invariante. Tais curvas são chamadas equigeodésicas. Fornecemos uma descrição algébrica para tais curvas e exibimos famílias de equigeodésicas em diversas famílias de variedades flag / Abstract: The main goal of this work is the study of harmonic maps in generalized flag manifolds. In the first part of the work, we consider maps whose domain is a Riemann surface. We prove that every holomorphic-horizontal map in the flag manifold is an equiharmonic map (i.e. harmonic with respect to each invariant metric in the flag manifold). We also obtain the Plucker formulae for holomorphic-horizontal curves in full flag manifolds. In the second part of the work, we consider harmonic maps whose domain has dimension one (i.e. geodesics) in the ag manifold. We prove that every generalized flag manifold admit curves that are geodesics with respect to each invariant metric. Such curves are called equigeodesics. We provide an algebraic characterization for such curves and exhibit families of equigeodesics in several families of flag manifolds / Doutorado / Doutor em Matemática

Orientador: Caio Jose Colletti Negreiros, Nir Cohen / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-05T00:38:44Z (GMT). No. of bitstreams: 1 Santos_EvandroCarlosFerreirados_D.pdf: 3261771 bytes, checksum: 1d6efb1b1f03e2549a501877f92a4952 (MD5) Previous issue date: 2005 / Resumo: O objetivo deste trabalho é contribuir para o estudo da geometria Hermitiana invariante das variedades bandeira. Estudamos a classe das métricas de Einstein sobre variedades bandeira. Neste trabalho apresentamos novas soluções para a equação de Einstein invariante sobre as variedades bandeira do tipo Az maximais e não-maximais. Considere W um subgrupo do grupo de WeyL Descrevemos uma ação natural de W sobre o conjunto das soluções da equação de Einstein invariante e provamos que esta ação deixa a equação e o conjunto solução invariantes. Determinamos a constante de Einstein de todas as métricas conhecidas e em alguns casos encontramos a métrica de Yamabe. Estudamos o funcional de Einstein- Hilbert e concluímos que toda métrica de Einstein invariante sobre uma variedade flag é estável. Usamos C- fibrações para provar que sobre JF(n), n > 4, uma métrica de Einstein (1,2)- simplética deve ser Kãhler. Fizemos uso da classificação das estruturas quase Hermitianas invariantes de San Martin- Negreiros e provamos que uma métrica de Einstein é Kãhler ou pertence à classe W1 EB W3. Isto implica em uma solução, no caso das variedades bandeira do tipo Az, para uma conjectura formulada por W. Ziller[17] / Abstract: The goal of this work is to contribute the study of invariant Hermitian geometry on flag manifolds. We study the class of Einstein metrics on flag manifolds. In this work we present new solutions for the invariant Einstein equation on flag manifolds, maximals or not, of Ai case. Let W a subgroup of the Weyl group. We described a natural action of W on the solution set of the Einstein equation, and we proved that W lefts the solution set invariant. We obtained the Einstein's constant of all the known metrics and in some cases we found the Yamabe metric. We studied the Einstein-Hilbert functional and we proved that all invariant Einstein metrics on a flag manifold are stable. Using C-fibrations we proved, in the case IF(n), n 2:: 4, if 9 is an invariant Einstein metric, and (1,2)-symplectic then 9 is Kãhler. According to San Martin-Negreiros's classification of all almost Hermitian structures on maximal flag manifolds we proved that an Einstein metric is Kãhler or belongs to W1 $ W3. This implies in a solution, in flag manifolds of Ai case, for a conjecture proposed by W. Ziller[17] / Doutorado / Geometria e Topologia / Doutor em Matemática

Orientador: Luiz Antonio Barrera San Martin / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica. / Made available in DSpace on 2018-08-19T06:18:00Z (GMT). No. of bitstreams: 1 Santos_ArianeLuziados_D.pdf: 829222 bytes, checksum: 870721241f42ea4a1c1748427ae28d99 (MD5) Previous issue date: 2011 / Resumo: Seja S um semigrupo com interior não vazio de um grupo de Lie simples G, conexo, complexo ou real. No caso em que o grupo G é real também considere-o não compacto, com centro finito e cuja álgebra de Lie é uma forma real, normal de uma álgebra clássica... ... Observação: O resumo, na íntegra, poderá ser visualizado no texto completo da tese digital / Abstract: Let S be a semigroup with nonempty interior of a complex or real connected simple Lie group G. In the case the group G is real also assume that G is non-compact, with finite center, whose algebra is a normal real form of a classic algebra. ... Note: The complete abstract is available with the full electronic digital thesis or dissertations / Doutorado / Matematica / Doutor em Matemática

Orientador: Luiz Antonio Barrera San Martin / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-21T00:23:49Z (GMT). No. of bitstreams: 1 Rabelo_Lonardo_D.pdf: 1560307 bytes, checksum: ad1323aadd78f3943acaf2c6e13b96d0 (MD5) Previous issue date: 2012 / Resumo: Esta tese apresenta uma abordagem para o estudo da topologia das variedades flag reais. A homologia é obtida pela determinação do operador fronteira da homologia celular. Isto se dá a partir de uma parametrização explícita das células de Shubert que fornecem a estrutura celular destas variedades. Para o anel de cohomologia de uma variedade flag maximal, encontram-se os seus geradores como classes de Stiefel-Whitney de um fibrado de linha sobre a variedade flag / Abstract: This thesis presents an approach for the study of topology of real flag manifolds. The homology is obtained by the determination of the boundary operator for the cellular homology. This follows from an explicit parametrization of the Schubert cells which gives a cellular structure for these manifolds. For the cohomology ring of a maximal flag manifold, its generators are found as Stiefel-Whitney classes of a line fiber bundle over the flag manifold / Doutorado / Matematica / Doutor em Matemática

Orientadores: Caio Jose Colleti Negreiros, Nir Cohen / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-14T14:44:13Z (GMT). No. of bitstreams: 1 Silva_NeitonPereirada_D.pdf: 4231710 bytes, checksum: af4dc57e0a7215547662f87d1744bb27 (MD5) Previous issue date: 2009 / Resumo: Neste trabalho encontramos todas as métricas de Einstein invariantes em quatro famílias de variedades bandeira do tipo B1 e C1. Os nossos resultados são consistentes com a conjectura de Wang e Ziller sobre a finitude das métricas de Einstein. O nosso método para resolver as equações de Einstein e baseado nas simetrias do sistema algébrico. Obtemos os sistemas algébricos de Einstein para variedades bandeira generalizadas do tipo B1 C1e G2. Estes sistemas são as condições necessárias e suficientes para métricas invariantes nessas variedades serem Einstein. Os sistemas algébricos que obtivemos generalizam as equações de Einstein obtidas por Sakane nos casos maximais. As equações nos casos Al e Dl foram obtidas por Arvanitoyeorgos. Calculamos o conjunto das trazes para as variedades bandeira generalizadas dos grupos de Lie clássicos. Assim estendemos à essas variedades certos resultados sobre estruturas Hermitianas invariantes obtidos por San Martin, Cohen e Negreiros. / Abstract: In this work we and all the invariant Einstein metrics on four families of ag manifolds of type Bl and Cl. Our results are consistent with the finiteness conjecture of Einstein metrics proposed by Wang and Ziller. Our approach for solving the Einstein equations is based on the symmetries of the algebraic system. We obtain the Einstein algebraic systems for the generalized ag manifolds of type Bl, Cl and G2. These systems are necessary and sufficient conditions for invariant metrics on these manifolds to be Einstein. The algebraic systems that we obtained generalize the Einstein equations obtained by Sakane in the maximal cases. The equations in the cases Al and Dl were obtained by Arvanitoyeorgos. We calculate all the t-roots on the generalized ag manifolds of the classical Lie groups. Thus we extend to these manifolds certain results on invariant structures Hermitian obtained by San Martin, Cohen and Negreiros. / Doutorado / Geometria Diferencial / Doutor em Matemática

Dans cette thèse, on s'intéresse au groupe de Brauer non ramifié des espaces homogènes à stabilisateur non connexe et à ses applications arithmétiques. On développe notamment différentes formules de nature algébrique et/ou arithmétique permettant de calculer explicitement, tant sur un corps fini que sur un corps de caractéristique 0, la partie algébrique du groupe de Brauer non ramifié d'un espace homogène G\G' sous un groupe linéaire G' semi-simple simplement connexe à stabilisateur fini G, le tout en donnant des exemples de calculs que l'on peut faire avec ces formules. Pour ce faire, on démontre au préalable (à l'aide d'un théorème de Gabber sur les altérations) un résultat décrivant la partie de torsion première à p du groupe de Brauer non ramifié d'une variété V lisse et géométriquement intègre sur un corps fini ou sur un corps global de caractéristique p au moyen de l'évaluation des éléments de Br(V) sur ses points locaux. Les formules pour un stabilisateur fini sont ensuite généralisées au cas d'un stabilisateur G quelconque via une réduction de la cohomologie galoisienne du groupe G à celle d'un certain sous-quotient fini. Enfin, pour K un corps global et G un K-groupe fini résoluble, on démontre sous certaines hypothèses sur une extension déployant G que l'espace homogène V:=G\G' avec G' un K-groupe semi-simple simplement connexe vérifie l'approximation faible (ces hypothèses assurant la nullité du groupe de Brauer non ramifié algébrique). On utilise une version plus précise de ce résultat pour démontrer ensuite le principe de Hasse pour des espaces homogènes X sous un K-groupe G' semi-simple simplement connexe à stabilisateur géométrique fini et résoluble, sous certaines hypothèses sur le K-lien défini par X. / This thesis studies the unramified Brauer group of homogeneous spaces with non connected stabilizer and its arithmetic applcations. In particular, we develop different formulas of algebraic and/or arithmetic nature allowing an explicit calculation, both over a finite field and over a field of characteristic 0, of the algebraic part of the unramified Brauer group of a homogeneous space G\G' under a semisimple simply connected linear group G' with finite stabilizer G. We also give examples of the calculations that can be done with these formulas. For achieving this goal, we prove beforehand (using a theorem of Gabber on alterations) a result describing the prime-to-p torsion part of the unramified Brauer group of a smooth and geometrically integral variety V over a global field of characteristic p or over a finite field by evaluating the elements of Br(V) at its local points. The formulas for finite stabilizers are later generalised to the case where the stabilizer G is any linear algebraic group using a reduction of the Galois cohomology of the group G to that of a certain finite subquotient.Finally, for a global field K and a finite solvable K-group G, we show under certain hypotheses concerning the extension splitting G that the homogeneous space V:=G\G' with G' a semi-simple simply connected K-group has the weak approximation property (the hypotheses ensuring the triviality of the unramified algebraic Brauer group). We use then a more precise version of this result to prove the Hasse principle forhomogeneous spaces X under a semi-simple simply connected K-group G' with finite solvable geometric stabilizer, under certain hypotheses concerning the K-kernel (or K-lien) defined by X.

Neste trabalho analisamos a positividade definida estrita de núcleos contínuos sobre um espaço compacto 2-homogêneo. R. Gangolli (1967) apresentou uma caracterização completa para os núcleos que são contínuos, isotrópicos e positivos definidos sobre um espaço compacto 2-homogêneo Md: a parte isotrópica do núcleo é uma série de Fourier uniformemente convergente, com coeficientes não negativos, em relação a certos polinômios de Jacobi atrelados a Md. Uma das contribuições de nosso trabalho é uma caracterização para a positividade definida estrita de tais núcleos, complementando a caracterização apresentada por Chen et al. (2003) no caso em que Md é uma esfera unitária de dimensão maior ou igual a 2. Outra contribuição do trabalho é uma extensão do resultado de Gangolli para núcleos sobre produtos cartesianos de espaços compactos 2-homogêneos, e a consequente caracterização para núcleos estritamente positivos definidos neste mesmo contexto. Por fim, a última contribuição do trabalho envolve a análise do grau de diferenciabilidade da parte isotrópica de um núcleo contínuo, isotrópico e positivo definido sobre Md e a aplicabilidade de tal análise em resultados envolvendo a positividade definida estrita. / In this work we analyze the strict positive definiteness of continuous kernels on compact two-point homogeneous spaces Md. R. Gangolli (1967) presented a complete characterization for continuous, isotropic and positive definite kernels on Md: the isotropic part of the kernel is a uniformly convergent Fourier series of certain Jacobi polynomials associated to Md, with nonnegative coefficients. One of the contributions of our work is a characterization for the strict positive definiteness of such kernels, completing that one presented by Chen et al. (2003) in the case Md is the unit sphere of dimension at least 2. Another contribuition of this work is an extension of Gangolli\'s result for kernels on a product of compact two-point homogeneous spaces, and the subsequent characterization of strict positive definiteness in this same context. Finally, the last contribution in this work involves the analysis of the differentiability of the isotropic part of a continuous, isotropic and positive definite kernel on Md and the applicability of such analysis in results involving the strict positive definiteness.

This PhD thesis consists of a summary and five papers which all deal with asymptotic problems on certain homogeneous spaces. In Paper I we prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. All our results are given with precise estimates on the rates of convergence to equidistribution. Papers II and III are concerned with statistical problems on the space of n-dimensional lattices of covolume one. In Paper II we study the distribution of lengths of non-zero lattice vectors in a random lattice of large dimension. We prove that these lengths, when properly normalized, determine a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the positive real line with intensity 1/2. In Paper III we complement this result by proving that the asymptotic distribution of the angles between the shortest non-zero vectors in a random lattice is that of a family of independent Gaussians. In Papers IV and V we investigate the value distribution of the Epstein zeta function along the real axis. In Paper IV we determine the asymptotic value distribution and moments of the Epstein zeta function to the right of the critical strip as the dimension of the underlying space of lattices tends to infinity. In Paper V we determine the asymptotic value distribution of the Epstein zeta function also in the critical strip. As a special case we deduce a result on the asymptotic value distribution of the height function for flat tori. Furthermore, applying our results we discuss a question posed by Sarnak and Strömbergsson as to whether there in large dimensions exist lattices for which the Epstein zeta function has no zeros on the positive real line.