The study of orderable groups is a topic that is all too often overlooked as a topic in algebra. The subject of orderable groups is a field of study which is directly associated with algebraic group theory, algebraic topology, and set theory. This paper will act as a guide into the world of orderable groups. It begins by enlightening the reader to the fundamental axioms of orderable groups, as well as, noting various important groups on which orders are often considered. We will then consider more interesting groups, on which the placement of orders is considered less often. After considering the orderings placed on various groups, we wish to consider in further detail the topologies of the sets of these orders. In particular, it is important to consider whether the set of orders placed on a particular group is finite or uncountable. We prove the latter by creating a homeomorphism from the group to the Cantor set, a set which is known for its uncountability. / Master of Science

We provide an L² theory for the local double Hilbert transform along an analytic surface (s, t ,φ(s, t )) in the Heisenberg group H¹, that is operator f ↦ Hφ f (x) := p.v.∫∣s∣,∣t∣≤1 f (x ∙ (s, t ,φ(s, t ))-¹) ds/s dt/t, where ∙ denotes the group operation in H1. This operator combines several features: it is amulti-parameter singular integral, its kernel is supported along a submanifold, and convolution is with respect to a homogeneous group structure. We reprove Hφ is always L²(H¹)→L²(H¹) bounded (a result first obtained in [Str12]) to illustrate the method and then refine it to characterize the largest class of polynomials P of degree less than d such that the operator HP is uniformly bounded when P ranges in the class. Finally, we provide examples of surfaces that can be treated by our method but not by the theory of [Str12].

In this thesis, we will examine the properties of subelliptic jets in the Engel group of step 3. Step-2 groups, such as the Heisenberg group, do not provide insight into the general abstract calculations. This thesis then, is the first explicit non-trivial computation of the abstract results.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / FundaÃÃo Cearense de Apoio ao Desenvolvimento Cientifico e TecnolÃgico / Nesta dissertaÃÃao, estudamos as superfÃcies mÃnimas do grupo de Heisenberg tridimensional, bem como a aplicaÃÃo de Gauss destas superfÃcies. Inicialmente à feito uma breve exposiÃÃo sobre a geometria do grupo de Heisenberg. EntÃo, mostramos que, em tal espaÃo: as Ãnicas superfÃcies com aplicaÃÃo de Gauss constante sÃo os planos verticais; nÃo existem superfÃcies totalmente umbÃlicas nem superfÃcies mÃnimas compactas; toda superfÃcie mÃnima Ã, necessariamente, estÃvel. Mostramos, ainda, que as Ãnicas superfÃcies mÃnimas verticais sÃo os planos verticais. Por fim, apresentamos uma classificaÃÃo das superfÃcies com aplicaÃÃo de Gauss de posto constante, igual a zero ou um. / In this report, we study minimal surfaces of the tridimensional Heisenberg group, as well as their Gauss maps. We begin with a short presentation of the geometry of the Heisenberg group. Then, we show that, in this space: the only surfaces with constant Gauss map are the vertical planes; there are no totally umbilical surfaces nor compact minimal surfaces; every minimal surface is, necessarily, stable. We also show that the only vertical minimal surfaces are vertical planes. Finally, we present a classification of the surfaces with Gauss map of constant rank, equal to zero or one.

In this thesis we study actions of countable groups on measure spaces underthe assumption that the dynamics are non-singular, with particular reference topointwise ergodic theorems and their relationship to the critical dimensions ofthe action.

In this thesis we deal with two problems in harmonic analysis. In the first problem we discuss weighted norm inequalities for Weyl multipliers satisfying Mauceri’s condition. As an application, we prove certain multiplier theorems on the Heisenberg group and also show in the context of a theorem of Weis on operator valued Fourier multipliers that the R-boundedness of the derivative of the multiplier is not necessary for the boundedness of the multiplier transform. In the second problem we deal with a variation of a theorem of Mauceri concerning the Lp bound-edness of operators Mwhich are known to be bounded on L2 .We obtain sufficient conditions on the kernel of the operaor Mso that it satisfies weighted Lp estimates. As an application we prove Lp boundedness of Hermite pseudo-multipliers.

We start with the recollection of basic results about differential manifolds and Lie groups. We also recall some preliminary terminologies in Lie algebra. Then we define the Lie algebra corresponding to a Lie group. In the next section, we define a strongly continuous representation of a Lie group on a Banach space. We further define the smooth, analytic and entire vectors for a given representation. Then, we move on to develop some necessary and sufficient criteria to characterize smooth, analytic and entire vectors. We, in particular, take into account of some specific representations of Lie groups like the regular representation of R, the irreducible representations of Heisenberg groups, the irreducible representations of the group of Affine transformations and finally the representations of non-compact simple Lie groups.

This work is concerned with two different problems in harmonic analysis, one on the Heisenberg group and other on Rn, as described in the following two paragraphs respectively. Let Hn be the (2n + 1)-dimensional Heisenberg group, and let K be a compact subgroup of U(n), such that (K, Hn) is a Gelfand pair. Also assume that the K-action on Cn is polar. We prove a Hecke-Bochner identity associated to the Gelfand pair (K, Hn). For the special case K = U(n), this was proved by Geller, giving a formula for the Weyl transform of a function f of the type f = Pg, where g is a radial function, and P a bigraded solid U(n)-harmonic polynomial. Using our general Hecke-Bochner identity we also characterize (under some conditions) joint eigenfunctions of all differential operators on Hn that are invariant under the action of K and the left action of Hn . We consider convolution equations of the type f * T = g, where f, g ε Lp(Rn) and T is a compactly supported distribution. Under natural assumptions on the zero set of the Fourier transform of T , we show that f is compactly supported, provided g is.

Dans cette thèse, nous nous intéressons à deux thèmes d'analyse géométrique. Le premier concerne le comportement asymptotique des fonctions harmoniques en relation avec la géométrie, sur des graphes et des variétés. Nous étudions des critères de convergence au bord des fonctions harmoniques, comme celui de la bornitude non-tangentielle, de la finitude de l'énergie ou encore de la densité de l'énergie. Nous nous plaçons pour cela dans différents cadres comme les graphes hyperboliques au sens de Gromov, les variétés hyperboliques au sens de Gromov, les graphes de Diestel-Leader ou encore dans un cadre abstrait pour obtenir des résultats pour les points du bord minimal de Martin. Les méthodes probabilistes utilisées exploitent le lien entre les fonctions harmoniques et les martingales. Le deuxième thème abordé dans cette thèse concerne l'étude des propriétés des ensembles rectifiables de dimension 1 dans le groupe de Heisenberg, en relation avec des opérateurs d'intégrales singulières. Nous étendons à ce contexte sous-riemannien une partie des résultats de la théorie des ensembles uniformément rectifiables de David et Semmes. Nous obtenons notamment un théorème géométrique du voyageur de commerce qui fournit une condition pour qu'un ensemble Ahlfors-régulier du premier groupe de Heisenberg soit contenu dans une courbe Ahlfors-régulière. / In this thesis, we are interested in two topics of geometric analysis. The first one is concerned with the asymptotic behaviour of harmonic functions in connection with geometry on graphs and manifolds. We study criteria for convergence at boundary of harmonic functions such as non-tangential boundedness, finiteness of non-tangential energy or finiteness of the energy density. We deal with Gromov hyperbolic manifolds, Gromov hyperbolic graphs, Diestel-Leader graphs and with an abstract frame to obtain criteria at minimal Martin boundary points. The methods, coming from probability theory and metric geometry, use the relation between harmonic functions and martingales. The second topic concerns the rectifiability properties of 1-dimensional sets in the Heisenberg group in connection with the boundedness of singular integral operators. We extend to this sub-Riemannian setting parts of the theory of uniformly rectifiable sets due to David and Semmes. In particular, we obtain a geometric traveling salesman theorem which provides a condition for an Ahlfors regular set of the first Heisenberg group to be contained in an Ahlfors regular curve.

Made available in DSpace on 2015-05-15T11:46:02Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 666060 bytes, checksum: 1ad661f6cc42df5f3ee67db9a939af86 (MD5) Previous issue date: 2011-07-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we obtain Weierstrass-type representations for immersed surfaces in Heisenberg space, endowed with a left-invariant metric. We will consider the Riemannian and Lorentzian case. We will define two complex functions (spinors) satisfying a linear Dirac-type equation, obtaining thus a representation for immersed surfaces with prescribed mean curvature. The same will enable us write a representation of minimal immersion in terms of a harmonic Gauss map. / Neste trabalho obtemos uma representações tipo Weierstrass para superfícies imersas no espaço de Heisenberg, dotado com uma métrica invariante à esquerda. Consideraremos os casos Riemanniano e Lorentziano. Definimos duas funções complexas (spinors), satisfazendo uma equação linear tipo Dirac que usamos para obter uma representação para superfícies imersas com curvatura média prescrita. A mesma possibilita escrever uma representação de imersões mínimas em termos de uma aplicação de Gauss harmônica.