Thèse, Département d'architecture, Ecole polytechnique fédérale de Lausanne, 1999. / Bibliogr.

This paper is a contribution to the study of countable Borel equivalence relations on standard Borel spaces. We concentrate here on the study of the nature of translation equivalence. We study these known hyperfinite spaces in order to gain insight into the approach necessary to classify certain variables as either being hyperfinite or not. In Chapter 1, we will give the basic definitions and examples of spaces used in this work. The general construction of marker sets is developed in this work. These marker sets are used to develop several invariant tilings of the equivalence classes of specific variables . Some properties that are equivalent to hyperfiniteness in the certain space are also developed. Lastly, we will give the new result that there is a continuous injective embedding from certain defined variables.

This paper examines the relationship between Borel sets and Baire functions.

This thesis examines measure functions. A measure function has as its domain of definition a class of sets. It also must satisfy a certain additive condition. To state a concise definition of a measure function, it is convenient to define set function and completely additive set function.

Soit F une famille de sous-ensembles d'un espace polonais X. Soient A et B deux sous-ensembles de X. A est séparable de B par un ensemble de F s'il existe C dans F qui contient A et n'intersecte pas B. Lorsque A et B sont des sous-ensembles analytiques et F est une classe de Wadge de boréliens, la non séparabilité de A et B a été caractérisée par Louveau et Saint Raymond. Ils ont fourni un exemple minimum, pour un certain quasi-ordre, ce qui est connu sous le nom de dichotomie à la Hurewicz. Dans les espaces polonais produit, de nouvelles familles naturelles d'ensembles apparaissent. Un exemple, qui est lié à l'étude des graphes définissables, est la classe des produits de deux ensembles dans deux classes Borel F et F'. On caractérise separabilité des ensembles analytiques, lorsque F et F' sont des classes de Borel de petit rang. Une autre classe naturelle est celle des boréliens qui sont dans une certaine classe F si on affine les topologies originales en d'autres topologies polonaises. Nous appelons ces ensembles potentiellement dans F. Lecomte a caractérisé la séparabilité des ensembles analytiques par des ensembles potentiellement dans F, lorsque F est une classe de Wadge de boréliens. Une question naturelle est de se demander si nous pouvons avoir cette dichotomie pour un quasi-ordre plus fin qui utilise des fonctions injectives. Dans la deuxième partie du manuscrit, on caractérise la separabilité pour ce quasi-ordre lorsque A, B sont des ensembles analytiques et F est la classe des ensembles potentiellement dans C. Des hypothèses plus fortes sont nécessaires, de sorte que nous développons une notion qui généralise à la fois l'acyclicité et la locale dénombrabilité. / Let F be a family of subsets of a Polish space X. Given two subsets A and B of X, one says that A is separable from B by a set in F if there is C in F which contains A and does not intersect B. When A and B are analytic subsets and F is a Wadge class of Borel sets, the non separability of A and B was characterized by Louveau and Saint Raymond. They provided a minimum example, for some quasi-order, in what is known as a Hurewicz dichotomy. If we consider the product of two Polish spaces, new natural families of sets (and thus new separation problems) arise. An example, that is related to the study of definable graphs, is the class of products of sets in two Borel classes F and F'. We obtain a characterization of the separability when F and F' are Borel classes of low rank. Another natural class is the class of Borel sets which are in a certain class F if one refines the original topologies into other Polish topologies. We call these sets potentially in F. A dichotomy characterizing separability of analytic sets by sets potentially in F was proved by Lecomte when F is any Wadge class of Borel sets. However, a natural question is to ask whether we can have this dichotomy for a finer quasi-order, involving injective functions. In the second part of the manuscript, we give a characterization involving this other quasi-order when A,B are analytic sets and F is the class of sets potentially in C. Stronger hypothesis are needed, so we develop a notion that generalizes both acyclicity and local countability.

acase@tulane.edu / Let G be a reductive group, B be a Borel subgroup, and let K be a symmetric subgroup of G. The study of B orbits in a symmetric variety G/K or, equivalently, the study of K orbits in a flag variety G/B has importance in the study of Harish-Chandra modules; it comes with many interesting Schubert calculus problems. Although this subject is very well studied, it still has many open problems from combinatorial point of view. The most basic question that we want to be able to answer is that how many B orbits there are in G/K. In this thesis, we study the enumeration problem of Borel orbits in the case of classical symmetric varieties. We give explicit formulas for the numbers of Borel orbits on symmetric varieties for each case and determine the generating functions of these numbers. We also explore relations to lattice path enumeration for some cases. In type A, we realize that Borel orbits are parameterized by the lattice paths in a pxq grid moving by only horizontal, vertical and diagonal steps weighted by an appropriate statistic. We provide extended results for type C as well. We also present various t-analogues of the rank generating function for the inclusion poset of Borel orbit closures in type A. / 1 / Ozlem Ugurlu

Techn. Universiẗat, Diss., 2005--München.

Thesis (Doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1990. / Includes bibliographical references.

Brandenburgische Techn. Universiẗat, Diss., 2003--Cottbus.

Orientador: Marcelo Otávio Caminha Gomes / Doutor