Let $\bk$ be a field of characteristic zero. A Pham-Brieskorn ring is a $\bk$-algebra of the form $B_{a_0,\dots,a_n} = \bk[X_0,\dots,X_n] / \lb X_0^{a_0} + \cdots + X_n^{a_n} \rb$, where $n \geq 2$ and $a_0, \dots, a_n$ are positive integers. A ring $B$ is rigid if the only locally nilpotent derivation $D : B \to B$ is the zero derivation. Consider the following conjecture. \begin{conjnonumber}\label{PBConjectureAbstract} Let $n \geq 2$, and let $B_{a_0, \dots, a_n} = \bk[X_0, \dots, X_n] / \langle X_0^{a_0} + \cdots + X_n^{a_n} \rangle$ be a Pham-Brieskorn ring. If $\min\{a_0, \dots,a_n \} \geq 2$ and at most one element $i$ of $\{0,\dots ,n\}$ satisfies $a_i = 2$, then $B_{a_0, \dots, a_n}$ is rigid. \end{conjnonumber} The $n = 2$ case of the Conjecture is known to be true. In this thesis, we make progress towards solving the above conjecture. Our main results are: \begin{enumerate}[\rm(1)] \item For any $n \geq 3$, in order to prove the above conjecture, it suffices to prove rigidity of $B_{a_0, \dots, a_n}$ in the cases where $\bk = \Comp$ and $\cotype(a_0, \dots, a_n) = 0$. \item For any $n \geq 2$, $X = \Proj B_{a_0, \dots, a_n}$ is a well-formed quasismooth weighted complete intersection if and only if $\cotype(a_0, \dots, a_n) = 0$. \item When $n = 3$ and $\cotype(a_0, a_1, a_2, a_3) = 0$, $B_{a_0, a_1, a_2, a_3}$ is rigid, except possibly in the cases where, up to a permutation of the $a_i$, $(a_0, a_1, a_2, a_3) \in \{(2,3,4,12), (2,3,5,30)\}$. \item We summarize the list of 3-dimensional Pham-Brieskorn rings $B_{a_0, a_1, a_2, a_3}$ for which rigidity is known. It follows in particular that if $B_{2,3,4,12}$ and $B_{2,3,5,30}$ are rigid then the $n = 3$ case of the above conjecture is true. \end{enumerate} In addition to the above, we develop techniques for proving rigidity of rings in general; prove rigidity of many Pham-Brieskorn rings whose dimension is greater than 3; give simple examples of rational projective surfaces with quotient singularities that have an ample canonical divisor and prove that the members of a certain family of singular hypersurfaces are not rational.

Este trabalho está dividido em duas partes distintas. Na primeira parte caracterizamos os números de Lê de polinômios que são rodutos de polinômios de Pham-Brieskorn de mesmo tipo, que denominamos de arranjos de Pham-Brieskorn, obtendo fórmulas para estes números somente utilizando o número de variáveis, os pesos e o grau de homogeneidade destes polinômios. Na segunda parte nos dedicamos a estabelecer relações entre os números de Lê, que é um conceito local, e as classes de Milnor, que são objetos globais que fornecem informações quanto a geometria e topologia de hipersuperfícies analíticas complexas. No contexto geral, usando a hipótese de especialização, relacionamos a classe de Milnor de dimensão máxima de uma hipersuperfície Z numa variedade compacta M com uma soma, sobre os estratos de uma estratificação de Whitney de Z (com estratos conexos) que estão contidos no conjunto singular, em termos do último número de Lê associado a cada estrato. Além disso, obtivemos uma caracterização da classe de Milnor de dimensão mínima via os números de Lê sem usar a hipótese de especialização. Esta classe coincide com o chamado número de Milnor de Parusinski que, assim como os números de Lê, também é uma generalização do número de Milnor / This work is divided into two distinct parts. In the first part we characterize the Lê numbers of polynomials that are products of Pham- Brieskorn polynomials of the same type that we call Pham-Brieskorn arrangements, obtaining formulas to these numbers only using the number of variables, weights and degree of homogeneity of these polynomials. In the second part we are dedicated to establishing relationships between Lê numbers, which is a local concept, and the Milnor classes, which are global objects that provide information about the geometry and topology of complex analytic hypersurfaces. In a general context, using the hypothesis of specialization we relate the top dimensional Milnor class of a hypersurface Z in a compact manifold M with a sum given in terms of the last Lê number associated to each stratum of a Whitney estratification of Z (with connected strata) that are contained in singular set. Moreover, we obtain a characterization of the Milnor class of minimum dimension via the Lê numbers without using the hypothesis of specialization. This class coincides with the Milnor number of Parusinski that, as the Lê numbers, it is also a generalization of the Milnor number

Este trabalho está dividido em duas partes distintas. Na primeira parte caracterizamos os números de Lê de polinômios que são rodutos de polinômios de Pham-Brieskorn de mesmo tipo, que denominamos de arranjos de Pham-Brieskorn, obtendo fórmulas para estes números somente utilizando o número de variáveis, os pesos e o grau de homogeneidade destes polinômios. Na segunda parte nos dedicamos a estabelecer relações entre os números de Lê, que é um conceito local, e as classes de Milnor, que são objetos globais que fornecem informações quanto a geometria e topologia de hipersuperfícies analíticas complexas. No contexto geral, usando a hipótese de especialização, relacionamos a classe de Milnor de dimensão máxima de uma hipersuperfície Z numa variedade compacta M com uma soma, sobre os estratos de uma estratificação de Whitney de Z (com estratos conexos) que estão contidos no conjunto singular, em termos do último número de Lê associado a cada estrato. Além disso, obtivemos uma caracterização da classe de Milnor de dimensão mínima via os números de Lê sem usar a hipótese de especialização. Esta classe coincide com o chamado número de Milnor de Parusinski que, assim como os números de Lê, também é uma generalização do número de Milnor / This work is divided into two distinct parts. In the first part we characterize the Lê numbers of polynomials that are products of Pham- Brieskorn polynomials of the same type that we call Pham-Brieskorn arrangements, obtaining formulas to these numbers only using the number of variables, weights and degree of homogeneity of these polynomials. In the second part we are dedicated to establishing relationships between Lê numbers, which is a local concept, and the Milnor classes, which are global objects that provide information about the geometry and topology of complex analytic hypersurfaces. In a general context, using the hypothesis of specialization we relate the top dimensional Milnor class of a hypersurface Z in a compact manifold M with a sum given in terms of the last Lê number associated to each stratum of a Whitney estratification of Z (with connected strata) that are contained in singular set. Moreover, we obtain a characterization of the Milnor class of minimum dimension via the Lê numbers without using the hypothesis of specialization. This class coincides with the Milnor number of Parusinski that, as the Lê numbers, it is also a generalization of the Milnor number

Thesis (M. Sc. (Hons.))--University of Western Sydney, 2001. / "Research thesis submitted in fulfillment of the requirements for the degree of Master of Science (Honours) Social Ecology, School of Social Ecology and Lifelong Learning, University of Western Sydney, August 2001." Includes bibliographical references (leaves 398-400).

La pratique du rituel de possession de hầu bóng au Vietnam a connu une période d’interdiction pendant plus de quarante ans (de 1954 au début des années quatre-vingt-dix). Ces travaux, basés sur des études sur le terrain réalisées à la fin de cette période d’interdiction tentent d’abord de faire une description détaillée du rituel et l’état des lieux de ses cung văn «maîtres musiciens» à travers l’étude du répertoire du plus grand maître, Pham Van Kiêm. Ils proposent d’explorer ensuite les questions techniques utilisées par les maîtres musiciens et leur rôle par rapport à la pratique du rituel ainsi que les liaisons entre paroles et musique, entre le répertoire musical et le panthéon des génies. Ce témoignage de cette période cruciale permet, par le biais de la musique, de mettre en perspective une pratique religieuse bien complexe en plein essor à ce jour. / The possession ritual practice in Vietnam has been prohibited during more than forty years (from 1954 to early nineties). The field-works are done during this important period. These studies try to portray an outline of the organisation and development of the rite and its music and musicians, especially the most important master at his time, Pham Van Kiêm. They also try to explore the musical technics used by masters and their role in the ritual practice, the music-language, musical directory-deities pantheon relationships. This crucial account allows to put into a perspective this complex practice, very popular nowadays.