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.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23681号 / 理博第4771号 / 新制||理||1683(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 加藤 周, 教授 雪江 明彦, 教授 平岡 裕章 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22972号 / 理博第4649号 / 新制||理||1668(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 藤原 耕二, 教授 山口 孝男, 教授 入谷 寛 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Les structures menues apparaissent dans les années 60 en lien avec la conjecture de Vaught. Les structures minces englobent à la fois les structures minimales et menues. Les ensembles définissables d'une structure mince sont rangés par le rang de Cantor-Bendixson. Nous présentons des propriétés de calcul de ce rang, une condition de chaîne descendante locale sur les groupes acl(0)-définissables ainsi qu'une notion de presque stabilisateur local, et en déduisons des propriétés algébriques des structures minces : un corps mince de caractéristique positive est localement de dimension finie sur son centre, et un groupe mince infini a un sous groupe abélien infini. Nous nous intéressons ensuite aux structures menues infiniment définissables, et montrons que les groupes d'arité finie infiniment 0-définissable sont l'intersection de groupes définissables. Nous étendons le résultat aux demi-groupes, anneaux, corps, catégories et groupoïdes infiniment 0-définissables, et donnons des résultats de définissabilité locale pour les groupes et corps simples et menus, infiniment définissables sur des paramètres quelconques. Enfin, nous réintroduisons le rang de Cantor dans son contexte topologique et montrons que la dérivée de Cantor peut être vue comme un opérateur de dérivation dans un semi-anneau d'espaces topologiques. Dans l'idée de trouver un rang de Cantor global pour les théories stables, nous essayons de nous débarrasser du mot dénombrable omniprésent lorsque l'on fait de la topologie, en le remplaçant par un cardinal régulier k. Nous développons une notion d'espace k-métrique, de k-topologie, de k-compacité etc. et montrons un k-analogue du lemme de métrisabilité d'Urysohn, et du théorème de Cantor-Bendixson. / Abstract. Small structures appear in the '60s together with Vaught's conjecture. Weakly small structures include both minimal and small structures. Definable sets in a weakly small structure are ranked by Cantor-Bendixson rank. We show computational properties of this rank, which imply a local descending chain condition on acl(0)-definable subgroups, and introduce a notion of local almost stabiliser. We deduce algebraic properties of weakly small structures. Among them, a weakly small field of positive characteristic is locally finite dimensional over its centre, and an infinite weakly small group has an infinite abelian subgroup. We then turn to small type-definable structures, showing that finitary small type 0-de_nable groups are the intersection of definable groups. We extend the result to finitary small type 0- definable monoids, rings, fields, categories and groupoids. We give local definability results concerning groups and fields type definable over an arbitrary set of parameters in small and simple theories. Finally, we reintroduce the Cantor Bendixson rank in its topological context, and show that the Cantor derivative can be seen as a derivation in a semi-ring of topological spaces. In an attempt to find a global Cantor rank for stable structures, we try to eliminate the word denumerable, omnipresent when one does topology, by replacing it by a regular cardinal k. We develop the notions of k-metrisable space, k-topology, k-compactness etc. and show an analogue of Urysohn's metrisability lemma and Cantor-Bendixson theorem.

Sejam $F$ um corpo de característica diferente de $2$ e $G$ um grupo. A partir da involução clássica, que envia cada elemento em seu inverso, e uma orientação do grupo $G$ é possível definir uma involução clássica orientada na álgebra de grupo $FG$. O objetivo desta tese é estudar propriedades de Lie do conjunto dos elementos simétricos $(FG)^+$ e, em alguns casos, do conjunto dos elementos anti-simétricos $(FG)^-$. Primeiro, abordamos o caso quando $G$ não tem elementos de ordem $2$. Aqui, mostramos que se $(FG)^+$ (ou $(FG)^-$) é Lie nilpotente ou Lie $n$-Engel, então $FG$ também é Lie nilpotente ou Lie $m$-Engel, respectivamente. Depois, consideramos o caso quando $G$ contém uma cópia do grupo quatérnio de ordem $8$. Neste caso, caracterizamos completamente as álgebras de grupo tais que $(FG)^+$ é fortemente Lie nilpotente, Lie nilpotente e Lie $n$-Engel. Como consequência, provamos que o conjunto das unidades simétricas deste tipo de grupos é nilpotente. Estudamos também o caso em que quando $G$ não contém uma cópia do grupo quatérnio de ordem $8$. Em particular, apresentamos um exemplo que mostra que os resultados obtidos em pesquisas anteriores, com a involução clássica, não devem ser esperados ao trabalhar com involuções clássicas orientadas. Não entanto, damos alguns casos especiais de grupos nos quais esses resultados são obtidos. Finalmente, estudamos o índice de Lie nilpotência de $(FG)^+$. Estabelecemos uma condição necessária e suficiente, para que o índice de Lie nilpotência de $(FG)^+$ e a classe de nilpotência das unidades simétricas de uma álgebra de grupo Lie nilpotente sejam o maior possível. Além disso, consideramos a situação em que o grupo $G$ contém uma cópia de $Q_8$. / Let $F$ be a field of characteristic different from $2$ and $G$ a group. From the classical involution, which sends each element in its inverse and an orientation of $G$, it is possible to define an oriented classical involution on the group algebra $FG$. The goal of this thesis is to study Lie properties of the set of symmetric elements $(FG)^+$ and, in some cases, of the set of skew-symmetric elements $(FG)^-$. We first deal with the case when $G$ does not have elements of order $2$. In this situation, we show that if $(FG)^+$ (or $(FG)^-$) is Lie nilpotent or Lie $n$-Engel, then the whole group algebra $FG$ satisfies the same property. Later we consider the case when $G$ contains a copy of the quaternion group of order $8$. In this instance, we give a complete description of the group algebras such that $(FG)^+$ is strongly Lie nilpotent, Lie nilpotent and Lie $n$-Engel. As a consequence, we get that the set of symmetric units of this kind of groups is nilpotent. Furthermore, we study the case when $G$ does not contain a copy of the quaternion group of order $8$. Here, we present an example that shows that the previews results obtained in former works, with the classical involution, may not hold with an oriented classical involution. However, we give some kinds of groups for which those results are achieved. Finally, we study the Lie nilpotency index of $(FG)^+$. It is given a necessary and sufficient condition to the Lie nilpotency index of $(FG)^+$ and the nilpotency class of the symmetric units to be maximal, in a Lie nilpotent group algebra. In addition, we consider the situation when $G$ contains a copy of the quaternion group of order $8$.

Sejam $F$ um corpo de característica diferente de $2$ e $G$ um grupo. A partir da involução clássica, que envia cada elemento em seu inverso, e uma orientação do grupo $G$ é possível definir uma involução clássica orientada na álgebra de grupo $FG$. O objetivo desta tese é estudar propriedades de Lie do conjunto dos elementos simétricos $(FG)^+$ e, em alguns casos, do conjunto dos elementos anti-simétricos $(FG)^-$. Primeiro, abordamos o caso quando $G$ não tem elementos de ordem $2$. Aqui, mostramos que se $(FG)^+$ (ou $(FG)^-$) é Lie nilpotente ou Lie $n$-Engel, então $FG$ também é Lie nilpotente ou Lie $m$-Engel, respectivamente. Depois, consideramos o caso quando $G$ contém uma cópia do grupo quatérnio de ordem $8$. Neste caso, caracterizamos completamente as álgebras de grupo tais que $(FG)^+$ é fortemente Lie nilpotente, Lie nilpotente e Lie $n$-Engel. Como consequência, provamos que o conjunto das unidades simétricas deste tipo de grupos é nilpotente. Estudamos também o caso em que quando $G$ não contém uma cópia do grupo quatérnio de ordem $8$. Em particular, apresentamos um exemplo que mostra que os resultados obtidos em pesquisas anteriores, com a involução clássica, não devem ser esperados ao trabalhar com involuções clássicas orientadas. Não entanto, damos alguns casos especiais de grupos nos quais esses resultados são obtidos. Finalmente, estudamos o índice de Lie nilpotência de $(FG)^+$. Estabelecemos uma condição necessária e suficiente, para que o índice de Lie nilpotência de $(FG)^+$ e a classe de nilpotência das unidades simétricas de uma álgebra de grupo Lie nilpotente sejam o maior possível. Além disso, consideramos a situação em que o grupo $G$ contém uma cópia de $Q_8$. / Let $F$ be a field of characteristic different from $2$ and $G$ a group. From the classical involution, which sends each element in its inverse and an orientation of $G$, it is possible to define an oriented classical involution on the group algebra $FG$. The goal of this thesis is to study Lie properties of the set of symmetric elements $(FG)^+$ and, in some cases, of the set of skew-symmetric elements $(FG)^-$. We first deal with the case when $G$ does not have elements of order $2$. In this situation, we show that if $(FG)^+$ (or $(FG)^-$) is Lie nilpotent or Lie $n$-Engel, then the whole group algebra $FG$ satisfies the same property. Later we consider the case when $G$ contains a copy of the quaternion group of order $8$. In this instance, we give a complete description of the group algebras such that $(FG)^+$ is strongly Lie nilpotent, Lie nilpotent and Lie $n$-Engel. As a consequence, we get that the set of symmetric units of this kind of groups is nilpotent. Furthermore, we study the case when $G$ does not contain a copy of the quaternion group of order $8$. Here, we present an example that shows that the previews results obtained in former works, with the classical involution, may not hold with an oriented classical involution. However, we give some kinds of groups for which those results are achieved. Finally, we study the Lie nilpotency index of $(FG)^+$. It is given a necessary and sufficient condition to the Lie nilpotency index of $(FG)^+$ and the nilpotency class of the symmetric units to be maximal, in a Lie nilpotent group algebra. In addition, we consider the situation when $G$ contains a copy of the quaternion group of order $8$.

Thèse numérisée par la Division de la gestion de documents et des archives de l'Université de Montréal.

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2014-09-18T15:33:16Z No. of bitstreams: 2 Dissertacao Yerko Contreras Rojas.pdf: 673331 bytes, checksum: 5359343f8c3a32e21369c3bc57917634 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2014-09-18T15:43:59Z (GMT) No. of bitstreams: 2 Dissertacao Yerko Contreras Rojas.pdf: 673331 bytes, checksum: 5359343f8c3a32e21369c3bc57917634 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2014-09-18T15:43:59Z (GMT). No. of bitstreams: 2 Dissertacao Yerko Contreras Rojas.pdf: 673331 bytes, checksum: 5359343f8c3a32e21369c3bc57917634 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2013-07-05 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This document presents an approach and development of some of the results of Shumyatsky in [14, 15, 16, 17, 18], where he worked with automorphisms of order two in finite groups of odd order, mainly showing the influence that the structure of the centralizer has on that of Group. Let G be a group with odd order, and ϕ an automorphism on G, of order two, where G = [G,ϕ], and given a limitation in the order of the centralizer of ϕ regard to G, CG(ϕ), which induces a limitation in the order of derived group G′ of group G, and we also verified that G has a normal subgroup H that is ϕ-invariant, such that H′ ≤ Gϕ and its index [G : H] is bounded with the initial limitation. With the same hypothesis of the group G and with the same limitation of the order of the centralizer of the automorphism, let V a abelian p-group such that G⟨ϕ⟩ act faithful and irreductible on V, then there is a bounded constant k, limitated by a function depending only on the parameter m, where m is tha limitation in the order of CG(ϕ), and elements x1, ...xk ∈ G−ϕ such that V = ρϕx 1,...,xk(V−ϕ). / O trabalho baseia-se na apresentação e desenvolvimento de alguns resultados expostos por Shumyatsky em [14, 15, 16, 17, 18], onde trabalha com automorfismos de ordem dois em grupos de ordem ímpar, mostrando fundamentalmente a influência da estrutura do centralizador do automorfismo na estrutura do grupo. Seja G um grupo de ordem ímpar e ϕ um automorfismo de G, de ordem dois, tal que G = [G,ϕ], dada uma limitação na ordem do centralizador de ϕ em G, CG(ϕ), a mesma induz uma limitação na ordem do grupo derivado G′ do grupo G, além disso verificamos que G tem um subgrupo H normal ϕ-invariante, tal que H′ ≤ Gϕ e o índice [G : H] é limitado dependendo da limitação inicial de CG(ϕ). Nas mesmas hipóteses do grupo G e com a mesma limitação da ordem do centralizador do automorfismo, seja V um p-grupo abeliano, tal que G⟨ϕ⟩ age fiel e irredutivelmente sobre V, então existe uma constante k, limitada por uma função que depende só da limitação de CG(ϕ), e elementos x1, ...xk ∈ G−ϕ, tal que V = ρϕx 1,...,xk(V−ϕ).

Dans ce travail nous nous intéressons aux groupes cristallographiques, i.e. aux sous-groupes du groupe des transformations affines qui agissent proprement discontinûment et de façon cocompacte sur l’espace affine. Ce sont les groupes fondamentaux des variétés affines compactes et complètes. Nous classifions les groupes cristallographiques dont la partie linéaire préserve une forme hermitienne de signature (n,1). Grunewald et Margulis ont prouvé que ces groupes cristallographiques sont virtuellement résolubles (la conjecture d’Auslander affirme que c’est toujours le cas). Notre classification est effectuée pour n ≤ 3. Elle correspond à la classification, à revêtement fini près, des variétés Hermite-Lorentz plates, compactes et complètes en dimension complexe inférieure ou égale à4. Ce travail est inspiré par ceux menés par Bieberbach, puis Fried, et enfin Grunewald et Margulis sur les groupes cristallographiques dont la partie linéaire préserve une forme quadratique définie positive ou lorentzienne. En effectuant cette classification, nous avons été amené à étudier certains familles d’algèbres de Lie nilpotentes de dimension 8. Nous avons ensuite étendu cette classification à celle de toutes les algèbres de Lie 3-nilpotentes de dimension 8 ayant l’algèbre de Lie libre 3-nilpotente à 3générateurs pour quotient. Ce résultat peut être vu comme un pas dans la direction d’une classification des algèbres de Lie nilpotentes de dimension 8. Ensuite nous nous sommes demandé lesquelles de ces algèbres admettent une métrique pseudo-riemannienne plate et nous avons donné une réponse partielle. / In this work we deal with crystallographic groups, i.e. the subgroups of the group of affine transformations that act properly discontinuously and cocompactly on affine space. In otherwords they are the fundamental groups of compact and complete affine manifolds. In this thesis we classify such groups with the additional hypothesis that the linear part preserves a Hermitian form of signature (n,1). Grunewald and Margulis proved that such crystallographic groups are virtually solvable (the Auslander conjecture states that this is always true). Our classification is for n ≤ 3. It corresponds to a classification, up to finite covering, and for complex dimension at most 4, of flat compact complete Hermite-Lorentz manifolds. This is inspired by the works done by Bieberbach,then Fried, and finally Grunewald and Margulis who classified crystallographic groups whose line arpart preserves a positive definite or Lorentzian quadratic form. Making this classification we had to classify a family of 8-dimensional nilpotent Lie algebras. We then extended this classification toall the 8-dimensional 3-step nilpotent Lie algebras having the free 2-step nilpotent Lie algebra on 3generators as quotient. This result can be seen as a step in the direction of a general classification of nilpotent Lie algebras of dimension 8. We then wondered which of these Lie algebras admit flat pseudo-Riemannian metrics and gave a partial answer to this question.

Este trabalho é sobre as geométrias finitas com 3 ou 4 pontos na cada reta e os loops e qiasigrupos relacionados. Em caso de 3 pontos na cada reta descrevemos o loop de Steiner correspondente livre e calculamos o grupo de automorfismos em caso de 3 geradores livres. Além disso descrevemos os loopos de Steiner nilpotentes de clase dois e classificamos estes loopos com 3 geradores. Em caso de 4 pontos na cada reta construimos as geometrias novas atraves de expanção central de um análogo não comutativo do quasigrupo de Steiner. Temos fortes indícios que esta construção é universal em algum sentido. / This work is about finite geometries with 3 or 4 points on every line and related loops and quasigroups. In the case of 3 points on any line we describe the structure of free loops in the variety of corresponding Steiner loops and we calculate the group of automorphisms of free Steiner loop with three generators. We describe the structure of nilpotent class two Steiner loops and classifiy all such loops with three generators. In the case of 4 points on a line we constructe new series of such geometries as central extension of corresponding non-commutative Steiner quasigroups. We conjecture that those geometries are universal in some sense.