The work of mathematical giants, such as Lobachevsky, Gauss, Riemann, Klein and Poincaré, to name a few, lies at the foundation of the study of the highly structured Riemann surfaces, which allow definition of holomorphic maps, corresponding to analytic maps in the theory of complex analysis. A topological result of Poincaré states that every path-connected Riemann surface can be realised by a construction of identifying congruent points in the complex plane, the Riemann sphere or the hyperbolic plane; just three simply connected surfaces that cover the underlying Riemann surface. This requires the discontinuous action of a discrete subgroup of the automorphisms of the corresponding space. In the hyperbolic plane, which is the richest source for Riemann surfaces, these groups are called Fuchsian, and there are several ways to study the action of such groups geometrically by computing fundamental domains. What is accomplished in this thesis is a combination of the methods found by Reidemeister & Schreier, Singerman and Voight, and thus provides a unified way of finding Dirichlet domains for subgroups of cofinite groups with a given index. Several examples are considered in-depth.

A distinguishing colouring of a graph G is a labelling of the vertices of G with colours such that no non-trivial automorphism of G preserves all colours. The distinguishing number of G is the minimum number of colours in a distinguishing colouring. This thesis presents a survey of the history of distinguishing colouring problems and proves new bounds and computational results about distinguishability. An algorithm to generate all labellings of a graph up to isomorphism is presented and compared to a previously published algorithm. The new algorithm is shown to have performance competitive with the existing algorithm, as well as being able to process automorphism groups far larger than the previous limit. A specialization of the algorithm is used to generate all minimal distinguishing colourings of a set of graphs with large automorphism groups and compute their distinguishing numbers. / Graduate / 0984 / 0405 / bbird@uvic.ca

Orientador: João Meidanis / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-21T17:01:05Z (GMT). No. of bitstreams: 1 Feijao_PedroCipriano_D.pdf: 1943126 bytes, checksum: 4c547e8c568bbd0f2eb8235dfde05524 (MD5) Previous issue date: 2012 / Resumo: Rearranjo de genomas é o nome dado a eventos onde grandes blocos de DNA trocam de posição durante o processo evolutivo. Com a crescente disponibilidade de sequências completas de DNA, a análise desse tipo de eventos pode ser uma importante ferramenta para o entendimento da genômica evolutiva. Vários modelos matemáticos de rearranjo de genomas foram propostos ao longo dos últimos vinte anos. Nesta tese, desenvolvemos dois novos modelos. O primeiro foi proposto como uma definição alternativa ao conceito de distância de breakpoint. Essa distância é uma das mais simples medidas de rearranjo, mas ainda não há um consenso quanto à sua definição para o caso de genomas multi-cromossomais. Pevzner e Tesler deram uma definição em 2003 e Tannier et al. a definiram de forma diferente em 2008. Nesta tese, nós desenvolvemos uma outra alternativa, chamada de single-cut-or-join (SCJ). Nós mostramos que, no modelo SCJ, além da distância, vários problemas clássicos de rearranjo, como a mediana de rearranjo, genome halving e pequena parcimônia são fáceis, e apresentamos algoritmos polinomiais para eles. O segundo modelo que apresentamos é o formalismo algébrico por adjacências, uma extensão do formalismo algébrico proposto por Meidanis e Dias, que permite a modelagem de cromossomos lineares. Esta era a principal limitação do formalismo original, que só tratava de cromossomos circulares. Apresentamos algoritmos polinomiais para o cálculo da distância algébrica e também para encontrar cenários de rearranjo entre dois genomas. Também mostramos como calcular a distância algébrica através do grafo de adjacências, para facilitar a comparação com outras distâncias de rearranjo. Por fim, mostramos como modelar todas as operações clássicas de rearranjo de genomas utilizando o formalismo algébrico / Abstract: Genome rearrangements are events where large blocks of DNA exchange places during evolution. With the growing availability of whole genome data, the analysis of these events can be a very important and promising tool for understanding evolutionary genomics. Several mathematical models of genome rearrangement have been proposed in the last 20 years. In this thesis, we propose two new rearrangement models. The first was introduced as an alternative definition of the breakpoint distance. The breakpoint distance is one of the most straightforward genome comparison measures, but when it comes to defining it precisely for multichromosomal genomes, there is more than one way to go about it. Pevzner and Tesler gave a definition in a 2003 paper, and Tannier et al. defined it differently in 2008. In this thesis we provide yet another alternative, calling it single-cut-or-join (SCJ). We show that several genome rearrangement problems, such as genome median, genome halving and small parsimony, become easy for SCJ, and provide polynomial time algorithms for them. The second model we introduce is the Adjacency Algebraic Theory, an extension of the Algebraic Formalism proposed by Meidanis and Dias that allows the modeling of linear chromosomes, the main limitation of the original formalism, which could deal with circular chromosomes only. We believe that the algebraic formalism is an interesting alternative for solving rearrangement problems, with a different perspective that could complement the more commonly used combinatorial graph-theoretic approach. We present polynomial time algorithms to compute the algebraic distance and find rearrangement scenarios between two genomes. We show how to compute the rearrangement distance from the adjacency graph, for an easier comparison with other rearrangement distances. Finally, we show how all classic rearrangement operations can be modeled using the algebraic theory / Doutorado / Ciência da Computação / Doutor em Ciência da Computação

Les travaux présentés dans cette thèse de doctorat relèvent de la combinatoire algébrique et de la théorie des groupes. Précisément, ils apportent une contribution au domaine de recherche qui étudie le comportement des profils des groupes oligomorphes.La première partie de ce manuscrit introduit la plupart des outils qui nous seront nécessaires, à commencer par des éléments de combinatoire et combinatoire algébrique.Nous présentons les fonctions de comptage à travers quelques exemples classiques, et nous motivons l'addition d'une structure d'algèbre graduée sur les objets énumérés dans le but d'étudier ces fonctions.Nous évoquons aussi les notions d'ordre et de treillis.Dans un second temps, nous donnons un aperçu des définitions et propriétés de base associées aux groupes de permutations, ainsi que quelques résultats de théorie des invariants. Nous terminons cette partie par une description de la méthode d'énumération de Pólya, qui permet de compter des objets sous une action de groupe.La deuxième partie est consacrée à l'introduction du domaine dans lequel s'inscrit cette thèse, celui de l'étude des profils de structures relationnelles, et en particulier des profils orbitaux. Si G est un groupe de permutations infini, son profil est la fonction de comptage qui envoie chaque entier n > 0 sur le nombre d'orbites de n-sous-ensembles, pour l'action induite de G sur les sous-ensembles finis d'éléments.Cameron a conjecturé que le profil de G est équivalent à un polynôme dès lors qu'il est borné par un polynôme. Une autre conjecture, plus forte, a été plus tard émise par Macpherson : elle implique une certaine structure d'algèbre graduée sur les orbites de sous-ensembles, créée par Cameron et baptisée algèbre des orbites, soutenant que si le profil est borné par un polynôme, alors l'algèbre des orbites est de type fini.Comme amorce de notre étude de ce problème, nous développons quelques exemples et faisons nos premiers pas vers une résolution en examinant les systèmes de blocs des groupes de profil borné par un polynôme --- que nous appelons P-oligomorphes ---,ainsi que la notion de sous-produit direct.La troisième partie démontre une classification des groupes P-oligomorphes, notre résultat le plus important et dont la conjecture de Macpherson se révèle un corollaire.Tout d'abord, nous étudions la combinatoire du treillis des systèmes de blocs,qui conduit à l'identification d'un système généralisé particulier, constituébde blocs ayant de bonnes propriétés. Nous abordons ensuite le cas particulier o`u il se limite à un seul bloc de blocs, pour lequel nous établissons une classification. La preuve emprunte à la notion de sous-produit direct pour gérer les synchronisations internes au groupe, et a requis une part d'exploration informatique afin d'être d'abord conjecturée.Dans le cas général, nous nous appuyons sur les résultats précédents et mettons en évidence la structure de G comme produit semi-direct impliquant son sous-groupe normal d'indice fini minimal et un groupe fini. Ceci permet de formaliser une classification complète des groupes P-oligomorphes,et d'en déduire la forme de l'algèbre des orbites : (à peu de choses près) une algèbre d'invariants explicite d'un groupe fini. Les conjectures de Macpherson et de Cameron en découlent, et plus généralement une compréhension exhaustive de ces groupes.L'annexe contient des extraits du code utilisé pour mener la preuve à bien,ainsi qu'un aperçu de celui qui a été produit en s'appuyant sur la nouvelle classification, qui permet de manipuler les groupes P-oligomorphes en usant d'une algorithmique adaptée. Enfin, nous joignons ici notre première preuve, plus faible, des deux conjectures. / This PhD thesis falls under the fields of algebraic combinatorics and group theory. Precisely,it brings a contribution to the domain that studies profiles of oligomorphic permutation groups and their behaviors.The first part of this manuscript introduces most of the tools that will be needed later on, starting with elements of combinatorics and algebraic combinatorics.We define counting functions through classical examples ; with a view of studying them, we argue the relevance of adding a graded algebra structure on the counted objects.We also bring up the notions of order and lattice.Then, we provide an overview of the basic definitions and properties related to permutation groups and to invariant theory. We end this part with a description of the Pólya enumeration method, which allows to count objects under a group action.The second part is dedicated to introducing the domain this thesis comes withinthe scope of. It dwells on profiles of relational structures,and more specifically orbital profiles.If G is an infinite permutation group, its profile is the counting function which maps any n > 0 to the number of orbits of n-subsets, for the inducedaction of G on the finite subsets of elements.Cameron conjectured that the profile of G is asymptotically equivalent to a polynomial whenever it is bounded by apolynomial.Another, stronger conjecture was later made by Macpherson : it involves a certain structure of graded algebra on the orbits of subsetscreated by Cameron, the orbit algebra, and states that if the profile of G is bounded by a polynomial, then its orbit algebra is finitely generated.As a start in our study of this problem, we develop some examples and get our first hints towards a resolution by examining the block systems ofgroups with profile bounded by a polynomial --- that we call P-oligomorphic ---, as well as the notion of subdirect product.The third part is the proof of a classification of P-oligomorphic groups,with Macpherson's conjecture as a corollary.First, we study the combinatorics of the lattice of block systems,which leads to identifying one special, generalized such system, that consists of blocks of blocks with good properties.We then tackle the elementary case when there is only one such block of blocks, for which we establish a classification. The proof borrows to the subdirect product concept to handle synchronizations within the group, and relied on an experimental approach on computer to first conjecture the classification.In the general case, we evidence the structure of a semi-direct product involving the minimal normal subgroup of finite index and some finite group.This allows to formalize a classification of all P-oligomorphic groups, the main result of this thesis, and to deduce the form of the orbit algebra: (little more than) an explicit algebra of invariants of a finite group. This implies the conjectures of Macpherson and Cameron, and a deep understanding of these groups.The appendix provides parts of the code that was used, and a glimpse at that resulting from the classification afterwards,that allows to manipulate P-oligomorphic groups by apropriate algorithmics. Last, we include our earlier (weaker) proof of the conjectures.

Cette thèse porte sur trois problèmes en combinatoire algébrique effective et algorithmique.Les premières parties proposent une approche alternative aux bases de Gröbner pour le calcul des invariants secondaires des groupes de permutations, par évaluation en des points choisis de manière appropriée. Cette méthode permet de tirer parti des symétries du problème pour confiner les calculs dans un quotient de petite dimension, et ainsi d'obtenir un meilleur contrôle de la complexité algorithmique, en particulier pour les groupes de grande taille. L'étude théorique est illustrée par de nombreux bancs d'essais utilisant une implantation fine des algorithmes. Un prérequis important est la génération efficace de vecteurs d'entiers modulo l'action d'un groupe de permutation, dont l'algorithmique fait l'objet d'une partie préliminaire.La quatrième partie cherche à déterminer, pour un certain quotient naturel d'une algèbre de Hecke affine, quelles spécialisations des paramètres aux racines de l'unité donne un comportement non générique.Finalement, la dernière partie présente une conjecture sur la structure d'une certaine $q$-déformation des polynômes harmoniques diagonaux en plusieurs paquets de variables pour la famille infinie de groupes de réflexions complexes.Tous ces chapitres s'appuient fortement sur l'exploration informatique, et font l'objet de multiples contributions au logiciel Sage. / This thesis concerns algorithmic approaches to three challenging problems in computational algebraic combinatorics.The firsts parts propose a Gröbner basis free approach for calculating the secondary invariants of a finite permutation group, proceeding by using evaluation at appropriately chosen points. This approach allows for exploiting the symmetries to confine the calculations into a smaller quotient space, which gives a tighter control on the algorithmic complexity, especially for large groups. The theoretical study is illustrated by extensive benchmarks using a fine implementation of algorithms. An important prerequisite is the generation of integer vectors modulo the action of a permutation group, whose algorithmic constitute a preliminary part of the thesis.The fourth part of this thesis is determining for a certain interesting quotient of an affine Hecke algebra exactly which root-of-unity specialization of its parameter lead to non-generic behavior.Finally, the last part presents a conjecture on the structure of certain q-deformed diagonal harmonics in many sets of variables for the infinite family of complex reflection groups.All chapters proceed widely by computer exploration, and most of established algorithms constitute contributions of the software Sage.