La théorie des groupes de tresses s'inscrit au croisement de plusieurs domaines des mathématiques, en particulier, l'algèbre et la géométrie. La recherche actuelle s'étend dans chacune de ces directions, et de riches développements naissent du mariage de ces deux aspects. D'un point de vue géométrique, le groupe des tresses à n brins est vu comme le groupe modulaire d'un disque à n trous, avec composante de bord. On peut représenter une tresse par un diagramme de courbes, c'est-à-dire l'image d'une famille fixée d'arcs sur le disque, par l'élément correspondant du groupe modulaire. Dans cette thèse est présenté l'algorithme de relaxations par la droite, qui permet de retrouver, étant donné un diagramme de courbes, la tresse à partir de laquelle il a été obtenu. Cet algorithme aide à faire le lien entre des propriétés géométriques du diagramme de courbes, et des propriétés algébriques du mot de tresse, en permettant de repérer de grandes puissances d'un générateur sous forme de spirales dans le diagramme de courbes. D'un point de vue algébrique, le groupe de tresses est l'exemple classique de groupe de Garside. L'un des objectifs actuels des recherches en théorie de Garside est d'obtenir un algorithme de résolution en temps polynomial du problème de conjugaison dans les groupes de tresses. À cette fin, on cherche à exploiter les propriétés de certains ensembles finis de conjugués d'une tresse, qui sont des invariants de conjugaison. L'un des résultats de cette thèse concerne la taille d'un de ces invariants, l'ensemble super-sommital : on exhibe une famille de tresses pseudo-anosoviennes dont l'ensemble super-sommital est de taille exponentielle. González-Meneses avait déjà établi le résultat similaire pour une famille de tresses réductibles. La conséquence de ces résultats est qu'on ne peut pas espérer résoudre le problème de conjugaison en temps polynomial au moyen de cet ensemble, et qu'il vaut mieux chercher à exploiter des invariants plus petits. Dans le cas des tresses pseudo-anosoviennes, des espoirs résident actuellement en l'ensemble des circuits glissants. Dans cette thèse, un algorithme en temps polynomial s'appuyant sur ce dernier ensemble résout génériquement le problème de conjugaison, c'est-à-dire qu'il le résout pour une proportion de tresses tendant exponentiellement vite vers 1 lorsque la longueur de la tresse tend vers l'infini. On montre également que, dans une boule du graphe de Cayley avec pour générateurs les tresses simples, une tresse générique est pseudo-anosovienne, ce qui était une conjecture bien connue des spécialistes de la théorie de Garside. / The theory of braid groups is at the intersection of several areas of mathematics, especially algebra and geometry. The current research extends in each of these directions, leading to rich developments. From a geometrical point of view, the braid group on n strands is seen as the mapping class group of a disc with n punctures, with boundary component. A braid can be represented by a curve diagram, that is to say, the image of a family of arcs attached to the disc, by the corresponding mapping class. In this thesis we present the algorithm of relaxations from the right, which, given a curve diagram, determines the braid from which it was obtained. This algorithm helps us to make the link between geometric properties of the curve diagram and algebraic properties of the braid word, allowing us to identify great powers of a generator as spirals in the curve diagram. From an algebraic point of view, the braid group is the classical example of a Garside group. One of the objectives of current research in Garside theory is to obtain a polynomial time algorithm to solve the conjugacy problem in braid groups. For this, a possibility is to exploit the properties of some finite sets of conjugates of a braid, which are invariants of the conjugacy classes. One of the results of this thesis concerns the size of one of these invariants, the super summit set: we construct a family of pseudo-Anosov braids whose super summit set has exponential size. González- Meneses had already established the similar result for a family of reducible braids. These results implies that we cannot hope to solve the conjugacy problem in polynomial time through this set, and it is better to try to use smaller invariants. In the case of pseudo-Anosov braids, one may hope that the so-called sliding circuit set is more useful. In this thesis, we present a polynomial time algorithm based on this last set which generically solves the conjugacy problem, that is to say, it solves it for a proportion of braids that tends exponentially fast to 1 as the length of the braid tends to infinity. We also show that, in a ball of the Cayley graph with generators the simple braids, a braid is generically pseudo-Anosov, which was a well-known conjecture for the specialists in Garside theory.

Bobbin lace is a 500-year-old art form in which threads are braided together in an alternating manner to produce a lace fabric. A key component in its construction is a small pattern, called a bobbin lace ground, that can be repeated periodically to fill a region of any size. In this thesis we present a mathematical model for bobbin lace grounds representing the structure as the pair (Δ(G), ζ (v)) where Δ(G) is a topological embedding of a 2-regular digraph, G, on a torus and ζ(v) is a mapping from the vertices of G to a set of braid words. We explore in depth the properties that Δ(G) must possess in order to produce workable lace patterns. Having developed a solid, logical foundation for bobbin lace grounds, we enumerate and exhaustively generate patterns that conform to that model. We start by specifying an equivalence relation and define what makes a pattern prime so that we can identify unique representatives. We then prove that there are an infinite number of prime workable patterns. One of the key properties identified in the model is that it must be possible to partition Δ(G) into a set of osculating circuits such that each circuit has a wrapping index of (1,0); that is, the circuit wraps once around the meridian of the torus and does not wrap around the longitude. We use this property to exhaustively generate workable patterns for increasing numbers of vertices in G by gluing together lattice paths in an osculating manner. Using a backtracking algorithm to process the lattice paths, we identify over 5 million distinct prime patterns. This is well in excess of the roughly 1,000 found in lace ground catalogues. The lattice paths used in our approach are members of a family of partially directed lattice paths that have not been previously reported. We explore these paths in detail, develop a recurrence relation and generating function for their enumeration and present a bijection between these paths and a subset of Motzkin paths. Finally, to draw out of the extremely large number of patterns some of the more aesthetically interesting cases for lacemakers to work on, we look for examples that have a high degree of symmetry. We demonstrate, by computational generation, that there are lace ground representatives from each of the 17 planar periodic symmetry groups. / Graduate / 0389 / 0984 / 0405 / veronikairvine@gmail.com

We prove that the non-k-equal configuration space of a graph has a discretized model, analogous to the discretized model for configurations on graphs. We apply discrete Morse theory to the latter to give an explicit combinatorial formula for the ranks of homology and cohomology of configurations of two points on a tree. We give explicit presentations for homology and cohomology classes as well as pairings for ordered and unordered configurations of two and three points on a few simple trees, and show that the first homology group of ordered and unordered configurations of two points in any tree is generated by the first homology groups of configurations of two points in three particular graphs, K_{1,3}, K_{1,4}, and the trivalent tree with 6 vertices and 2 vertices of degree 3, via graph embeddings.

Neste trabalho, estudamos os grupos de tranças em superfícies visando encontrar apresentações para estes grupos em superfícies fechadas orientáveis de gênero g >= 1 ou superfícies fechadas não orientáveis de gênero g >= 2. Uma consequência destas apresentações é resolvermos o problema da palavra, que consiste em encontrar um algoritmo para decidir quando uma dada palavra num grupo definido por seus geradores e suas relações é a palavra trivial / In this work, we find presentations for surface braid groups either in closed orientable surfaces of genus g >= 1 or in closed non-orientable surfaces of genus g >= 2. A consequence of this presentations is to solve the word problem, which consists in finding an algorithm to decide when a given word in a group defined by its generators and its relations is the trivial word

Dada uma superfície M, definiremos os grupos de tranças de M, denotado por \'B IND. n\' (M), geometricamente e usando a noção de espaços de confiuração. Mostraremos a equivalência das definições. Na mesma linha de raciocínio, definiremos os grupos de tranças puras de superfícies \'P IND. n\' (M). Apresentaremos as propriedades mais importantes dos grupos de tranças do plano e mostraremos que \'B IND. n\' (\'R POT. 2\') injeta em \'B IND. n\' (M), para muitas superfícies M. Mais detalhadamente, obteremos a apresentação de \'B IND. n\' (\'RP POT. 2\' ) e \'P IND. n\'(\'RP POT. 2\') / For a surface M, we define the braid groups of M, \'B IND. n\'(M), geometricaly and using the notion of configuration spaces. We show the equivalence of these definitions. In the sequence, we define the pure braid group of M, \'P IND. n\' (M). We present the most important properties of braid groups of the plane and we show that \'B IND. n\'\'(\'R POT. 2\') embedds in \'B IND. n\' (M), for almost all M. In a more detailed fashion, we present \'B IND. n\' (\'RP POT. 2\') and \'P IND. n\' (\'RP POT. 2)

Thesis by publication. / Thesis (PhD)--Macquarie University (Division of Information & Communication Sciences, Dept. of Mathematics), 2007. / Bibliography: p. 119-123. / Introduction -- Mackey functors on compact closed categories -- Lax braidings and the lax centre -- On centres and lax centres for promonoidal catagories -- Pullback and finite coproduct preserving functors between categories of permutation representations -- Conclusion. / This thesis studies the theory of Mackey functors as an application of enriched category theory and highlights the notions of lax braiding and lax centre for monoidal categories and more generally promonoidal categories ... The third contribution of this thesis is the study of functors between categories of permutation representations. / x,123 p. ill

Υπάρχει μια ιδιαίτερη κατηγορία φυσικών συστημάτων, τα οποία καλούνται ολοκληρώσιμα. Η ολοκληρωσιμότητα ενός συστήματος συνεπάγεται άμεσα πως αυτό είναι ακριβώς επιλύσιμο, ενώ συνήθως το σύστημα παρουσιάζει μεγάλη συμμετρία. Η θεωρία των ο- λοκληρώσιμων συστημάτων, κλασικών και κβαντικών, παρέχει τα κατάλληλα εργαλεία για τη μελέτη των εν λόγω προτύπων με συστηματικό τρόπο. Στην παρούσα διατρι- βή μελετούμε τέτοιου είδους συστήματα, δίνοντας έμφαση στις αλγεβρικές δομές και τις συμμετρίες που βρίσκονται πίσω από αυτά. Στο πρώτο μέρος, περιγράφονται στοιχεία της θεωρίας των κλασικών ολοκληρώσιμων συστημάτων. Ο συστηματικός τρόπος περιγρα- φής τους επιτρέπει και την επέκταση αυτών, εισάγοντας για παράδειγμα μη τετριμμένες συνοριακές συνθήκες ή τοπικές ατέλειες, έτσι ώστε η ολοκληρωσιμότητα του συστήματος να διατηρείται. Στο δεύτερο κεφάλαιο περιγράφεται η θεωρία της ολοκληρωσιμότητας σε κβαντικό επίπεδο και το πλαίσιο ακριβούς επίλυσης τέτοιων συστημάτων μέσω ισχυρών μεθόδων, όπως η τεχνική Bethe ansatz. Σημαντικό ρόλο στο πεδίο αυτό διαδραματίζει η ομάδα braid και τα υποσύνολά της, καθώς εξασφαλίζουν την παραγωγή συμμετρικών λύσεων των εξισώσεων της κβαντικής ολοκληρωσιμότητας, με συστηματικό τρόπο. Στο κεφάλαιο αυτό περιγράφεται το πλαίσιο παραγωγής τέτοιων λύσεων, και συγκεκριμένα δημοσιευμένα αποτελέσματα. Τέλος, στο κεφάλαιο 3 περιγράφονται εμβαπτίσεις μεμβρα- νών σε σφαιρικές υποπολλαπλότητες, όπως αυτές υπεισέρχονται στη θεωρία των χορδών. Εκτός της κατασκευής των συγκεκριμένων εμβαπτίσεων, παρουσιάζεται και η σχέση τους με συστήματα της φυσικής της συμπυκνωμένης ύλης, χρησιμοποιώντας το ισχυρό πλαίσιο της αντιστοιχίας AdS/CFT. / There is a special category of physical systems, called integrable. The integrability of a system implies directly that this is exactly solvable, while there usually exists a large amount of symmetry. The theory of integrable systems, both classical and quantum, provides the appropriate tools for the study of these models in a systematic way. In this dissertation we study such systems, giving emphasis on the underlying algebraic structures and symmetries. In the first part, we describe elements of the theory of classic integrable systems. The systematic way of describing them leads to natural extensions, for example by introducing non-trivial boundary conditions or local defects, in a way that the integrability of the system is preserved. In the second chapter the theory of integrability at the quantum level is described, as well as the framework for exactly solving such systems through powerful methods, such as Bethe ansatz method. Important role in this framework is played by the braid group and its quotients, as they provide a systematic way of obtaining solutions of the equations of quantum integrability in a systematic manner. This chapter describes the framework for the construction of such solutions, and particular published results. Finally, chapter 3 describes brane embeddings in sphere submanifolds, which exist within string theory. Besides the construction of these embeddings, their relation with systems of physics of condensed matter is presented, using the powerful framework of the AdS/CFT correspondence.

abstract: There are many applications for polymer matrix composite materials in a variety of different industries, but designing and modeling with these materials remains a challenge due to the intricate architecture and damage modes. Multiscale modeling techniques of composite structures subjected to complex loadings are needed in order to address the scale-dependent behavior and failure. The rate dependency and nonlinearity of polymer matrix composite materials further complicates the modeling. Additionally, variability in the material constituents plays an important role in the material behavior and damage. The systematic consideration of uncertainties is as important as having the appropriate structural model, especially during model validation where the total error between physical observation and model prediction must be characterized. It is necessary to quantify the effects of uncertainties at every length scale in order to fully understand their impact on the structural response. Material variability may include variations in fiber volume fraction, fiber dimensions, fiber waviness, pure resin pockets, and void distributions. Therefore, a stochastic modeling framework with scale dependent constitutive laws and an appropriate failure theory is required to simulate the behavior and failure of polymer matrix composite structures subjected to complex loadings. Additionally, the variations in environmental conditions for aerospace applications and the effect of these conditions on the polymer matrix composite material need to be considered. The research presented in this dissertation provides the framework for stochastic multiscale modeling of composites and the characterization data needed to determine the effect of different environmental conditions on the material properties. The developed models extend sectional micromechanics techniques by incorporating 3D progressive damage theories and multiscale failure criteria. The mechanical testing of composites under various environmental conditions demonstrates the degrading effect these conditions have on the elastic and failure properties of the material. The methodologies presented in this research represent substantial progress toward understanding the failure and effect of variability for complex polymer matrix composites. / Dissertation/Thesis / Doctoral Dissertation Mechanical Engineering 2016

En 2004, motivés par des constructions de quasimorphismes sur des groupes d'homéomorphismes et de difféomorphismes, Gambaudo et Ghys démontrèrent une formule liant les ω-signatures d'un entrelacs et les propriétés symplectiques d'une représentation du groupe de tresses.Le but de la thèse est d’étendre le résultat de Gambaudo et Ghys en termes d’un invariant algébrique associé à une tresse : la classe de Witt de sa forme de Blanchfield. Il est en effet possible de définir des invariants d'entrelacs en étudiant l'homologie des revêtements cycliques. Les groupes d’homologie et de cohomologie mis en jeu sont munis de structures de modules sur l’anneau du groupe Λ = Z[π].La forme de Blanchfield d’un entrelacs est ainsi la généralisation de la forme d’enlacement définie sur la partie de torsion du premier groupe d’homologie d’une variété fermée de dimension 3. Elle définit alors pour chaque tresse β une classe L(β) dans un groupe de Witt WT(Λ) .Théorème. Soit α et β deux tresses. On a l’égalité suivante, dans WT(Λ) : L(αβ) － L(α) － L(β) = －∂ Meyer(Burau(α), Burau(β)), où le cocycle de Meyer est maintenant défini sur le sous-groupe des éléments de GLn(Λ) préservant la forme de Squier, à valeurs dans le groupe de Witt W(Q(t)). On retrouve essentiellement le résultat original en spécifiant t = ω dans la formule précédente. / In 2004, Gambaudo and Ghys proved a formula establishing a connection between the ω-signatures of a link and the symplectic features of a representation of the braid group. Their main motivation was the construction on quasimorphisms on homeomorphism and diffeomorphism groups.The main goal of this thesis is to extend this result in terms of an algebraic invariant of a braids: the Witt class of the Blanchfield form. Some link invariants are defined through the cyclic covering spaces of their exterior. (Co)homology groups are then equipped with module structures over the ring Λ = Z[π]. For example, the Blanchfield form of a link is a generalisation of the linking form of a 3-manifold, which is a bilinear form on the torsionpart of its first homology group. In particular, every braid β defines a class L(β) in a Witt group WT(Λ) .Theorem. Let α and β be two braids. Then, in WT(Λ):L(αβ) － L(α) － L(β) = －∂ Meyer(Burau(α), Burau(β)),where the Meyer cocycle is defined on the sub group of GLn(Λ) whose elements preserve the Squier form.The result by Gambaudo and Ghys can essentially be recovered from this equality.

Dada uma superfície M, definiremos os grupos de tranças de M, denotado por \'B IND. n\' (M), geometricamente e usando a noção de espaços de confiuração. Mostraremos a equivalência das definições. Na mesma linha de raciocínio, definiremos os grupos de tranças puras de superfícies \'P IND. n\' (M). Apresentaremos as propriedades mais importantes dos grupos de tranças do plano e mostraremos que \'B IND. n\' (\'R POT. 2\') injeta em \'B IND. n\' (M), para muitas superfícies M. Mais detalhadamente, obteremos a apresentação de \'B IND. n\' (\'RP POT. 2\' ) e \'P IND. n\'(\'RP POT. 2\') / For a surface M, we define the braid groups of M, \'B IND. n\'(M), geometricaly and using the notion of configuration spaces. We show the equivalence of these definitions. In the sequence, we define the pure braid group of M, \'P IND. n\' (M). We present the most important properties of braid groups of the plane and we show that \'B IND. n\'\'(\'R POT. 2\') embedds in \'B IND. n\' (M), for almost all M. In a more detailed fashion, we present \'B IND. n\' (\'RP POT. 2\') and \'P IND. n\' (\'RP POT. 2)