Rozprawy doktorskie na temat „APD topology”

Mémoire rédigé en vue de l'obtention de l'habilitation à diriger les recherches. Il donne un résumé de mon activité de recherche (anneaux de Chow, classes de Stiefel-Whitney, algèbres de Hopf, entrelacs, K-théorie de Milnor).

La integració de tots els components bàsics per a la generació, manipulació i detecció de llum en xips òptics està impulsant avenços científics i tecnològics, per exemple, en el desenvolupament de tecnologies de la informació o de dispositius de detecció per a les tecnologies quàntiques. Degut a la seva flexibilitat, escalabilitat i la possibilitat d’observar directament l’evolució de la funció d’ona utilitzant senzilles tècniques de tractament d’imatges, les estructures fotòniques integrades són una plataforma ideal per a la simulació quàntica, és a dir, per emular fenòmens quàntics que apareixen en altres branques de la física. A més, aquestes analogies òptiques-quàntiques també permeten dissenyar circuits fotònics integrats amb propietats excepcionals. En aquesta tesi aprofitem propietats no trivials de la física quàntica per dissenyar nous dispositius fotònics integrats amb funcionalitats avançades i rendiments millorats, així com nous simuladors fotònics. Específicament, explotem les similituds entre les equacions de Helmholtz i de Schrödinger, que permeten reproduir la dinàmica temporal d’una partícula atrapada en un potencial periòdic amb l’evolució espacial de la llum propagant-se en guies d’ona acoblades, per aplicar transformacions supersimètriques i processos adiabàtics així com explorar geometries topològiques no trivials en sistemes de guies d’ona òptiques acoblades. En aquesta línia, la primera part de la tesi està dedicada a introduir els conceptes físics i matemàtics que descriuen les guies d’ona òptiques acoblades, les analogies òptiques-quàntiques i la supersimetria en òptica. La segona part de la tesi engloba el disseny de nous dispositius fotònics integrats combinant l’aplicació de transformacions supersimètriques per manipular modes espacials amb tècniques de passatge adiabàtic per introduir la robustesa. Primer presentem un nou mètode per a la multiplexació de modes espacials basat en guies d’ona supersimetriques, que filtren els modes, en combinació amb la tècnica de passatge adiabàtic espacial que es fa servir per transmetre eficient i robustament els modes escollits entre guies. De manera similar, mantenint-nos en la idea d’aplicar protocols d’enginyeria quàntica per dissenyar nous dispositius fotònics amb rendiments millorats, proposem connectar de manera adiabàtica estructures supersimètriques al llarg de la distància de propagació. En particular, aquesta tècnica l’utilitzem per dissenyar guies d’ona còniques, filtres de modes, divisors de feixos i interferòmetres, eficients i robustos. Finalment, la tercera part de la tesi està dedicada a la simulació de diferents fenòmens quàntics utilitzant sistemes fotònics. Per començar aquesta part, explorem els efectes que les transformacions supersimètriques indueixen en sistemes amb propietats topologies no trivials, les quals estan intrínsecament lligades a les simetries internes del sistema. Amb aquest objectiu, considerem el sistema més simple amb propietats topològiques no trivials i demostrem en sistemes de guies d’ona acoblades com la protecció topològica d’un estat pot ser suspesa i restablerta utilitzant transformacions supersimètriques. A més, per accedir a aquestes fases topològiques no trivials, un element clau és la introducció de camps artificials gauge (AGF) que controlen la dinàmica de partícules no carregades que d’una altra manera eludeixen la influència dels camps electromagnètics estàndards. En aquesta línia, investiguem la possibilitat d’induir AGF utilitzant llum amb moment orbital angular en comptes de manipular la geometria del sistema. Específicament, mesurem l’efecte de gàbia d’Aharonov-Bohm que està lligat amb la presència d’un camp magnètic. Aquesta tècnica permet accedir a diferent règims topològics en una sola estructura, un pas important per a la simulació quàntica utilitzant sistemes fotònics.
La integración de todos los componentes básicos para la generación, manipulación y detección de luz en chips ópticos está impulsando avances científicos y tecnológicos, por ejemplo, en el desarrollo de tecnologías de la información o en los dispositivos de detección para las tecnologías cuánticas. Debido a su flexibilidad, escalabilidad y a la posibilidad de observar directamente la evolución de la función de onda utilizando senzillas técnicas de trata, las estructuras fotónicas son ideales para la simulación cuántica, es decir, para emular fenómenos cuánticos que aparecen en otras ramas de la física. Es más, estas analogías ópticas-cuánticas también permiten diseñar nuevos circuitos fotónicos integrados con propiedades excepcionales. En esta tesis, aprovechamos propiedades no triviales que emergen de la física cuántica para diseñar nuevos dispositivos fotónicos integrados con funcionalidades avanzadas y rendimientos mejorados, así como nuevos simuladores fotónicos. Específicamente, explotamos las similitudes entre las ecuaciones de Helmholtz y de Schrödinger, que permiten reproducir la dinámica temporal de una particula atrapada en un potencial periódico con la evolución espacial de la luz propagándose en guías de onda, para aplicar transformaciones supersimétricas y procesos adiabáticos así como explorar geometrías topológicas no triviales en sistemas de guías de onda ópticas acopladas. La primera parte de la tesis está dedicada a introducir los conceptos matemáticos y físicos que describen las guías de onda ópticas acopladas, las analogías ópticas-cuánticas y la supersimetria óptica. La segunda parte de la tesis engloba el diseño de nuevos dispositivos fotónicos integrados basados en combinar transformaciones supersimétricas para manipular los modos espaciales con las técnicas adiabáticas para introducir robustez. Primero presentamos un nuevo método para la multiplexación de modos espaciales basado en guías de onda supersimétricas, que filtran los modos, en combinación con la técnica de pasaje adiabático espacial que se usa para transmitir de manera eficiente y robusta los modos escogidos entre guías. De manera similar, manteniéndonos en la idea de aplicar protocolos de ingeniería cuántica para diseñar nuevos dispositivos fotónicos con rendimientos superiores, proponemos conectar de manera adiabática estructuras supersimétricas a lo largo de la propagación. En particular, ésta técnica la utilizamos para diseñar guías de onda cónicas, filtros modales, divisores de haz e interferómetros. Finalmente, la tercera parte de la tesis está dedicada a la simulación de diferentes fenómenos físicos utilizando sistemas fotónicos. Para empezar, exploramos los efectos que las transformaciones supersimétricas inducen en sistemas con propiedades topológicas no triviales, las cuales están intrínsecamente ligadas a las simetrías internas del sistema. Con este objetivo, consideramos el sistema más simple con propiedades topológicas no triviales y demostramos en un sistema de guías de onda acopladas cómo la protección topológica de un estado puede ser suspendida y restablecida utilizando transformaciones supersimétricas. Además, para acceder a las fases topológicas no triviales, un elemento clave es la introducción de campos artificiales de gauge (AGF) que controlan la dinámica de partículas no cargadas que de otra manera eluden la influencia de los campos electromagnéticos. Es esta línea, investigamos la posibilidad de inducir AGF utilizando luz con momento orbital angular en lugar de manipular la geometría del sistema. Específicamente, medimos el fenómeno de jaula de Aharonov-Bohm que está ligado a la presencia de un campo magnético sintético. Esta técnica permite acceder a diferentes regímenes topológicos en una sola estructura, un paso importante para la simulación cuántica utilizando sistemas fotónicos.
The integration of all the basic components for light generation, manipulation and detection in optical chips is boosting scientific and technological advances, for instance, in the development of information technology and data communications or of sensing devices for quantum technologies. Due to its flexibility, scalability and of the possibility of directly observing the wavefunction evolution using simple imaging techniques, integrated photonic structures are an ideal playground for quantum simulation i.e., for emulating quantum phenomena appearing in other branches of physics. Moreover, these quantum-optical analogies also allow to design novel integrated photonic circuits with exceptional properties. In this context, in this thesis we harness non-trivial properties stemming from quantum physics to design novel integrated photonic devices with advanced functionalities and enhanced performances as well as to engineer novel photonic simulators. Specifically, we exploit the similarities between the Helmholtz and the Schrödinger equations, which allow to mimic the temporal dynamics of a single particle trapped in a lattice potential with the spatial evolution of a light beam propagating in an array of optical waveguides, to apply supersymmetric (SUSY) transformations and adiabatic passage processes as well as to explore non-trivial topological geometries in systems of coupled optical waveguides. In this vein, the first part of the thesis is devoted to introduce the mathematical concepts and physical ideas behind coupled optical waveguides, quantum-optical analogies and optical SUSY. After that, the second part of the thesis encompasses the design of novel integrated photonic devices by combining the spatial modal content manipulation offered by SUSY transformations with the robustness supplied by adiabatic passage techniques. In this regard, we start by presenting a novel method for mode division (de)multiplexing rooted on SUSY waveguides, which provide the mode filtering capabilities, in combination with a Spatial Adiabatic Passage protocol, which is used to efficiently and robustly transfer the desired modes between waveguides. Similarly, keeping on the idea of applying quantum engineering protocols to design novel photonic devices with enhanced performances, we also propose to connect, in an adiabatic fashion, SUSY structures along the propagation direction. In particular, this technique is used to engineer efficient and robust tapered waveguides, mode filters, beam splitters and interferometers. Finally, the third part of the thesis is dedicated to the photonic simulation of different phenomena. We explore first the effect that SUSY transformations induce in systems with non-trivial topological properties, which are intrinsically connected with the system's internal symmetries. To this aim, we consider the simplest system with non-trivial topological properties and demonstrate in waveguide arrays how the topological protection of a targeted state can be suspended and reestablished by applying SUSY transformations. Moreover, to access these non-trivial topological phases, a key step is the introduction of Artificial Gauge Fields (AGF) controlling the dynamics of uncharged particles that otherwise elude the influence of standard electromagnetic fields. To this end, we investigate the possibility of inducing AGF by injecting light beams carrying Orbital Angular Momentum, rather than manipulating the geometry of the system. Specifically, we measure the Aharonov-Bohm caging effect, which is directly related with the presence of a synthetic magnetic flux, in an array of coupled optical waveguides. This technique paves the way towards accessing different topological regimes in one single structure, representing an important step forward for quantum simulation in photonic structures.

This PhD thesis is the result of our research on duality theory and completions for partially ordered sets. A first main aim of this dissertation is to propose different kind of topological dualities for some classes of partially ordered sets and a second aim is to try to use these dualities to obtain completions with nice properties. To this end, we intend to follow the line of the classical dualities for bounded distributive lattices due to Stone and Priestley. Thus, we will need to consider a notion of distributivity on partially ordered sets. Also we propose a topological duality for the class of all partially ordered sets and we use this duality to study some properties of partially ordered sets like its canonical extension, order-preserving maps and the extensions of n-ary maps that are order-preserving in each coordinate. Moreover, to attain these aims we will study the partially ordered sets from an algebraic point of view.
Esta tesis doctoral es el resultado de nuestra investigación sobre la teoría de la dualidad y completaciones de conjuntos parcialmente ordenados. Un primer objetivo general de este trabajo es proponer diferentes tipos de dualidades topológicas para algunas clases de conjuntos parcialmente ordenados y un segundo objetivo es tratar de utilizar estas dualidades para obtener diferentes completaciones con buenas propiedades. Para este fin, nos proponemos seguir la línea de las dualidades clásicas para retículos distributivos acotados debidas a Stone y a Priestley. Por lo tanto, necesitaremos considerar una noción de distributividad sobre conjuntos parcialmente ordenados. También proponemos una dualidad topológica para la clase de todos los conjuntos parcialmente ordenados y usamos esta dualidad para estudiar algunas propiedades de los conjuntos parcialmente ordenados como su extensión canónica, funciones que preservan orden y las extensiones de funciones n-arias que preservan orden en cada coordenada. Por otra parte, para alcanzar estos objetivos vamos a estudiar los conjuntos parcialmente ordenados desde un punto de vista algebraico.

In the last years, there has been done research in using topology as a new tool for studying data sets, typically high dimensional data. These studies have brought new methods for qualitative analysis, simplification, and visualization of high dimensional data sets. One good example, where these methods are useful, is in the study of microarray data (DNA data). To be able to use these methods, one needs to acquire knowledge of different topics in topology. In this paper we introduce simplicial homology, persistent homology, Mapper, and some simplicial complex constructions.

Today there is an immense production of data, and the need for better methods to analyze data is ever increasing. Topology has many features and good ideas which seem favourable in analyzing certain datasets where statistics is starting to have problems. For example, we see this in datasets originating from microarray experiments. However, topological methods cannot be directly applied on finite point sets coming from such data, or atleast it will not say anything interesting. So, we have to modify the data sets in some way such that we can work on them with the topological machinery. This way of applying topology may be viewed as a kind of discrete version of topology. In this thesis we present some ways to construct simplicial complexes from a finite point cloud, in an attempt to model the underlying space. Together with simplicial homology and persistent homology and barcodes, we obtain a tool to uncover topological features in finite point clouds. This theory is tested with a Java software package called JPlex, which is an implementation of these ideas. Lastly, a method called Mapper is covered. This is also a method for creating simplicial complexes from a finite point cloud. However, Mapper is mostly used to create low dimensional simplicial complexes that can be easily visualized, and structures are then detected this way. An implementation of the Mapper method is also tested on a self made data set.

This thesis is devoted to the study of the effect of Legendrian surgery on contact manifolds. In particular, we study the effect of this surgery on the Reeb dynamics of the contact manifold on which we perform such a surgery along Legendrian links. We obtain an exact sequence of cyclic Legendrian homology for the Legendrian links. Then we present the applications in 3-dimension and higher dimensions.
Doctorat en Sciences
info:eu-repo/semantics/nonPublished

Un réseau mobile ad hoc (Mobile Ad hoc Network, MANET) est un réseau sans fil, formé dynamiquement par un ensemble d'utilisateurs équipés de terminaux mobiles, sans l'utilisation d'une infrastructure préexistante, ou d'une administration centralisée. Les équipements utilisés dans les MANETs sont limités par la capacité de la batterie, la puissance de calcul et la bande passante. Les utilisateurs des MANETs sont libres de se déplacer, ce qui induit à des topologies dynamiques dans le temps. Toutes ces contraintes ajoutent plus de challenges aux protocoles et services de communications afin de fonctionner dans les MANETs. L'évolution des réseaux de 4ème génération (4G) est appelée à intégrer les MANETs avec les autres types de réseaux afin d'étendre leurs portées. Nous nous sommes intéressés dans la première partie de cette thèse à quelques challenges connus dans les MANETs en proposant des solutions novatrices utilisant des propriétés intéressantes des topologies de graphes. Dans un premier temps, nous avons effectué une étude sur la prédiction de la mobilité afin de maintenir une topologie d'ensemble dominant connecté dans les MANETs. Nous avons proposé dans un autre travail comment construire des topologies de graphes ayant des propriétés globales en se basant seulement sur des informations locales des nœuds mobiles. Ces topologies servent comme overlay aux MANETs. Nous avons proposé des algorithmes distribués pour construire des alliances offensives et défensives globales minimales. Nous avons aussi défini des heuristiques pour ces algorithmes afin de réduire les tailles des alliances obtenues. La première partie de cette thèse est achevée par la proposition d'un framework pour la conception et l'analyse des protocoles de contrôle de topologie dans les MANETs. Nous avons identifié les points communs des algorithmes de contrôle de topologie conçus pour les réseaux mobiles ad hoc et nous avons enrichi le simulateur NS-2 avec un ensemble d'extensions pour supporter le contrôle de topologie
We are interested in this thesis to graph-based approaches to deal with some challenges in networking, namely, graph topologies of mobile ad hoc networks (MANETs) and process model matchmaking in large scale web service. We propose in the first part: (1) a generic mechanism using mobility information of nodes to maintain a graph topology of the network. We show particularly, how to use the prediction of future emplacements of nodes to maintain a connected dominating set of a given MANET. (2) distributed algorithms to construct minimal global offensive alliance and global defensive alliance sets in MANETs. We also introduce several heuristics to get a better approximation of the cardinality of the alliance sets which is a desirable property for practical considerations. (3) a framework to facilitate the design and evaluation of topology control protocols in MANETs. We propose in the framework, a common schema for topology control based on NS-2 simulator and inspired from the commonalities between the components of the topology control algorithms in MANETs. In the second part, we focus on process model matchmaking. We propose two graph-based solutions for process model inexact matching to deal with high computational time of existing work in the literature. In the first solution, we decompose the process models into their possible execution sequences. After, we propose generic graph techniques using string comparator metrics for process model matchmaking based on this decomposition. In order to get better optimization of the execution time and to deal with process model matching in large scale web services, the second solution combines a spectral graph matching with structural and semantic proposed approaches. This solution uses an eigen-decomposition projection technique that makes the runtime faster

In order to utilise network resources efficiently, we need a strong knowledge of how the resources are shared and provisioned. However,this information is often unavailable due to the complexity of modern networks, the restrictive access to information describing their configurations and accuracy/reliability issues regarding information provisioning methods. Here, we propose the concept of functional topologies to deduce how resources are shared between different traffic flows. A functional topology describes the dependencies between traffic flows as a graph of interactions; this is in contrast to typical network graphs that model the physical connections between network components (routers and hosts). Unlike other work relying on in-network data, this topology is constructed solely at end hosts by measuring interdependencies of traffic flows via cross-correlation analysis. In order to measure the complete sets of interdependencies of traffic flows, different time intervals are used for sampling time series data. It is shown that these time intervals are related to maximum delays of traffic flows in network. The results of cross-correlation analysis are validated using well-known inverse participation ratio (IPR). As a part of the validation process, the results are analysed and compared with dominant/important flows of the network obtained by a new technique that uses eigen decomposition and spanning tree algorithm. The methodology of measuring interdependencies of traffic flows is validated and evaluated using real world data from a sensor network,as well as detailed simulation modelling different network topologies e.g. local area network. All the dependency measurements of traffic flow results are fed into a novel algorithm to construct functional topology of the network. Result shows that the algorithm constructs accurate functional topology of the network. Functional topology simplifies network topology by considering only nodes that create dependencies among traffic flows. With the help of this topology, end hosts can gain insight into resource provisioning of a network without requiring ISP assistance.

Dans cette thèse, nous étudions d'un point de vue théorique différents aspects de la matière topologique. Ces systèmes présentent des propriétés résistantes aux éventuelles perturbations grâce à une topologie non-triviale de leur structure de bandes. En particulier, des excitations exotiques, par exemple des fermions de Majorana, peuvent apparaitre à leurs bords.L'entropie d'intrication, ainsi que le spectre d'intrication ont été fondamentaux dans l'étude théorique de ces systèmes, et plus généralement des phases libres. Il est cependant difficile de les mesurer expérimentalement. L'étude des fluctuations de charge bipartites a été proposée afin de remédier à ce problème, et celles-ci permettent une mesure faible de l'intrication, en particulier pour des modèles unidimensionnels libres. Nous généralisons les précédents travaux sur les Liquides de Luttinger à des familles génériques de supraconducteurs et isolants topologiques en une et deux dimensions, systèmes dans lesquels la charge observée n'est plus conservée. Nous montrons que les transitions de phases topologiques sont caractérisées par certains coefficients universels dans les fluctuations et les fonctions de corrélations. Les systèmes bidimensionnels que nous étudions présentent des cônes de Dirac, et ces coefficients dépendent de leur enroulement. Cela nous permet de caractériser la topologie de ces points critiques. Dans tous les cas, les fluctuations suivent une loi de volume, qui a un comportement non-analytique aux transition de phase.Dans un second temps, nous nous intéressons aux systèmes en interactions. Nous montrons tout d'abord que certaines des signatures des transitions topologiques survivent en leur présence, dans les supraconducteurs topologiques. Nous étudions ensuite le diagramme de phase de deux fils supraconducteurs couplés par une interaction Coulombienne. Celle-ci mène à la création de phases exotiques grâce à la compétition avec la supraconductivité non-conventionnelle. Nous montrons en particulier l'apparition de phases de Mott brisant spontanément la symétrie de renversement du temps et présentant des courant orbitaux non-triviaux, ainsi que celle d'une phase de fermions libres, qui est l'extension de deux chaînes de Majorana critiques en interaction.Enfin, nous nous intéressons aux effets de la présence de fermions de Majorana sur le transport électronique. Nous étudions un îlot supraconducteur où plusieurs de ces fermions existent. Ce système pourrait être l'un des composants élémentaires d'un éventuel ordinateur quantique. Les fermions de Majorana changent les statistiques d'échange des porteurs de charges, ce qui se traduit par une fractionnalisation de la conductance. Celle-ci se révèle très robuste face aux anisotropies et autres perturbations. Nous étendons les études précédentes au cas où le nombre d'électrons dans la boîte peut fluctuer, et montrons l'équivalence de ce problème avec le modèle Kondo à plusieurs canaux. Nous réinterprétons alors ce modèle en terme du déplacement d'une particule dans un réseau fictif dissipatif
In this thesis, we study theoretically different aspects of topological systems. These models present resilient properties due to a non-trivial topology of their band structures, and in particular exotic edge excitations such as Majorana fermions.Entanglement entropy and entanglement spectrum have been fundamental to the study of these systems and of gapless systems in general, but are difficult to measure experimentally. Bipartite charge fluctuations were proposed as a weak measurement of this entanglement, in particular for one-dimensional gapless phases. We extend previous results on standard Luttinger Liquids to generic families of one- and two-dimensional non-interacting topological systems. Through exact computations, we show that their critical points are characterized by universal coefficients that reveal the topological aspect of the transitions. In two dimensions, the Dirac cones give quantized contributions to the fluctuations and various correlation functions. These contributions depend on their winding numbers, allowing for a precise determination of the topological structure of the gapless points. A volume law is also present and linked to the Quantum Fisher information, with characteristic non-analyticities at the phase transitions.In a second time, we include interactions and show that some of these signatures are preserved in topological superconductors even in their presence. Through analytical (bosonization, renormalization group) and numerical (exact diagonalization and DMRG) methods, we study the phase diagram of two Coulomb-coupled topological superconducting wires. We are interested in their behavior when the interactions are strong enough to break the topological protection: the interplay between unconventional superconductivity and interactions leads to exotic phases. We show the appearance of phases spontaneously breaking the time-reversal symmetry, with non-trivial orbital currents, and of an unusual gapless phase that is the extension of two critical interacting Majorana modes.Finally, we are interested in electronic transport mediated by Majorana fermions. We study a floating superconducting island carrying several such impurities. This device is thought to be a potential building block for a quantum computer. The Majorana fermions affect the statistics of the charge carriers, which leads to very resilient fractionalized transport. We extend previous studies to the charge degenerate case, where the total number of fermions in the island is not fixed, and map it to the well-known Multi-Channel Kondo model at large interaction. We reinterpret this standard model in terms of a particle moving in a highly dimensional, dissipative lattice

This thesis concerns the study of string topology, a relatively new branch of algebraic topology. We begin with a survey of the background of string topology. In particular, this includes a summary of the papers [4] by Chas and Sulivan, [15] by Jones and [5] by Cohen and Jones that provide the background for the original work of this thesis. We then proceed to give a new e cient technique to do systematic computations of the full structure of the string topology for a large family of manifolds. For this, we rst use the results of Jones [15] and Cohen and Jones [5] to reduce the problem to calculating Hochschild homology and cohomology. Secondly, we use the concept of models to compute Hochschild homology and cohomology and obtain some further Hochschild structure. Thus, most of this work is devoted to developing this technique for calculating Hochschild homology and cohomology via models. This research contributes to the area by providing the rst general and systematic method of computing the full structure of string topology. In addition, we give multiple, transparent examples of our new theory.

In this thesis connections between categories of interior algebras and categories of topological spaces, and generalizations of topological concepts to interior algebras, are investigated. The following are some of the most significant results we obtain: The establishment of a duality between topological spaces and complete atomic interior algebras formalized in terms of a category-theoretic co-equivalence between the category of topological spaces and continuous maps and the category of complete atomic interior algebras and maps known as complete topomorphisms (Theorem 2.1.7). Under this co-equivalence, continuous open maps correspond to complete homomorphisms (Theorem 2.1.8). We also establish a duality between arbitrary interior algebras and structures known as Stone fields in terms of a co-equivalence between the category of interior algebras and topomorphisms (see Definition 1.1.8) and the category of Stone fields and their morphisms the field maps (Theorem 2.2.14). Under this co-equivalence weakly open field maps (see Definition 2.2.17) correspond to homomorphisms (Theorem 2.2.18). The well-known connection between pre-ordered sets and interior algebras is shown to be a special case of topological duality (see section 4 of chapter 2). The topological concepts of neighbourhoods, convergence and accumulation are generalized to interior algebras (Chapter 3), and are used to generalize the topological separation and compactness properties to interior algebras (Chapter 4 and Chapter 5). What is particularly interesting with regard to the separation properties is that most of them are first order properties of interior algebras (see Theorem 4.5.11). This should be contrasted with the situation for frames/locales [12] and topological model theory [10]. By generalizing the concept of α-separation to interior algebras we obtain an ω chain of strictly elementary classes of interior algebras all of which have hereditarily undecidable first order theories (Theorem 4.3.14). Characterizations of irreducibility properties for interior algebras are also found. These properties (subdirect irreducibility, finite subdirect irreducibility, direct indecomposability, simplicity and semi-simplicity) can be characterized in many different ways. Characterizations in terms of open elements (fixed points of the interior operator) are found (Theorem 1.3.18 and Theorem 1.3.21) and these are used to obtain further characterizations. In particular a characterization in terms of topological properties of Stone spaces of interior algebras is obtained (Theorem 2.3.9). We also find characterizations of the irreducibility properties in the power set interior algebras of topological spaces (Theorem 2.1.15) and in interior algebras obtained from pre-ordered sets (Theorem 2.4.16). What is particularly striking is that the irreducibility properties correspond to very natural topological properties. (Other results characterizing or related to the irreducibility properties are 2.4.11, 2.4.17, 5.1.13, and 5.1.15). Bibliography: pages 134-135.

This thesis has four chapters. After a brief introduction in Chapter 1, the $AJ$-conjecture is introduced in Chapter 2. The $AJ$-conjecture for a knot $K \subset S^3$ relates the $A$-polynomial and the colored Jones polynomial of $K$. If $K$ satisfies the $AJ$-conjecture, sufficient conditions on $K$ are given for the $(r,2)$-cable knot $C$ to also satisfy the $AJ$-conjecture. If a reduced alternating diagram of $K$ has $\eta_+$ positive crossings and $\eta_-$ negative crossings, then $C$ will satisfy the $AJ$-conjecture when $(r+4\eta_-)(r-4\eta_+)>0$ and the conditions of Theorem 2.2.1 are satisfied. Chapter 3 is about quantum curves and their relation to the $AJ$ conjecture. The variables $l$ and $m$ of the $A$-polynomial are quantized to operators that act on holomorphic functions. Motivated by a heuristic definition of the Jones polynomial from quantum physics, an annihilator of the Chern-Simons section of the Chern-Simons line bundle is found. For torus knots, it is shown that the annihilator matches with that of the colored Jones polynomial. In Chapter 4, a tangle functor is defined using semicyclic representations of the quantum group $U_q(sl_2)$. The semicyclic representations are deformations of the standard representation used to define Kashaev's invariant for a knot $K$ in $S^3$. It is shown that at certain roots of unity the semicyclic tangle functor recovers Kashaev's invariant.

Master of Science
Department of Mathematics
David Auckly
This report introduces the fiber bundles. It includes the definitions of fiber bundles such as vector bundles and principal bundles, with some interesting examples. Reduction of the structure groups, and covering homotopy theorem and some specific computation using obstruction classes, Cech cohomology, Stiefel-Whitney classes, and first Chern classes are included.

The measurability of Mobile ad hoc network (MANET) is the precondition of itsmanagement, performance optimization and network resources re-allocations. However, MANET is an infrastructure-free, multi-hop, andself-organized temporary network, comprised of a group of mobile nodes with wirelesscommunication devices. Not only does its topology structure vary with time going by, butalso the communication protocol used in its network layer or data link layer is diverse andnon-standard.In order to solve the problem of interior links performance (such as packet loss rate anddelay) measurement in MANET, this thesis has adopted an external measurement basedon network tomography (NT). To the best of our knowledge, NT technique is adaptable for Ad Hoc networkmeasurement.This thesis has deeply studied MANET measurement technique based on NT. The maincontributions are:(1) An analysis technique on MANET topology dynamic characteristic based onmobility model was proposed. At first, an Ad Hoc network mobility model formalizationis described. Then a MANET topology snapshots capturing method was proposed to findand verify that MANET topology varies in steady and non-steady state in turnperiodically. At the same time, it was proved that it was practicable in theory to introduceNT technique into Ad Hoc network measurement. The fitness hypothesis verification wasadopted to obtain the rule of Ad Hoc network topology dynamic characteristic parameters,and the Markov stochastic process was adopted to analyze MANET topology dynamiccharacteristic. The simulation results show that the method above not only is valid andgenerable to be used for all mobility models in NS-2 Tool, but also could obtain thetopology state keeping experimental formula and topology state varying probabilityformula.IV(2) An analysis technique for MANET topology dynamic characteristic based onmeasurement sample was proposed. When the scenario file of mobile models could notbe obtained beforehand, End-to-End measurement was used in MANET to obtain thepath delay time. Then topology steady period of MANET is inferred by judging whetherpath delay dithering is close to zero. At the same time, the MANET topology wasidentified by using hierarchical clustering method based on measurement sample of pathperformance during topology steady period in order to support the link performanceinference. The simulation result verified that the method above could not only detect themeasurement window time of MANET effectively, but also identify the MANETtopology architecture during measurement window time correctly.(3) A MANET link performance inference algorithm based on linear analysis modelwas proposed. The relation of inequality between link and path performance, such as lossrate of MANET, was deduced according to a linear model. The phenomena thatcommunication characteristic of packets, such as delay and loss rate, is more similarwhen the sub-paths has longer shared links was proved in the document. When the rankof the routing matrix is equal to that of its augmentation matrix, the linear model wasused to describe the Ad Hoc network link performance inference method. The simulationresults show that the algorithm not only is effective, but also has short computing time.(4) A Link performance inference algorithm based on multi-objectives optimizationwas proposed. When the rank of the routing matrix is not equal to that of its augmentationmatrix, the link performance inference was changed into multi-objectives optimizationand genetic algorithm is used to infer link performance. The probability distribution oflink performance in certain time t was obtained by performing more measurements andstatistically analyzing the hypo-solutions. Through the simulation, it can be safelyconcluded that the internal link performance, such as, link loss ratio and link delay, can beinferred correctly when the rank of the routing matrix is not equal to that of itsaugmentation matrix.

Le but principal de la topologie algébrique dirigée est d’étudier des systèmes qui évoluent avec le temps à travers leur géométrie. Ce sujet émergea en informatique, plus particulièrement en vraie concurrence, où Pratt introduisit les automates de dimension supérieure (HDA) en 1991 (en réalité, l’idée de la géométrie de la concurrence peut être retracée jusque Dijkstra en 1965). Ces automates sont géométriques par nature: chaque ensemble de n processus exécutant des actions indépendantes en parallèle peuvent être modélisées par un cube de dimension n, et un tel automate donne naissance à un espace topologique, obtenu en recollant ces cubes. Cet espace a naturellement une direction du temps provenant du flot d’exécution. Il semble alors totalement naturel d’utiliser des outils provenant de la topologie algébrique pour étudier ces espaces: les chemins modélisent les exécutions et les homotopies de chemins, c’est-à-dire les déformations continues de chemins, modélisent l’équivalence entre exécutions modulo ordonnancement d’actions indépendantes, mais ces notions géométriques doivent préserver la direction du temps, d’une façon ou d’une autre. Ce caractère dirigé apporte des complications et la théorie doit être refaite, essentiellement depuis le début. Dans cette thèse, j’ai développé des théories de l’homotopie et de l’homologie pour ces espaces dirigés. Premièrement, ma théorie de l’homotopie dirigée est basée sur la notion de rétracts par déformations, c’est-à-dire de déformations continues d’un gros espaces sur un espace plus petit, suivant des chemins inessentiels, c’est-à-dire qui ne changent pas le type d’homotopie des « espaces d’exécutions ». Cette théorie est reliée aux catégories de composantes et catégories de dimension supérieures. Deuxièmement, ma théorie de l’homologie dirigée suit l’idée que l’on doit regarder les « espaces d’exécutions » et comment ceux-ci évoluent avec le temps. Cette évolution temporelle est traitée en définissant cette homologie comme un diagramme des « espaces d’exécutions » et en comparant de tels diagrammes en utilisant une notion de bisimulation. Cette théorie homologique a de très bonnes propriétés: elle est calculable sur des espaces simples, elle est un invariant de notre théorie homotopique, elle est invariante par des raffinements d’actions simples et elle une théorie des suites exactes
Studying a system that evolves with time through its geometry is the main purpose of directed algebraic topology. This topic emerged in computer science, more particularly in true concurrency, where Pratt introduced the higher dimensional automata (HDA) in 1991 (actually, the idea of geometry of concurrency can be tracked down Dijkstra in 1965). Those automata are geometric by nature: every set of n processes executing independent actions can be modeled by a n-cube, and such an automaton then gives rise to a topological space, obtained by glueing such cubes together. This space naturally has a specific direction of time coming from the execution flow. It then seems natural to use tools from algebraic topology to study those spaces: paths model executions, homotopies of paths, that is continuous deformations of paths, model equivalence of executions modulo scheduling of independent actions, and so on, but all those notions must preserve the direction. This brings many complications and the theory must be done again.In this thesis, we develop homotopy and homology theories for those spaces with a direction. First, my directed homotopy theory is based on deformation retracts, that is continuous deformation of a big space on a smaller space, following directed paths that are inessential, meaning that they do not change the homotopy type of spaces of executions. This theory is related to categories of components and higher categories. Secondly, my directed homology theory follows the idea that we must look at the spaces of executions and those evolves with time. This evolution of time is handled by defining such homology as a diagram of spaces of executions and comparing such diagrams using a notion of bisimulation. This homology theory has many nice properties: it is computable on simple spaces, it is an invariant of our homotopy theory, it is invariant under simple action refinements and it has a theory of exactness

A técnica de sinterização por plasma, também conhecida como processo SPS (Spark Plasma Sintering), é um processo para consolidação e sinterização de pós, em que corrente elétrica alternada pulsada e pressão de compactação são aplicadas simultaneamente aos componentes ferramentais (molde, punções, etc.). O molde utilizado neste processo tradicionalmente é cilíndrico, composto por grafite e permite a fabricação de amostras com geometria circular. Esse processo também possibilita a sinterização de um grande número de materiais, em especial, Materiais com Gradação Funcional (MGF). Tendo em vista os aspectos de geometria e composição da amostra, um projeto de otimização de moldes pode ser desenvolvido visando a fabricação de amostras com geometrias e gradação complexas. Com isso, é possível adequar a geometria do molde ao formato e composição da amostra que se deseja sinterizar, visando uma sinterização uniforme. Portanto, o objetivo deste trabalho é o desenvolvimento de uma metodologia para projetos de moldes utilizados na sinterização por plasma. Esta metodologia consiste na implementação de um algoritmo de otimização baseado no Método de Otimização Topológica (MOT), considerando três tipos de abordagem: a primeira abordagem, a qual visa a geometria da amostra, busca obter um molde prismático considerando amostras com geometria arbitrária, como por exemplo quadrada, triangular ou em cruz, com o objetivo de uniformizar o campo de temperaturas na amostra: na segunda abordagem, que considera moldes para a fabricação de amostras (MGF, os moldes podem ser projetados de modo a produzirem um gradiente de temperatura, na direção axial, através da variação da espessura da parede do molde; a terceira abordagem considera um molde constituído por material compósito. Nesta última abordagem é proposto um novo conceito de molde, onde se busca trabalhar não apenas com a geometria, como também com a microestrutura do molde dada por um material anisotrópico. Para a implementação do algoritmo de otimização, um modelo computacional baseado no Método dos Elementos Finitos (MEF), é desenvolvido considerando o processo SPS como um problema de acoplamento eletrotérmico. Na implementação do MOT utiliza-se um modelo de material baseado no SIMP (Solid Isotropic Material with Penalization) e Programação Linear Sequencial (PLS) para resolver o problema de otimização do molde. Todo algoritmo de otimização é implementado na linguagem própria do ambiente Matlab® e o pós-processamento, para verificação e validação dos resultados, é executado no software comercial Comsol®.
The Spark Plasma Sintering (SPS) technique is a powder consolidating and sintering process, in which pulsed DC electric current and pressure loads are applied simultaneously in the tool system components (graphite die, punchers, etc.) in order to perform the sintering process. Generally, a cylindrical graphite die is used for circular samples manufacturing and through this process the sinterization of a large number of materials, including Functionally Graded Materials (FGM), is possible. Considering the geometry and sample material aspects, an optimization die design technique can be developed based on the manufacturing of samples with complex geometry and gradation. Thus, it is possible to adjust the die geometry to the sample geometry or gradation in order to achieve a uniform sinterization. Therefore, the aim of this work is the development of a methodology to be applied in the design of dies used in SPS sintering process. This methodology consists of implementing an optimization algorithm based on the Topology Optimization Method (TOM), considering three approaches: in the first one a prismatic die is designed to process a sample with arbitrary geometry, for example square, triangular and cross sample; in the second approach the change of the die wall thickness is considered to achieve a predefined temperature gradient in the gradation direction of MGF samples and the third approach the same previous objective is considered, however the focus is the optimization of thermal conductive fibers. In the latest approach, a new die concept is proposed, where the objective is to optimize not only the die geometry but he microstructure considering a die composed by an anisotropic material. T implement the optimization algorithm a computational model based on the Finite Element Method (FEM) is developed considering the SPS process as an electrothermal coupled problem. In the TOM implementation a material model based on SIMP (Solid Isotropic Material with Penalization) is adopted and the Sequential Linear Programming is used to solve the optimization problem. The optimization algorithm is implemented using the Matlab® environment and the pos-processing, for verification and validation of the obtained results is carried out by using Comsol®.

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2013.
Cataloged from PDF version of thesis. "June 2013."
Includes bibliographical references (p. 51-52).
The thesis presents a technique for producing economical solutions for conventional structural steel frames using topology optimization on the lateral bracing scheme. The study focuses mainly on minimizing the amount of material used and optimizing the placement of bracing elements in typical steel frame structures to achieve economical and realistic solutions. Linear structural analysis is performed on steel frame structures while considering static gravity and wind loading. The optimization scheme uses a "multi-level design" approach with two distinct optimization loops. The optimal beam and column sizes in a structural steel frame system are generated in the first optimization loop and a bracing removal criterion is derived in the second loop to optimize the lateral bracing topology. A space constraint is imposed on the steel frame structure to enable designers to specify large empty spaces. A performance index is proposed to compare the cost between structural steel frames designed using conventional approaches, which rely on engineering experience and trial-and-error, and the approach specified in this study, which uses a multi-step optimization scheme. Two case studies are made, comparing steel frame structures designed using the proposed method with one designed using the traditional method. Keywords: Topology optimization, steel frame optimization, space-constrained optimization, multilevel optimization
by Zhen John Goo.
M.Eng.

Thesis: M. Eng., Massachusetts Institute of Technology, Department of Civil and Environmental Engineering, 2016.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 117-120).
With ever-increasing pollution and scarcity of resources, structural optimization, the science of finding the optimal structural arrangements under equilibrium constraints, is becoming an increasing necessity in engineering practice. However, designers are hesitant to adopt a method that is by nature a limit state and thus potentially unreliable. This thesis embeds a level of safety, namely redundancy, within the structural optimization process. Redundancy is the ability to remove a certain number of elements from the structure without losing stability. The thesis translates this constraint into a linear mathematical optimization problem. Then, a topology optimization algorithm is developed that identifies the least volume structure with the ability to remove any element(s) while maintaining stability under the initial loading. Besides the developed algorithm, this thesis shows the relation between the internal forces of redundant structures and their substructures, and in fact shows that it can be expressed linearly when only 1 level of redundancy is provided, and polynomial for higher levels. The algorithm is eventually implemented and extensively analyzed for a series of configurations, showing that redundant optimal shapes have considerably less volume than twice that of the pure volumetric optimal, and hence effectively combine safety with material efficiency. Overall, this thesis constitutes the early stage of a novel structural optimization algorithm that is unique to its volumetric optimization objectives.
by Dimitrios Pagonakis.
M. Eng.

Bibliography: leaves 99-106.
We introduce and investigate the concept of a nearness structure on a σ-frame. Analogues of the Samuel Compactification, Uniform Coreflection and Completion in the nearness σ-frame setting are obtained. Convergence in uniform frames is also a subject of this thesis integrating compactness, precompactness and paracompactness. Finally, the notion of uniform paracompactness is introduced and its relation with convergence is investigated.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2001.
Includes bibliographical references (p. 142-149).
The properties of biological polymers are controlled by two fundamental aspects: the energetic interactions, and the topological properties due to the fact that they are chains of connected monomers. In this thesis I study the interplay of topological and energetic properties in heteropolymers. We first study the problem of aggregation of polymers in solution. We find that aggregation is slowed by a transient topological force that exists on time scales shorter than the reptation time. We then study the translocation of a polymer through a membrane. Our key finding is that the dynamics of translocation occur on the same time scale as diffusion, and are affected only weakly by the entropic barrier created by the hole. Next we survey the energetics of random sequence heteropolymers, searching for "typical" properties of a sequence using lattice models. By enumerating the states and energies of compact 18, 27, and 36mers on a 3-d square lattice with an ensemble of random sequences, we test the validity of the self-averaging approximation. We find that fluctuations in the free energy between sequences are weak, and that self-averaging is a valid approximation at the length scale of real proteins. We then examine the validity of the Random Energy Model for these same lattice heteropolymers, through an analysis of the density of states. We see that the shape of the density of states is dependent on the sequence, and that this dependence is stronger for certain types of interactions between monomers.
(cont.) More generally, the validity of the random energy model depends strongly on the monomer-monomer interaction matrix. Finally, we consider a system in which the energy and topology of polymers can be quantified together more explicitly - a heteropolymer gel. We derive a theory for the affinity of heteropolymer gels for target molecules placed in solution with them, based on the the composition of the gel and solvent, and predict the composition of a gel which will switch from single-point to multi-point adsorption of target molecules as it undergoes its volume phase transition.
by Jeffrey Hsu-Min Chuang.
Ph.D.

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 53-55).
In this thesis, we look at some of the first topological results for Lagrangian immersions and embeddings. In particular, we state and consider some applications of the h-principle of Gromov which gives a homotopy classification of Lagrangian immersions. We outline a proof of Matsushima's theorem which states that there is no Lagrangian embedding ... We define the notions of the Malsov class and of monotone Lagrangian immersions or embeddings and we give some examples.
by Christian Hilaire.
M.Eng.

Thesis: S.M., Massachusetts Institute of Technology, Department of Department of Electrical Engineering and Computer Science, 2017.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 63-65).
A distributed computation in which nodes are connected by a partial communication graph is called topology-hiding if it does not reveal information about the graph beyond what is revealed by the output of the function. Previous results have shown that topology-hiding computation protocols exist for graphs of constant degree and logarithmic diameter in the number of nodes [Moran-Orlov-Richelson, TCC'15; Hirt et al., Crypto'16] as well as for other graph families, such as cycles, trees, and low circumference graphs [Akavia-Moran, Eurocrypt'17], but the feasibility question for general graphs was open. In this work we positively resolve the above open problem: we prove that topology-hiding computation is feasible for all graphs under the Decisional Diffie-Hellman assumption. Our techniques employ random or deterministic walks to generate paths covering the graph, upon which we apply the Akavia-Moran topology-hiding broadcast for chain-graphs (paths). To prevent topology information revealed by the random-walk, we design multiple graph-covering sequences that, together, are locally identical to receiving at each round a message from each neighbor and sending back a processed message from some neighbor (in a randomly permuted order).
by Rio LaVigne.
S.M.

Intermetallic borides are characterized by a great variety of crystal structures and bonding interactions, however, a comprehensive rationalisation of the electronic structure is missing. A more general interpretation will be targeted towards comparing several boride phases of one particular transition metal on one hand side, but also isostructural borides of various metals at the other side. Finally, a concise model should result from a detailed analysis of excellent data both from experimental charge density analysis and quantum chemical methods. Ultimate target is a transferability model based on typical building blocks. Experimental investigations of the electron density derived from diffraction data are very rare for intermetallic compounds. One of the main reasons is that the suitability of such compounds for charge density analysis is estimated to be relatively low as compared to organic compounds. In the present work, X-ray single crystal diffraction measurements up to high resolution were carried out for TiB2, VB2, V3B4, and VB crystals. The respective experimental electron densities were reconstructed using the multipole model introduced by Hansen and Coppens [1]. The topological aspects of the experimental electron density were analysed on the basis of the multipole parameters using Bader’s Quantum Theory, Atoms in Molecules [2] and compared with theoretical calculations. References [1] Hansen, N.K.; Coppens, P. Acta Crystallogr. 1978, A34, 909 [2] Bader, R.F.W. Atoms in Molecules─A Quantum Theory; Oxford University Press: Oxford, 1990

This thesis concerns structural optimization in conceptual design stages, for which constraints that are adapted to industrial requirements have been developed for topology optimization problems. The objective of the project has been to identify and solve problems that today prevent structural optimization from being used in a broader sense in the avionic industry; the main focus has been on stress and fatigue constraints in topology optimization. The thesis consists of two parts. The rst part gives an introduction to topology optimization and describes the developed methods for stress and fatigue constraints. In the second part, two papers are included, where the stress and fatigue constraints are evaluated, respectively. In the rst paper, a clustered approach is presented, where stress constraints are applied to stress clusters, rather than points on the structure. This allows for a trade-o between computational time and accuracy, as the number of clusters and thus constraints can be varied. Dierent approaches for how to sort stress evaluation points into clusters and how to update the clusters, such that the results are suciently accurate for conceptual designs, are developed and evaluated. The two-dimensional examples conrm the theoretical discussions and the designs that are obtained have managed to avoid large stress concentrations, even for problems with an initial stress singularity. Compared to the traditional stiness based designs, the stress constrained designs are considered to be closer to a nal design, which will decrease the total product development time. The second paper uses the methodology developed in the rst paper and applies it to high-cycle fatigue constraints. Using loads described by a variable load spectrum and material data from fatigue tests, the tensile principal stresses are constrained by a limit that is determined such that fatigue failure will not occur. In the examples, where the mass is minimized subjected to fatigue and static stress constraints, simple topologies are obtained and the structural parts are sized with respect to the critical fatigue stress and the yield limit. Stress concentrations are again avoided, for example by the creation of a radius around an internal corner. A comparison between static stress constraints based on the von Mises criterion and the highest tensile principal stresses is given and the examples clearly show the characteristics of the two formulations.

Les études et les résultats présentés dans ce manuscrit ont pour but de développer une meilleure compréhension des principes à la base de l'auto-organisation dans les systèmes biologiques. La théorie topologique des transitions de phase est l'un des approches possibles pour fournir une généralisation de la description des transitions de phase dans les systèmes petits ou mésoscopiques. Cette théorie a été rigoureusement enracinée dans deux théorèmes: un contre exemple à l'un de ces théorèmes a été récemment découvert. La première partie de ce manuscrit est donc consacré à mieux comprendre ce «contre-exemple » pour verifier si et comment la théorie peut être sauvé.Dans la deuxieme parte de ce manuscrit les résultats des recherches théoriques, numériques et expérimentales sur la condensation à la Fr "ohlich sont reportés. Ceci est une condition préalable à l'activation des oscillations dipolaires géantes qui entraînent des interactions électrodynamiques à long portée entre les molécules coresonnantes. Dans cette thèse, on montre que les interactions à longue portée affectent sensiblement les propriétés de diffusion des molécules en solution. Une empreinte des interactions à long portée pourrait être un phénomène de «transition» en ce qui concerne le coefficient de diffusion en fonction d'un paramètre de contrôle proportionnel à l’intensité d'interaction. Simulations analogues ont été réalisées afin de valider une approche expérimentale visant à trouver une telle «empreinte» dans les systèmes avec interactions à longue portée
The studies and results reported in this manuscript are aimed to develop a deeper understanding of the principles at the basis of self-organization in biological system.The Topological Theory of phase transitions is one of the possible approaches to provide a generalization of description of phase transitions in small or mesoscopic systems. This theory has been rigorously rooted in two theorems: a counterexample to one of these theorems has been recently found. The first part of this manuscript is devoted to investigation of the "counterexample" to understand if and how the theory can be saved. In the second part of this manuscript the results of theoretical, numerical and experimental investigations on Fr"ohlich-like condensation for normal modes of biomolecules are reported. This is a prerequisite for the activation of giant dipole oscillations in biomolecules which entail long-range electrodynamic interactions between coresonant molecules. In this thesis is shown that long-range interactions markedly affect the self-diffusion properties of molecules in solution. A fingerprint of long-range interactions could be a "transitional" phenomenon concerning the self-diffusion coefficient as a function of a control parameter proportional to interaction strength. Analogous simulations have been performed to validate an experimental approach aimed at finding such "fingerprint" in systems with built-in long-range interactions

Thesis: M. Eng., Massachusetts Institute of Technology, Department of Civil and Environmental Engineering, May, 2020
Cataloged from the official PDF of thesis.
Includes bibliographical references (pages 48-50).
Domes are an efficient way to span long distances and resist gravity loads. The two kinds of classical domes prevalent in architecture are continuous shells and grid shells. Continuous shells are monolithic concrete or masonry; grid shells reorient material in lattice members to create depth throughout the shell thickness. This thesis considers the design of a topology optimized hemisphere, a hemispherical continuous shell, and a hemispherical grid shell, and compares the structural performance of these three shells under asymmetric loads. First, a novel topology optimized dome is defined and generated with the objective to minimize strain energy. Then a numerical study is conducted on the three shells, continuous, grid, and optimized hemisphere, to investigate the different structural behavior of each material design scheme. This includes a linear elastic finite element analysis of each hemisphere's response under its own buckling load. Finally, plastic analyses are presented including the effects of large deformations and material yielding to determine the optimized hemisphere's response in comparison to the classical hemispheres. The proposed method uses topology optimization over the new domain of a thin shell lofted into space, as opposed to the more common planar and three-dimensional spaces for structural optimization. This thesis demonstrates good correlation of load capacity between Timoshenko's theoretical predictions and numerical analysis using Abaqus. The proposed topology optimized hemisphere has a seven-fold increase in load capacity under asymmetrical loading, when compared to a grid shell of the same volume.
by Grace Melcher.
M. Eng.
M.Eng. Massachusetts Institute of Technology, Department of Civil and Environmental Engineering

Thesis: M. Eng., Massachusetts Institute of Technology, Department of Civil and Environmental Engineering, 2019
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 32-33).
Topology optimization is a field extending to the built environment. Traditionally, optimization focuses mainly on monolithic structures but recently, developments have been made toward determining algorithms for multi-material optimization. A preexisting algorithm is modified to broaden the type of design possible with the method. The algorithm uses a three-phase design problem, a void phase and two other materials, and implements Heaviside Projection Method (HPM) and Rational Approximation of Material Properties (RAMP) method and employs the Method of Moving Asymptotes (MMA) as the gradient based optimizer. Three distinct object projection shapes are proposed, a horizontal, a vertical and a diagonal. The horizontal shaped inclusion enables designs such as, longitudinal reinforced concrete beam design of variable length bars. The vertical shaped inclusion enables designs of columns. The diagonal shaped inclusion allows for design of rebar within more slanted sections of optimized topology. The proposed algorithm is tested on two examples, the cantilever beam and the MBB beam, showing that it works as expected.
by Oluwanifemi O. Ajayi.
M. Eng.
M.Eng. Massachusetts Institute of Technology, Department of Civil and Environmental Engineering

It is said that a topologist is a mathematician who can not tell the difference between a doughnut and a coffee cup. The surfaces of the two objects, viewed as topological spaces, are homeomorphic to each other, which is to say that they are topologically equivalent. In this thesis, we acknowledge some of the most well-known examples of surfaces: the sphere, the torus, and the projective plane. We then observe that all surfaces are, in fact, homeomorphic to either the sphere, the torus, a connected sum of tori, a projective plane, or a connected sum of projective planes. Finally, we delve into algebraic topology to determine that the aforementioned surfaces are not homeomorphic to one another, and thus we can place each surface into exactly one of these equivalence classes.

Thesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2018.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 63-65).
Creating and updating road maps is currently an expensive and often manual process, and thus maps today are outdated or have poor coverage in large regions of the world. Automatically inferring the road network graph from aerial imagery provides a promising avenue to reducing the cost of maintaining road maps, but existing inference methods have poor precision. This thesis develops a novel iterative graph construction process for extracting graph structures from images, and applies this process to automatic road topology inference to significantly reduce error rates.
by Favyen Bastani.
S.M.

Optical singularities are points in complex scalar and vector fields where a property of the field becomes undefined (singular). In complex scalar fields these are phase singularities and in vector fields they are polarisation singularities. In the former the phase of the field is singular and in the latter it is the polarisation ellipse axes. In three dimensions these singularities are lines and natural light fields are threaded by these lines. The interference between three, four and five waves is investigated and inequalities are given which establish the topology of the singularity lines in fields composed of four plane waves. Beyond several waves, numerical simulations are used, supported by experiments, to establish that optical singularties in speckle fields have the fractal properties of a Brownian random walk. Approximately 73% of singularity lines percolate random optical fields, the remainder forming closed loops. The statistical results are found to be similar to those of vortices in random discrete lattice models of cosmic strings, implying that the statistics of singularities in random optical fields exhibit universal behavior. It is also established that a random superposition of plane-waves, such as optical speckle, form singularities which not only map out fractal lines, but create topological features within them. These topological features are rare and include vortex loops which are threaded by infinitely long lines and pairs of loops that form links. Such structures should be not only limited to optical fields but will be present in all systems that can be modeled as random wave superpositions such as those found in cosmic strings and Bose-Einstein condensates. Also reported are results from experiments that generated compact vortex knots and links in real Gaussian beams. These results were achieved through the use of algebraic knot theory and random search optimisation algorithms. Finally, polarisation singularity densities are measured experimentally which confirm analytic predictions.

Ginzburg-Landau theory is the phenomenological theory through which phase transitions are understood. It is within this framework that the first insights into superconductivity were obtained. More recently superconductivity in systems with no inversion symmetry has become a topic of interest. In such systems lack of inversion symmetry is believed to allow singlet and triplet order parameters to be admixed although the pairing mechanism in some cases is unclear. This thesis us.es Ginzburg-Landau theory to analyse the materials LaNiC2l LaNiGa2l Li2(PdxPtl -xhB and Mo2AhC. In the case of centrosymmetric LaNiGa2 a term coupling superconductivity to magnetism is introduced and shown to encourage the time-reversal symmetry (TRS) breaking non-unitary triplet state. In the case of Li2(PdxPtl-xhB an investigation of the nodal topologies arising from the Al irrep of the 0 group and corresponding specific heat signatures reveals a number of unique power law dependencies which can be used to detect topological transitions arising from triplet admixture in this system.

We analyse the perturbative expansion of knot invariants related with infinite dimensional representations of sl(2,R) and the Lorentz group taking as a starting point the Kontsevich Integral and the notion of central characters of infinite dimensional unitary representations of Lie Groups. The prime aim is to define C-valued knot invariants. This yields a family of C([h])-valued knot invariants contained in the Melvin-Morton expansion of the Coloured Jones Polynomial. It is verified that for some knots, namely torus knots, the power series obtained have a zero radius of convergence, and therefore we analyse the possibility of obtaining analytic functions of which these power series are asymptotic expansions by means of Borel re-summation. This process is complete for torus knots, and a partial answer is presented in the general case, which gives an upper bound on the growth of the coefficients of the Melvin-Morton expansion of the Coloured Jones Polynomial. In the Lorentz group case, this perturbative approach is proved to coincide with the algebraic and combinatorial approach for knot invariants defined out of the formal R-matrix and formal ribbon elements in the Quantum Lorentz Group, and its infinite dimensional unitary representations.

Since the metamaterials ethos of geometry over chemistry was first conceived at the end of the last century, a great deal of effort has been directed towards the conceptual, computational and experimental development of myriad effective electromagnetic media. Taking inspiration from quantum mechanics, here we exploit the possibility of independently controlling the individual elements of an effective polarizability matrix to reveal unique polarisation based phenomena. Firstly, by employing resonant “meta-atoms” to selectively absorb specific polarisation states of THz radiation, while tuning the polarisation conversion efficiency via near-field coupling, Parity Time symmetry breaking has been proposed, based on analytical and numerical modelling, and observed, using THz-Time Domain Spectroscopy, in polarisation space for the first time. We also reveal that anisotropic material as well radiative loss can be highly useful for tailoring the response of resonant metamaterials. Secondly, the possibility of achieving a topologically non-trivial phase within an effectively homogeneous photonic medium is discussed. Originating from the inherent spin-orbit interaction for light, three dimensional metamaterials with chirality and hyperbolicity are shown to be topologically non-trivial, resulting in one-way surface waves that are immune to back-scattering. Building on the effective medium calculations, our predictions are confirmed by numerical studies of realistic meta-structures.

>Magister Scientiae - MSc
This dissertation is a presentation to generalize boundedness and pseudocompactness in pointfree topology. We rst obtain and introduce a boundedness notion for elements of a frame. This is then further inspiration to introduce a de nition of bounded frame homomorphism whose domain may be any frame E, not just the frame of open sets of the reals.