Abstract Iris The power of quantum computing lies in its ability to perform certain calculations and solve complex problems exponentially faster than classical computers. This potential has profound implications for a wide range of fields, including particle physics. This thesis lays a fundamental foundation for understanding quantum computing. Particular emphasis is placed on the intricate process of quantum gate decomposition, an elementary lynchpin that underpins the development of quantum algorithms and plays a crucial role in this research. In particular, this concerns the implementation of quantum algorithms designed to simulate the dynamic evolution of multi-particle quantum systems - so-called Hamiltonian simulations. The concept of quantum gate decomposition is introduced and linked to quantum circuit optimisation. The decomposition of quantum gates plays a crucial role in fault-tolerant quantum computing in the sense that an optimal implementation of a quantum gate is essential to efficiently perform a quantum simulation, especially for near-term quantum computers. Part of this thesis aims to propose a new explicit tensorial notation of quantum computing. Two notations are commonly used in the literature. The first is the Dirac notation and the other standard formalism is based on the so-called computational basis. The main disadvantage of the latter is the exponential growth of vector and matrix dimensions and the fact that it hides some relevant quantum properties of the operations by increasing the apparent number of independent variables. A third possible notation is introduced here, which describes qubit states as tensors and quantum gates as multilinear or quasi-multilinear maps. Some advantages for the detection of separable and entangled systems and for measurement techniques are also shown. Finally, this thesis demonstrates the advantage of quantum computing in the description of multi-particle quantum systems by proposing a quantum algorithm to simulate collective neutrino oscillations. Collective flavour oscillations of neutrinos due to forward neutrino-neutrino scattering provide an intriguing many-body system for time evolution simulations on a quantum computer. These phenomena are of particular interest in extreme astrophysical settings such as core-collapse supernovae, neutron star mergers and the early universe. A detailed description of the physical phenomena and environments in which collective flavor oscillations occur is first reported, and the derivation of the Hamiltonian governing the evolution of flavor oscillations is detailed. The aim is to reproduce this evolution using a quantum algorithm. To manage the computational complexity, we use the Trotter approximation of the time evolution operator, which mitigates the exponential growth of circuit complexity. The quantum algorithm was designed to work on a trapped-ion based testbed (the theory of which is presented in detail). After machine-aware optimisation, the quantum circuit implementing the algorithm was run on the real quantum machine 'Quantinuum', and the results are presented and discussed.

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<p dir="ltr">A bath of free electrons interacting with a local quantum impurity leads to various exotic non-Fermi liquid behaviors, such as the non-integer effective ground state degeneracy of the impurity and the correction to the zero temperature conductance, which is temperature to the power of a fractional number. The former indicates emergent anyons, which are the key ingredients for achieving topological protected quantum computations. The latter can be used for experimentally probing non-Fermi liquid physics. It was recently proposed that a Coulomb blockaded M-Majorana island coupled to normal metal leads realizes a novel type of Kondo effect where the effective impurity “spin” transforms under the orthogonal group SO(M). Inspired by the multichannel generalization of the original Kondo model, we introduce a physically motivated N-channel generalization of this topological Kondo model whose impurity spin stems from the non-local topological ground state degeneracy of the island. This multichannel topological Kondo model supports Z3 parafermion and Fibonacci anyon (not supported by one-channel topological Kondo model) but may be limited to experiments because it is unstable to channel anisotropy. Therefore, we propose a Majorana-free meso- scopic setup which implements the Kondo effect of the symplectic Lie group and can harbor emergent anyons (including Majorana fermions, Fibonacci anyons, and Z3 parafermions) even in the absence of perfect channel symmetry. Besides, I comment on the future work such as the strong tunneling case that is beyond the topological Kondo regime and the two-impurity Kondo physics.</p>

The work presented in this thesis has two parts. The first part deals with methodological developments in the area of solid-state NMR, relevant to the study of partially ordered systems. Liquid crystals are best examples of such partially ordered systems and they are easily oriented by the magnetic field used for the NMR study. They provide spectra rich in information useful for the study of structure and dynamic s of the oriented molecule. Dipolar couplings and anisotropic chemical shifts are relatively easy to obtain for these systems. However, the methodologies used for extracting the required information are constantly undergoing change, with newer ideas being used for optimal use of the technique and increasing the sensitivity of the methodology. In this thesis, existing methods used for obtaining dipolar couplings from oriented liquid crystalline samples are examined in detail and conditions for optimal use of the methods are investigated. Different approaches for enhancing the sensitivity of the techniques are also proposed. Estimation of chemical shift anisotropy of carbons for a molecule that is used as a building block for several mesogens has been obtained and its utility for estimating the order parameters of the system have been examined. The second part of the thesis deals with the application of solid state NMR methods to the study of a number of novel liquid crystalline systems and for the estimation of dynamics, order and orientation of the mesogenic molecules in the magnetic field. Chapter-2 deals with a detailed and systematic study to improve the sensitivity of cross-polarization based separated local field (SLF) NMR spectroscopy techniques such as PISEMA(Polarization Inversion Spin Exchange at the Magic Angle) and PDLF(Pro-ton Detected Local Field). The chapter has been further divided into three sections. Section-A describes the optimization procedure for cross-polarization period for reducing zero-frequency peaks in SLF experiments. Polarization Inversion(PI) is one of the important components of PISEMA and plays a crucial role in enhancing the dipolar cross-peaks and suppressing the axial-peaks. Shortening this period has the advantage of less r.f. power input into the system, thus less susceptibility to sample heating. Therefore it is crucial to arrive at the optimum condition for which maximum sensitivity and resolution are obtained. A detailed experimental investigation of the role of the initial po1arization period has been carried out for two different samples of static oriented liquid crystalline material at two different temperatures and a contact time of 2ms has been found to be optimal for such samples. Insection-B of this chapter, the initial preparation period of the experiment is considered as a possible means of increasing the sensitivity of the experiment. Thus the use of cross-polarization via the dipolar bath by the use of a diabatic demagnetization in the rotating frame(ADRF-CP) has been proposed to be incorporated into PISEMA. To understand the CP dynamics, magnetization in double-and zero-quantum reservoirs of an ensemble of spin-1/2 nuclei and their role in determining the sensitivity the experiments have been theoretically examined. Experimentally, a modification incorporating ADRF-CP is shown to result in enhancement of signal-to-noise by as much as 90% in the case of rigid single crystals of a model peptide and up to 50% in non-rigid, partially ordered liquid crystalline systems. In section-C another useful SLF technique known as PDLF spectroscopy has also been examined. In this case a sweep of one of the r.f. amplitudes(RAMP-CP),rather than ADRF is found to work well. The reason for the different behaviours has been discussed. Chapter-3 highlights two experimental approaches used to extract the chemical shift anisotropy(CSA) tensor information from rotating solids. Section-A is devoted to the measurement of the CSA values of thiophene by using MAS side band analysis, by extracting the principal values from the intensities of just a few spinning side bands. Experiments have been performed on thiophene-2 carboxylic acid and thiophene-3 carboxylic acid samples and the carbon CSA values have been obtained. In section-B, CSA values of carbons of the core unit of the liquid crystal4- hexyloxybenzoic acid (HBA) have been obtained by using the recoupling pulse sequence SUPER(Separation of Undistorted Powder patterns by Effortless Recoupling).HBA belongs to an important class of thermotropic liquid crystals which are structurally simple and often used as starting materials for many novel mesogens. As this molecule could serve as an ideal model compound, high resolution13C NMR studies of HBA in solution, solid and liquid crystalline phases have been also undertaken. The CSA values obtained from the 2D SUPER experiment showed good agreement to those computed by DFT calculations. The CSA values were used for obtaining the order parameter of the system at different temperatures. These matched well the order parameter obtained from the 13C-1 H dipolar couplings in the nematic phase determined by SLF spectroscopy at various temperatures. A knowledge of the CSA of the carbons is thus very useful, as they can be used for gaining knowledge about the system from the chemical shifts obtained from a simple 1D spectrum. In chapter-4, 1-and 2-dimensional13CNMR techniques have been utilized to obtain extensive information about some novel mesogenic molecules. Four molecules of different structure and topology have been taken up for study. These molecules have the following features. Mesogen-1 has a terminal hydroxyl group. Such systems with further modification can result in mesogenic monomers for side chain liquid crystalline polymers. Mesogen-2 has a dimethyl amino group at one end and has three phenyl groups connected by appropriate linking units that form the core. In the third case, mesogen-3, the terminal hydroxyl group of mesogen-1 is replaced with a hydrogen such that13C-1 H dipolar couplings provide directly information on molecular ordering and orientation. In the fourth case, mesogen-4, the core is built with four phenyl rings. Here the fourth ring is linked to other three rings via a flexible chain unit. In each of these cases the 2DSLFNMR experiments have been carried out where13C-1 H dipolar couplings as well as13C chemical shifts were used for obtaining the order parameters of various segments of the molecule. The data provide useful insight into the phase behaviour, ordering and orientation of the molecules. Chapter-5 discusses the applications of the natural abundance 13CNMR techniques to thiophene based mesogens, that have the potential for use in molecular electronics material. Typically, these molecules consist of phenyl rings appropriately connected by linking units with thiophene. Different core units as well as different linking units to thiophene have been considered. The six mesogens thus obtained have been investigated in detail using 1D and 2D NMR methods.13C-1 H dipolar couplings have been used to obtain ordering information, that show interesting correlation to the molecular orientation and dynamics.

Ce mémoire porte sur l'étude d'un système de gestion d'énergie électrique dans un système multi-sources soumis à des perturbations exogènes. L'application visée est l'alimentation d'une propulsion hybride diesel/électrique équipée d'un système d'absorption des pulsations de couple. Les perturbations exogènes considérées peuvent être transitoires ou persistantes. Une perturbation transitoire correspond à une variation rapide du couple de charge, due par exemple à une accélération ou une décélération du véhicule. Une perturbation persistante provient du système de compensation des pulsations de couple générées par le moteur thermique. Le premier objectif du contrôle est de maintenir constante la tension du bus continu. Le deuxième objectif est d'absorber dans un système de stockage rapide constitué de super condensateur ces perturbations qui peuvent à terme provoquer une usure prématurée de la batterie. Le troisième objectif est de compenser l'auto-décharge dans le super condensateur en maintenant constante sa tension nominale. Les deux sources (batterie et super condensateur) sont reliées au bus continu par l'intermédiaire de deux convertisseurs boost DC/DC. La commande consiste à piloter les rapports cycliques de chaque convertisseur. C'est un système non linéaire où la commande est multiplicative de l'état. L'approche classique consistant à résoudre les équations Francis-Byrnes-Isidori ne s'applique pas directement dans ce cas où la sortie et la matrice d'interconnection dépendent de la commande. De plus, si cette approche est bien adaptée au rejet de perturbations persistantes, elle montre ces limites pour le rejet de perturbations non persistantes combiné à des objectifs de régulation. Notre approche a consisté à écrire le système sous un formalisme Port-Controlled Hamiltonian et à s'affranchir de la contrainte de la dépendance de la matrice d'interconnection avec la commande en utilisant la théorie des perturbations singulières. La commande du système dégénéré peut ensuite être calculée par une approche passive. Les performances de cette commande ont été testées en simulation et à l'aide d'un banc d'essai expérimental. Les résultats montrent l'efficacité du système d'absorption des différents types de perturbation tout en respectant les deux objectifs de régulation. / This thesis presents the research of energy management in a battery/ultracapacitor hybrid energy storage system with exogenous disturbance in hybrid electric vehicular application. Transient and harmonic persistent disturbances are the two kinds of disturbances considered in this thesis. The former is due to the transient load power demand during acceleration and deceleration, and the latter is introduced from the process of the internal combustion engine torque ripples compensation. Our control objective is to absorb the disturbances causing battery wear via the ultracapacitor, and meanwhile, to maintain a constant DC voltage and to compensate the self-discharge in the ultracapacitor to maintain it operating at the nominal state of charge. The object system is nonlinear due to the multiplicative relation between the input and the state. The traditional approach to solve Francis-Byrnes-Isidori equations cannot be directly applied in this case since the interconnect matrix depends on the control input. Besides, even if this approach is well suited to the rejection of persistent disturbances, it shows the limits for the case of non-persistent disturbances which is also our object. Our contributed control method is realized through a cascade control structure based on the singular perturbation theory. The ultracapacitor current with the fastest motion rate is controlled in the inner fast loop through which we impose the desired dynamic to the system. The reduced system controlled in the outer slow loop is a Hamiltonian system and the controller is designed via interconnection and damping assignment. Simulations and experiments have been carried out to evaluate the control performance. A contrast of the system responses with and without the control algorithm shows that, with the control algorithm, the ultracapacitor effectively absorbs the disturbances; and verifies the effectiveness of the control algorithm.

Durant ces dernières décennies, d'importants efforts et beaucoup de café ont été dépensés en vue de caractériser les instances faciles des problèmes NP-difficiles. Dans ce domaine de recherche, une approche s'avère être redoutablement efficace : la théorie de la complexité paramétrée introduite par Downey et Fellows dans les années 90.Dans cette théorie, la complexité d'un problème n'est plus mesurée uniquement en fonction de la taille de l'instance, mais aussi en fonction d'un paramètre .Dans cette boite à outils, la largeur arborescente est sans nul doute un des paramètres de graphe les plus étudiés.Ce paramètre mesure à quel point un graphe est proche de la structure topologique d'un arbre.La largeur arborescente a de nombreuses propriétés algorithmiques et structurelles.Néanmoins, malgré l'immense intérêt suscité par la largeur arborescente, seules les classes de graphes peu denses peuvent avoir une largeur arborescente bornée.Mais, de nombreux problèmes NP-difficiles s'avèrent faciles dans des classes de graphes denses.La plupart du temps, cela peut s'expliquer par l'aptitude de ces graphes à se décomposer récursivement en bipartitions de sommets $(A,B)$ où le voisinage entre $A$ et $B$ possède une structure simple.De nombreux paramètres -- appelés largeurs -- ont été introduits pour caractériser cette aptitude, les plus remarquables sont certainement la largeur de clique , la largeur de rang , la largeur booléenne et la largeur de couplage induit .Dans cette thèse, nous étudions les propriétés algorithmiques de ces largeurs.Nous proposons une méthode qui généralise et simplifie les outils développés pour la largeur arborescente et les problèmes admettant une contrainte d'acyclicité ou de connexité tel que Couverture Connexe , Dominant Connexe , Coupe Cycle , etc.Pour tous ces problèmes, nous obtenons des algorithmes s'exécutant en temps $2^{O(k)}\cdot n^{O(1)}$, $2^{O(k \log(k))}\cdot n^{O(1)}$, $2^{O(k^2)}\cdot n^{O(1)}$ et $n^{O(k)}$ avec $k$ étant, respectivement, la largeur de clique, la largeur de Q-rang, la larguer de rang et la largueur de couplage induit.On prouve aussi qu'il existe un algorithme pour Cycle Hamiltonien s'exécutant en temps $n^{O(k)}$ quand une décomposition de largeur de clique $k$ est donné en entrée.Finalement, nous prouvons qu'on peut compter en temps polynomial le nombre de transversaux minimaux d'hypergraphes $\beta$-acyclique ainsi que le nombre de dominants minimaux de graphes fortement triangulés.Tous ces résultats offrent des pistes prometteuses en vue d'une généralisation des largeurs et de leurs applications algorithmiques. / In the last decades, considerable efforts have been spent to characterize what makes NP-hard problems tractable. A successful approach in this line of research is the theory of parameterized complexity introduced by Downey and Fellows in the nineties.In this framework, the complexity of a problem is not measured only in terms of the input size, but also in terms of a parameter on the input.One of the most well-studied parameters is tree-width, a graph parameter which measures how close a graph is to the topological structure of a tree.It appears that tree-width has numerous structural properties and algorithmic applications.However, only sparse graph classes can have bounded tree-width.But, many NP-hard problems are tractable on dense graph classes.Most of the time, this tractability can be explained by the ability of these graphs to be recursively decomposable along vertex bipartitions $(A,B)$ where the adjacency between $A$ and $B$ is simple to describe.A lot of graph parameters -- called width measures -- have been defined to characterize this ability, the most remarkable ones are certainly clique-width, rank-width, and mim-width.In this thesis, we study the algorithmic properties of these width measures.We provide a framework that generalizes and simplifies the tools developed for tree-width and for problems with a constraint of acyclicity or connectivity such as Connected Vertex Cover, Connected Dominating Set, Feedback Vertex Set, etc.For all these problems, we obtain $2^{O(k)}\cdot n^{O(1)}$, $2^{O(k \log(k))}\cdot n^{O(1)}$, $2^{O(k^2)}\cdot n^{O(1)}$ and $n^{O(k)}$ time algorithms parameterized respectively by clique-width, Q-rank-width, rank-width and mim-width.We also prove that there exists an algorithm solving Hamiltonian Cycle in time $n^{O(k)}$, when a clique-width decomposition of width $k$ is given.Finally, we prove that we can count in polynomial time the minimal transversals of $\beta$-acyclic hypergraphs and the minimal dominating sets of strongly chordal graphs.All these results offer promising perspectives towards a generalization of width measures and their algorithmic applications.

Avant-propos: Cet ouvrage se base en partie sur le travail réalisé en collaboration avec Alexia Jolicoeur-Martineau, Ioannis Mitliagkas et Rémi Tachet des Combes, réalisé en 2020 et publié à la conférence internationale d'apprentissage de représentations (ICLR 2021). Les analyses présentées dans les prochaines pages approfondissent, corrigent et ajoutent à cet ouvrage de manière substantive, sans toutefois reposer sur cet ouvrage ou quelconque connaissance couverte par ce texte. / Ce mémoire a pour but de présenter des analyses pertinentes au sujet des méthodes génératives dites Denoising Score Matching dans le but de mieux comprendre leur fonctionnement et d'améliorer les techniques existantes. Ces méthodes consistent à graduellement réduire le bruit dans une image en usant de réseaux neuraux profonds à des fins de synthèse. Tandis que les premiers chapitres contextualisent le problème du Denoising Score Matching, les chapitres suivants s’affairent à reformuler l’objectif d’entraînement du réseau neuronal, puis à analyser le processus itératif générateur. J’introduis par la suite les concepts fondateurs des méthodes de Monte Carlo par chaînes de Markov (MCMC) pour dynamiques Hamiltoniennes, que j’adapte ensuite à la synthèse d’image par réduction graduelle de bruit. Tandis que les dynamiques de Langevin ont jusqu’alors eut monopole des processus génératifs dans la littérature de synthèse par le score, les dynamiques Hamiltoniennes font l'objet d’un engouement quant à leur vitesse de convergence supérieure. Je démontre leur efficacité dans les sections suivantes et précise, dans le cas de la génération d'images complexes, les contextes dans lesquels leur usage est avantageux. Lors d’une étude d’ablation complète, je présente les gains indépendants et jumelés des améliorations proposées, et par le fait même, je contribue à notre compréhension des modèles basés sur le score. / This thesis presents pertinent analysis around generative modeling of the Denoising Score Matching family with the goals of better understanding how they work and improving existing methods. These methods work by gradually reducing noise in images using deep neural networks. While the first chapters contextualize the problem of Denoising Score Matching, the following chapters focus on reformulating the training objective of the neural network and analysing the iterative generative process. I introduce the founding concepts of Markov Chain Monte Carlo (MCMC) for Hamiltonian Dynamics and adapt them to our framework of image synthesis by annealing of Gaussian noise. While Langevin Dynamics have thus far dominated generative processes in the Denoising Score Matching literature, Hamiltonian Dynamics sustained interest from their superior convergence rate. I demonstrate their efficiency in the next chapters and elaborate on the contexts in which their use is advantageous to complex image generation. In a complete ablation study, I present the independent and coupled gains from every proposed improvements and thereby elevate our comprehension of Denoising Score Matching methods.

Dans cette thèse, nous développons de nouveaux outils pour relier les dynamiques qualitatives des systèmes hamiltoniens sur des surfaces aux propriétés algèbriques de leurs complexes de Floer - un objet algébrique qui encode l'information sur la façon dont les orbites 1-périodiques d'un système sont reliées par des cylindres satisfaisant une équation différentielle partielle elliptique appelée l'équation de Floer. L'idée principale est de considérer --- pour un hamiltonian \(H \in C^\infty(S^1 \times \Sigma)\) sur une surface symplectique \((\Sigma, \omega)\) --- les graphes des orbites contractiles 1-périodiques de l'isotopie \((\phi^H_t)_{t \in [0,1]}\) comme définissant une tresse \(P^H\) dans \(S^1 \times \Sigma\). En choisissant des capuchons pour chacune de ces orbites 1-périodiques, nous obtenons un objet que nous appelons une tresse encapuchonnée \(\hat{P}^H\), qui est muni d'une fonction d'indexation \(\mu_{CZ}: \hat{P}^H \rightarrow \mathbb{Z}\) obtenue en assignant à chaque brin encapuchonné l'indice de Conley-Zehnder de l'orbite encapuchonnée associée. L'idée est alors de s'interroger sur la relation entre l'information topologique encodée dans la tresse encapuchonnée indexée \((\hat{P}^H,\mu_{CZ})\) et la structure du complexe de Floer \(CF_*(H,J)\) pour une structure presque complexe générique \(J\). À cette fin, nous aurons recours à: un nouvel invariant relatif pour les paires de tresses encapuchonnées que nous appelons le nombre d'enlacement homologique, un cercle d'idées concernant le comportement asymptotique des courbes pseudo-holomorphes développé par Hofer-Wysocki-Zehnder dans leur série d'articles [8], [10], [12] et aussi [11] (ainsi qu'un raffinement supplémentaire dans le cas relatif dû à Siefring dans [32]), et une nouvelle technique en basses dimensions pour la construction de morphismes de continuation de Floer qui ont un comportement prescrit. En conséquence de ces techniques, nous établissons l'existence --- pour des systèmes hamiltoniens génériques sur une surface fermée arbitraire --- de certaines feuilletages singulières spéciaux sur \(S^1 \times \Sigma\) dont le comportement est étroitement lié à la fois à la dynamique sous-jacente et à la structure du complexe de Floer du système. La construction de tels feuilletages dans le cas particulier des pseudo-rotations d'un disque, par des méthodes très différentes des nôtres, a été au coeur des progrès significatifs récents de Bramham dans [3] sur une célèbre question de Katok concernant les systèmes conservatifs de basse dimension et d'entropie nulle. Ces feuilletages fournissent également, pour les systèmes hamiltoniens lisses génériques, une construction Floer-théorique des feuilletages positivement transversaux sur \(\Sigma\) qui ont été construits originellement (pour les homéomorphismes de surface généraux) par Le Calvez à travers d'une extension substantielle de la théorie de Brouwer classique pour les homéomorphismes de surface dans [16]. En plus de fournir un pont géométrique entre la dynamique d'une isotopie hamiltonienne et l'information algébrique contenue dans son complexe de Floer, les techniques développées dans cette thèse permettent également de donner une caractérisation --- purement en termes de la dynamique de l'isotopie hamiltonienne sous-jacente --- des cycles de Floer dans \(CF_*(H,J)\) qui représentent la classe fondamentale de la surface et qui de plus se trouvent dans l'image d'un morphisme de PSS au niveau des chaines. Finalement, ces techniques permettent de définir une nouvelle famille d'invariants d'un système hamiltonien (sur une variété symplectique arbitraire) qui se comporte formellement de manière similaire à une famille bien étudiée de tels invariants connue comme les invariants spectraux de Oh-Schwarz. L'avantage de nos nouveaux invariants est que nous sommes capable de calculer explicitement les plus importants d'entre eux pour des systèmes hamiltoniens génériques sur des surfaces arbitraires, ce uniquement en termes de topologie relative des orbites périodiques du système (avec leurs indices de Conley-Zehnder). Ceci généralise un résultat de Humilière-Le Roux-Seyfaddini dans [13] dans lequel ils ont donné une caractérisation dynamique du principal invariant spectral de Oh-Schwarz dans le cas de systèmes hamiltoniens autonomes sur des surfaces de genre positif. / In this thesis, we develop novel tools for relating the qualitative dynamics of Hamiltonian systems on surfaces to the algebraic properties of their Floer complexes --- an algebraic object which encodes information about the ways in which a system’s 1-periodic orbits are connected by cylinders satisfying an elliptic partial differential equation known as Floer’s equation. The main idea is to consider --- for a generic Hamiltonian \(H \in C^\infty(S^1 \times \Sigma)\) on a symplectic surface \((\Sigma, \omega)\) --- the graphs of the contractible time-1 periodic orbits of the isotopy \((\phi^H_t)_{t \in [0,1]}\) as defining a braid \(P^H\) in \(S^1 \times \Sigma\). Upon choosing cappings for each such 1-periodic orbit, we obtain an object which we term a capped braid \(\hat{P}^H\), which comes equipped with an indexing function \(\mu_{CZ}: \hat{P}^H \rightarrow \mathbb{Z}\) given by assigning to each (capped) strand of the braid the Conley-Zehnder index of the associated capped orbit. The idea is then to enquire into the relation of the topological information encoded in the indexed capped braid \((\hat{P}^H,\mu_{CZ})\) and the structure of the Floer complex \(CF_*(H,J)\) for a generic \(J\). The main tools employed to this end are: a novel relative invariant for pairs of capped braids which we term the homological linking number, a circle of ideas about the asymptotic behaviour of pseudo-holomorphic curves pioneered by Hofer-Wysocki-Zehnder in their series of papers [8], [10], [12] as well as in [11] (along with a further refinement to the relative case by Siefring in [32]), and a novel technique for the construction of regular Floer continuation maps in low-dimensions having prescribed behaviour. As a consequence of these techniques, we establish the existence --- for generic Hamiltonian systems on an arbitrary closed surface \(\Sigma\) --- of certain special singular foliations on \(S^1 \times \Sigma\) whose behaviour is tightly related to both the underlying dynamics, as well as the structure of the system’s Floer complex. The construction of such foliations (by very different methods) in the particular case of pseudo-rotations on a disk was the crux of Bramham’s recent significant progress in [3] on a famous question due to Katok about low-dimensional conservative systems with vanishing entropy. These foliations also provide, for generic smooth Hamiltonian systems, 7 a Floer-theoretic construction of the positively transverse foliations on \(\Sigma\) which were originally constructed (for general surface homeomorphisms) by Le Calvez through a significant extension of classical Brouwer theory for surface homeomorphisms in [16]. In addition to providing a geometric bridge between the dynamics of a Hamiltonian isotopy and the algebraic information contained in its associated Floer complex, the techniques developed in this dissertation also permit a characterization --- purely in terms of the dynamics of the underlying Hamiltonian isotopy --- of those Floer cycles in \(CF_*(H,J)\) which represent the fundamental class of the surface, and which moreover lie in the image of some chain-level PSS map. Finally, these techniques permit the definition of a new family of invariants of a Hamiltonian system (on an arbitrary symplectic manifold) which behave formally similarly to a well-studied family of such invariants known as ‘Oh-Schwarz spectral invariants’ (and which agree with them in all known cases). The advantage of these novel spectral invariants is that we are able to explicitly compute the most important of these spectral invariants for generic Hamiltonian systems on arbitrary surfaces purely in terms of the relative topology of the system’s periodic orbits (together with their Conley-Zehnder indices). This considerably generalizes a result by Humilière-Le Roux-Seyfaddini in [13] in which they gave a dynamical characterization of the main Oh-Schwarz spectral invariant in the case of time-independent Hamiltonian systems on surfaces with positive genus.

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