PURPOSE: To demonstrate the feasibility of using a knowledge base of prior treatment plans to generate new prostate intensity modulated radiation therapy (IMRT) plans. Each new case would be matched against others in the knowledge base. Once the best match is identified, that clinically approved plan is used to generate the new plan. METHODS: A database of 100 prostate IMRT treatment plans was assembled into an information-theoretic system. An algorithm based on mutual information was implemented to identify similar patient cases by matching 2D beam's eye view projections of contours. Ten randomly selected query cases were each matched with the most similar case from the database of prior clinically approved plans. Treatment parameters from the matched case were used to develop new treatment plans. A comparison of the differences in the dose-volume histograms between the new and the original treatment plans were analyzed. RESULTS: On average, the new knowledge-based plan is capable of achieving very comparable planning target volume coverage as the original plan, to within 2% as evaluated for D98, D95, and D1. Similarly, the dose to the rectum and dose to the bladder are also comparable to the original plan. For the rectum, the mean and standard deviation of the dose percentage differences for D20, D30, and D50 are 1.8% +/- 8.5%, -2.5% +/- 13.9%, and -13.9% +/- 23.6%, respectively. For the bladder, the mean and standard deviation of the dose percentage differences for D20, D30, and D50 are -5.9% +/- 10.8%, -12.2% +/- 14.6%, and -24.9% +/- 21.2%, respectively. A negative percentage difference indicates that the new plan has greater dose sparing as compared to the original plan. CONCLUSIONS: The authors demonstrate a knowledge-based approach of using prior clinically approved treatment plans to generate clinically acceptable treatment plans of high quality. This semiautomated approach has the potential to improve the efficiency of the treatment planning process while ensuring that high quality plans are developed. / Dissertation

Neste trabalho estabelecemos o Teorema Central do Limite para o modelo O(N) de Heisenberg hierárquico na criticalidade via equação a derivadas parciais no limite N -> infinito. Por simplicidade consideramos apenas o caso d = 4, sendo o teorema também válido para d > 4. Pelo estudo de uma dada equação a derivadas parciais (EDP) determinamos a temperatura inversa crítica do modelo esférico hierárquico contínuo para um d > 2 qualquer, havendo conexão entre criticalidade e o ponto fixo da EDP. Por meio de uma análise geométrica da trajetória crítica obtemos informações sobre a dinâmica e distribuição dos zeros de Lee-Yang. / In this work we stablish the Central Limit Theorem for the hierarchical O(N) Heisenberg model at criticality via partial differential equation in the limit N -> infinity. For simplicity we only treat the d = 4 case but the theorem is still valid for d > 4. By studying a given partial differential equation (PDE) we determine for any d > 2 the critical inverse temperature of the continuum hierarchical spherical model, and we show a connection between criticality and the fixed point of PDE. By means of a geometric analysis of the critical trajectory we obtain some informations about Lee-Yang zeros´s dynamics and distribution.

The study of conformal symmetry is motivated through an example in statistical mechanics and then rigorously developed in quantum field theories in general spatial dimensions. In particular, primary fields are introduced as the fundamental objects of such theories and then studied in the formalism of radial quantization. The implications of conformal invariance on the functional form of correlation functions are studied in detail. Conformal blocks are defined and various approaches to their analytical and numerical calculation are presented with a special emphasis on the one-dimensional case. Building on these preliminaries, a modern formulation of the conformal bootstrap program and its various extensions are discussed. Examples are given in which bounds on the scaling dimensions in a one-dimensional theory are derived numerically. Using these results I motivate the technique of using the extremal functional bootstrap which I then develop in more detail. Many technical details are discussed and examples shown. After a brief discussion of conformal field theories with a boundary I apply numerical methods to find constraints on the spectrum of the 3D Ising model. Another application is presented in which I study the 4-point function on the boundary of a particular theory in Anti-de-Sitter space in order to approximate the mass spectrum of the theory. / O estudo da simetria conforme é motivado através de um exemplo em mecânica estatística e em seguida rigorosamente desenvolvido em teorias de campos quânticos em dimensões espaciais gerais. Em particular, os campos primários são introduzidos como os objetos fundamentais de tais teorias e então estudados através do formalismo de quantização radial. As implicações da invariância conforme na forma funcional das funções de correlação são estudadas em detalhe. Blocos conformes são definidos e várias abordagens para seu cálculo analítico e numérico são apresentadas com uma ênfase especial no caso unidimensional. Com base nessas preliminares, uma formulação moderna do programa de bootstrap conforme e suas várias extensões são discutidas. Exemplos são dados em que limites nas dimensões de escala em uma teoria unidimensional são derivados numericamente. Usando esses resultados, motivei a técnica de usar o bootstrap funcional extremo, que depois desenvolvo em mais detalhes. Diversos detalhes técnicos são discutidos e exemplos são apresentados. Após uma breve discussão das teorias de campo conformes com fronteiras, eu aplico métodos numéricos para encontrar restrições no espectro do modelo de Ising em 3D. Outra aplicação é apresentada em que eu estudo a função de 4 pontos na fronteira de uma teoria particular no espaço Anti-de-Sitter, a fim de aproximar o espectro de massa da teoria.

Auf sehr kleinen Längenskalen erlaubt die Weltflächenbeschreibung über zweidimensionale konforme Feldtheorien eine störungstheoretische Definition der String-Theorie. Viele strukturelle Eigenschaften und phänomenologische Implikationen der letzteren können mit Hilfe von D(irichlet)-Branen untersucht werden, die in der zugrunde liegenden Weltflächentheorie durch konforme Randbedingungen beschrieben werden. Etliche interessante Hintergründe für die String-Theorie erhält man über Gruppenmannigfaltigkeiten und Coset-Modelle. Neben wichtigen Beispielen wie SL(2,R), SU(2) und Gepner-Modellen, die für AdS- und Calabi-Yau-Kompaktifizierungen eine Rolle spielen, beinhalten sie außerdem weitere Beispiele wie den Nappi-Witten-Hintergrund oder den Raum T^11, die über eine asymmetrische Wirkung der Eichgruppe definiert sind und eine kosmologische Raumzeit mit Urknall- und Weltsturz-Singularitäten bzw. die Basis des Conifolds beschreiben. Die vorliegende Arbeit bietet eine umfassende, auf den exakten Methoden der konformen Feldtheorie beruhende Analyse von asymmetrischen Coset-Modellen. Wegen der heterotischen Natur der zugrundeliegenden Symmetriealgebra erlauben diese Modelle nur Randbedingungen, die einen Teil der Symmetrie brechen. Nach einer allgemeinen Erläuterung der Grundidee für die Konstruktion von symmetriebrechenden Randbedingungen richtet sich das Hauptaugenmerk auf WZNW- und asymmetrische Coset-Modelle, die das Fundament nahezu aller bekannten konformen Feldtheorien bilden. Mit Hilfe der erzielten Ergebnisse werden die Struktur sowie die Geometrie von D-Branen in den Gruppen SL(2,R) und SU(2), im Hintergrund AdS_3 x S^3, in der kosmologischen Nappi-Witten-Raumzeit und in T^pq-Räumen untersucht. Die Techniken, die in dieser Arbeit entwickelt werden, erlauben jedoch ebenso die Behandlung von Rändern und Kontaktstellen in (1+1)- oder 2-dimensionalen kritischen Systemen, die in der Festkörpertheorie oder der statistischen Physik auftreten. Insbesondere können Defektlinien beschrieben werden, die weder totale Reflexion noch völlige Transmission aufweisen. / At very small length scales, the world sheet approach in terms of two-dimensional conformal field theories provides a perturbative definition of string theory. Many structural properties and phenomenological implications of the latter can be explored using D(irichlet)-branes which may be identified with conformal boundary conditions in the underlying world sheet theory. Several interesting backgrounds in string theory arise from group manifolds and coset theories. Apart from prominent examples such as SL(2,R), SU(2) and Gepner models which play a role in AdS and Calabi-Yau compactifications, they also include further instances like the Nappi-Witten background or the space T^11 which are constructed using an asymmetric action of the gauge group and which describe a cosmological space-time with big-bang and big-crunch singularities and the base of the conifold, respectively. The present thesis provides a comprehensive analysis of asymmetric cosets based on the exact methods of boundary conformal field theory. Due to the heterotic nature of the underlying symmetry algebra, the models only allow for conformal boundary conditions which break parts of the bulk symmetry. The universal ideas for the construction of symmetry breaking boundary conditions are indicated and applied in detail to WZNW and asymmetric coset theories which provide the basic building blocks of almost all known conformal field theories. The general results are used to investigate the structure and shape of D-branes in the group manifolds SL(2,R) and SU(2), the background AdS_3 x S^3, the cosmological Nappi-Witten space-time and T^pq-spaces. The techniques developed in this thesis also allow for a treatment of boundaries and junctions in (1+1)- or 2-dimensional critical systems in condensed matter theory and statistical physics. In particular, they enable us to describe defect lines which go beyond full reflection or transmission.

Eine almost Einstein-Struktur (M,g,sigma) ist eine n-dimensionale zusammenhängende Mannigfaltigkeit M mit einer pseudo-riemannschen Metrik g und einer glatten Skalenfunktion sigma deren almost Einstein-Tensor A[g,sigma] (der spurfreie Anteil von Hess[g] sigma + sigma P[g], wobei P[g] den Schouten-Tensor bezeichnet) verschwindet. Sie verallgemeinert die Idee einer Einsteinmannigfaltigkeit in dem Sinne, dass die konform geänderte Metrik 1/sigma^2 g außerhalb der Nullstellenmenge Sigma = sigma^(-1)(0) eine Einstein-Metrik ist. Ziel dieser Doktorarbeit ist es, ein detailiertes Bild von Sigma in Lorentzsignatur (-+...+) zu erhalten. Teil dieser Arbeit ist zudem eine indexfreie Darstellung ausgewählter Resultate für konform kompaktifizierbare Einsteinmannigfaltigkeiten in Lorentzsignatur im Rahmen von almost Einstein-Strukturen. Diese Umformulierung wird dann benutzt, um eine Verallgemeinerung der konformen Wellengleichungen für beliebige gerade Dimensionen n = 2m > 4 vorzuschlagen. / An almost Einstein structure (M,g,sigma) is an n-dimensional connected manifold M equipped with a pseudo-Riemannian metric g and a scale factor sigma in C^infty(M) such that the almost Einstein tensor A[g,sigma] (the trace-free part of Hess[g] sigma + sigma P[g], with Schouten tensor P[g]) vanishes. It generalises the idea of an Einstein manifold in the way that 1/sigma^2 g is an Einstein metric away from the singularity set Sigma = sigma^(-1)(0). The purpose of this thesis is to get a detailed picture of Sigma in Lorentzian signature (-+...+). Part of this thesis is also an index-free survey of selected results on conformally compact Einstein manifolds in Lorentzian signature in the framework of almost Einstein structures. This reformulation is used to suggest a generalisation of the conformal wave equations to arbitrary even dimensions n = 2m > 4.

Diese Arbeit beschäftigt sich mit drei Aspekten der AdS/CFT-Korrespondenz, die alle einen Schritt über die klassische SUGRA-Näherung hinausgehen. Zuerst diskutieren wir den BMN Grenzfall der Korrespondenz und untersuchen insbesondere das Verhalten der quantenfeldtheoretischen Propagatoren. Dabei weisen wir nach, dass die Propagatoren im für den BMN Fall relevanten Hintergrund ebener Wellen semiklassisch (WKB) exakt beschrieben werden. Danach wird im Rahmen der AdS/CFT-Korrespondenz der Grenzfall verschwindender Kopplung der konformen Feldtheorie betrachtet. Zur technischen Vereinfachung geschieht dies für das Beispiel des O(N)-Vektormodells. Dabei wird die OPE der Vierpunktfunktionen so umgeschrieben, dass sie strukturelle Ähnlichkeit mit Witten-Diagrammen einer korrespondierenden Theorie von Strömen mit höherem Spin hat. Außerdem wird das O(N)- Vektormodell bei großem N am wechselwirkenden Infrarot-Fixpunkt untersucht. Im letzten Punkt wenden wir uns schließlich der ursprünglichen AdS/CFTDualität unter Mitnahme der nächstführenden Ordnung der 1/N-Entwicklung zu. Für die Deformationen der CFT durch relevante Doppelspur-Operatoren finden wir bei Zustandssummen und konformen Anomalien exakte Übereinstimmung zwischen direkter und AdS-seitiger indirekter Rechnung. Damit wird ein nicht trivialer Test der Korrespondenz über die SUGRA-Näherung hinaus erbracht. / This thesis deals with three corners of the AdS/CFT Correspondence that lie one step beyond the classical supergravity (SUGRA) approximation. We first explore the BMN limit of the duality and study, in particular, the behavior of field theoretic propagators in the corresponding Penrose limit. We unravel the semiclassical (WKB-) exactness of the propagators in the resulting plane wave background metric. Then, we address the limit of vanishing coupling of the conformal field theory (CFT) at large N. In the simplified scenario of Higher Spin/O(N) Vector Model duality, the conformal partial wave (CPW) expansion of scalar four-point functions are reorganized to make them suggestive of a bulk interpretation in term of a consistent truncated massless higher spin theory and their corresponding exchange Witten graphs. We also explore the connection to the interacting O(N) Vector Model at its infra-red fixed point, at leading large N. Finally, coming back to the gauge theory, we study the effect of a relevant double-trace deformations of the boundary CFT on the partition function and its dual bulk interpretation. We show how the one-loop computation in the Anti-de Sitter (AdS) space correctly reproduces the partition function and conformal anomaly of the boundary theory. In all, we get a clean test of the duality beyond the classical SUGRA approximation in the AdS bulk and at the corresponding next-to-leading 1/N order of the CFT at the conformal boundary.

We study three types of theories, using supersymmetry and ideas from string theory as tools to gain understanding of systems of more general interest. Firstly, we introduce non-relativistic Chern-Simons-matter field theories in three dimensions and study their anyonic spectrum in a conformal phase. These theories have supersymmetric completions, which in the non-relativistic case suffices to protect certain would-be BPS quantities from corrections. This allows us to compute one-loop exact anomalous dimensions of various bound states of non-Abelian anyons, analyse some interesting unitarity bound violations, and test some recently proposed bosonization dualities. Secondly, we turn on a chemical potential and break conformal invariance, putting the theory into the regime of the Fractional Quantum Hall Effect (FQHE). This is illustrated in detail: the theory supports would-be BPS vortices which model the electrons of the FQHE, and they form bag-like states with the appropriate filling fractions, Hall conductivities, and anyonic excitations. This formalism makes possible some novel explicit computations: an analytic calculation of the anyonic phases experienced by Abelian quasiholes; analytic relationships to the boundary Wess-Zumino-Witten model; and derivations of a wide class of QHE wavefunctions from a bulk field theory. We also further test the three-dimensional bosonization dualities in this new setting. Along the way, we accumulate new descriptions of the QHE. Finally, we turn away from flat space and investigate a problem in (3+1)-dimensional quantum gravity. We find that even as an effective theory, the theory has enough structure to suggest the inclusion of certain gravitational instantons in the path integral. An explicit computation in a minimally supersymmetric case illustrates the principles at work, and highlights the role of a hitherto unidentified scale in quantum gravity. It also is an interesting result in itself: a non-perturbative quantum instability of a flat supersymmetric Kaluza-Klein compactification.

Cette thèse est consacrée à l'étude des valeurs propres de l'opérateur de Laplace et de l'opérateur de Steklov sur des variétés riemanniennes. On cherche à donner des bornes optimales parmi l'ensemble des métriques, dans une classe conforme donnée ou non, et à caractériser, si elles existent, les métriques qui atteignent ces bornes. Ces métriques extrémales ont des propriétés qui s'inscrivent dans la théorie des surfaces minimales. On s'intéresse d'abord à la borne supérieure des valeurs propres de Laplace parmi des métriques conformes entre elles, appelées valeurs propres conformes. Dans le chapitre 1, on estime la deuxième valeur propre conforme de la sphère standard. Dans les chapitres 2 et 3, on montre que la première valeur propre conforme d'une variété riemannienne est plus grande que celle de la sphère standard de même dimension avec égalité seulement pour la sphère standard. Ensuite, on cherche à démontrer l'existence et la régularité de métriques qui maximisent les valeurs propres sur des surfaces, dans une classe conforme donnée ou non. Dans les chapitres 3 et 4, on démontre un résultat d'existence pour les valeurs propres de Laplace. Dans le chapitre 6, le travail est fait pour les valeurs propres de Steklov. Enfin, dans le chapitre 5, fruit d'un travail réalisé en collaboration avec Paul Laurain, on démontre un résultat de régularité et de quantification des applications harmoniques à bord libre sur une surface Riemannienne. C'est un élément clé pour le chapitre 6 / This thesis is devoted to the study of the Laplace eigenvalues and the Steklov eigenvalues on Riemannian manifolds. We look for optimal bounds among the set of metrics, lying in a conformal class or not. We also characterize, if they exist the metrics which reach these bounds. These extremal metrics have properties from the theory of minimal surfaces. First, we are interested in the upper bound of Laplace eigenvalues in a class of conformal metrics, called the conformal eigenvalues. In Chapter 1, we estimate the second conformal eigenvalue of the standard sphere. In Chapters 2 and 3, we prove that the first conformal eigenvalue of a Riemannian manifold is greater than the one of the standard sphere of same dimension, with equality only for the standard sphere. Then, we look for existence and regularity results for metrics which maximize eigenvalues on surfaces, in a given conformal class or not. In Chapters 3 and 4, we prove an existence result for Laplace eigenvalues. In Chapter 6, the work is done for Steklov eigenvalues. Finally, in Chapter 5, obtained in collaboration with Paul Laurain, we prove a regularity and quantification result for harmonic maps with free boundary on a Riemannian surface. It is a key component for Chapter 6

La réalisation expérimentale de la condensation de Bose-Einstein (BEC) a ouvert un nouveau champ d'investigation très fertile dans l'étude des atomes froids. En particulier, la possibilité de synthétiser des gaz de fermions piégés dans des réseaux optiques représente un développement de la plus haute importance pour la physique de la matière condensée. Ceci ouvre notamment sur la perspective d'étudier des phases quantiques exotiques stabilisées dans des systèmes d'électrons fortement corrélés.Récemment, les gaz atomiques d'alcalino-terreux ou d'ytterbium ont suscité un vif intérêt et ont été refroidis jusqu'à la dégénérescence quantique. La structure atomique particulière de ces systèmes leur confère de très hauts degrés de symétrie, grâce au découplage entre le spin nucléaire et le moment angulaire électronique. Une physique exotique conduisant à de multiple applications peut résulter de ces systèmes de hautes symétries qui ne peut être sondée que par les solides bases de la matière condensée.Dans cette thèse, on se propose d'étudier les propriétés physiques de basse énergie d'un gaz de fermions de type alcalino-terreux, piégé dans un réseau optique à une dimension. À une dimension, il est possible d'analyser les effets des interactions de manière non-perturbative par des approches de théorie des champs comme la bosonisation ou la théorie des champs conformes, et numériquement par le groupe de renormalisation de la matrice densité (DMRG). L'ensemble de ces outils sera notamment utilisé pour déterminer le diagramme de phase des gaz de fermions d'alcalino-terreux ou d'ytterbium à une dimension. / Experimental realization of Bose-Einstein condensate (BEC) opened a new and rich field of investigation for the study of the cold atoms. In particular, the possibility of creating trapped fermionic gases in optical lattices represent one of the most important development for the condensed matter physics. This open the outlook of studying exotic and stabilized quantum phases in strongly correlated systems of electrons.Recently, alkline-earth or ytterbuim atomic gases have given rise to great interest and have been cooled down up to quantum degenaracy. The specific atomic structure of these systems confer them very high degrees of symetry, thanks to the decoupling beetwin the nuclear spin and the electronic angular momentum. An exotic physics which is only probe thanks to the strong fundament of the condensed matter.In this thesis, we propose to study the physical properties at low energy of a alkaline-earth-like fermionic gas, trapped in a one dimensional optical lattice. In one dimension, we are able to analyse effects of interactions in a non-pertubative way with conformal field theory or bosonization, and numerically with Density Matrix Renormalization Group (DMRG) approach. All of these tools will be used to provide the phase diagram of these alkaline-earth-like fermionic gases in one dimension.

This work is focused on visualisation of Schwarzschild, Reissner- Nordström and Kerr black hole. The two-dimensional conformal diagram was constructed. In the case of Kerr black hole, the causal structure was visualized by intersection of chronological future of given point in spacetime with hyper- surfaces of constant value of Boyer-Lindquist coordinate t. Conformal diagram for Kerr black hole was constructed only in the neighbourhood of outer event horizon. Then the causal diagram, which is analogous to conformal diagram for Reissner-Nordström black hole was constructed. In all cases two-dimensional spa- celike hypersurfaces were chosen that were embedded into Euclidean space. The interpretation of time evolution of black hole universe was given to a sequence of such embedded hypersurfaces. In the case of Kerr black hole the embedding of outer ergosphere and outer event horizon were also constructed. 1