京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20899号 / 理博第4351号 / 新制||理||1624(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)准教授 福間 將文, 教授 川合 光, 教授 青木 慎也 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

In this thesis, we provide a self contained introduction to the theory of random matrices and matrix models. Our analysis has a chronological order and it begins with the study of nuclear energy levels and ensemble averaging which yields the famous Wigner surmise. Then, the standard Gaussian theory of the orthogonal, unitary and symplectic ensemble is derived from symmetry arguments of the corresponding physical system, which is then followed by the explicit calculation of Wigner's semicircle distribution. Moreover, interactions are introduced to the free theory which leads to the topological expansion, a diagrammatic way of evaluating certain expectation values. Also, it is shown that there exists a duality between the resulting graphs and the quantization of a two dimensional surface through mapping as well as a method for solving a specific family of potentials. Finally, the numerical confirmation for some observables is carried out using the Hamburger moment problem, and the derivation of critical points for some theories.

Entre os grandes desafios da ecologia está a previsão e o entendimento das flutuações que ocorrem nas populações biológicas. Quando agregamos as intervenções humanas a esse entendimento, podemos, ao mesmo tempo em que fazemos previsões acerca do futuro das populações, entender melhor seus processos naturais de regulação e controle. Neste trabalho, realizamos um estudo da ecologia de duas espécies arbóreas tolerantes à sombra (Guapira opposita (Vell.) Reitz, pertencente ao dossel e Rudgea jasminoides (Cham.) Müll.Arg., pertencente ao sub-bosque) a fim de responder à questão de como a estrutura, a dinâmica populacional e o desempenho de plântulas respondem à heterogeneidade ambiental causada por distúrbios relacionados à fragmentação e à sucessão secundária. No Capítulo 1, testamos se a estrutura populacional, medida pela distribuição de tamanhos, está relacionada ao tamanho e/ou ao estádio sucessional do fragmento. A partir das conclusões geradas nesse primeiro estudo, no Capítulo 2 verificamos se parâmetros da dinâmica populacional, como a taxa assintótica de crescimento e as taxas vitais, explicam as variações encontradas na densidade e na estrutura. No Capítulo 3, testamos se o desempenho das plântulas destas espécies é afetado pelos fatores abertura do dossel e profundidade da serapilheira, os quais são reconhecidamente promotores da heterogeneidade ambiental em fragmentos florestais secundários. Para tanto, marcamos e acompanhamos por dois anos (de 2007 a 2009) populações das duas espécies em seis fragmentos de floresta secundária de distintos estádios sucessionais no Planalto Atlântico Paulista. Quanto à estrutura populacional, Guapira mostrou uma variação associada ao grau de estruturação da floresta, enquanto que para Rudgea o tamanho do fragmento foi a variável explicativa mais importante, o que nos levou a hipotetizar que Rudgea corre risco de extinção local nos fragmentos médios. Com relação às análises de dinâmica populacional, Rudgea apresentou um tempo de duplicação da população 4,5 vezes maior nos fragmentos médios do que nos grandes, o que nos leva a concluir que o risco de extinção local pode ser atenuado ou mesmo revertido. Para Guapira, as taxas não diferiram entre os estádios sucessionais, sendo previstas populações estáveis para ambos (955;8776;1). As análises do efeito da abertura do dossel e da profundidade da serapilheira no desempenho das plântulas demonstraram a importância da interação entre os fatores e a relevância de se considerar as diferentes fases ontogenéticas das plântulas. Através de uma abordagem inédita, demonstramos a importância do emprego de enfoques complementares (estrutura e dinâmica) para que se possa de fato entender o que ocorre em nível populacional. Além disso, destacamos a importância dos estudos de dinâmica para se elucidar os mecanismos demográficos atuantes em cada população. Esse conhecimento é uma ferramenta fundamental para o planejamento de ações direcionais de manejo e conservação. / One of the greatest challenges of ecology is to understand and predict the fluctuations in the biological populations. When we consider the human intervention in this understanding, we can, at the same time, make predictions about the future of the populations and understand better their natural processes of regulation and control. In this work we carried out a study about the ecology of populations of two shade tolerant tree species (Guapira opposita (Vell.) Reitz, that occupies the canopy and Rudgea jasminoides (Cham.) Müll.Arg., that occupies the understory) to answer the question of how the structure, population dynamics and seedling performance respond to environmental heterogeneity caused by disturbances related with fragmentation and secondary succession. In the first chapter we tested if the population structure, measured by size distributions, is related to the size and/or to the successional stage of the fragment. From the conclusions generated in this first study, in the second chapter we evaluated if population dynamics parameters, such as asymptotic growth rate and vital rates explain the variation in density and structure. In the third chapter we tested if the seedling performance of these species is affected by canopy openness and litter depth, which are known promoters of environmental heterogeneity in secondary forest fragments. In order to achieve this goal we marked and followed by two years (2007- 2009) populations of both species in six secondary forest fragments of different successional stages in the Southeast Atlantic Plateau. Regarding the population structure, Guapira showed variation associated to the degree of forest structuring, while for Rudgea the fragment size was the most important explanatory variable, what lead us to hypothesize that Rudgea is in risk of local extinction in the medium fragments. Regarding the analysis of population dynamics, Rudgea showed a population doubling time in the medium fragments 4.5 times greater than in the large ones, what lead us to conclude that this risk of local extinction can be mitigated or even reversed. For Guapira, the rates did not differ between the successional stages, being predicted stable populations in both (955;8776;1). The analysis of the effect of the canopy openness and the litter depth in the seedling performance showed the importance of the interaction between these factors and the relevance of considering the different seedling ontogenetic stages. Through a new approach, we demonstrated how important is to consider complementary studies (structure and dynamics) in order to really understand what happens at population level. Besides, we highlight the importance of studies with dynamics in order to elucidate the demographic mechanisms that occur in each population. This knowledge is a fundamental tool for planning more directional management and conservation actions.

[Truncated abstract] Most ecological research investigates the demography of organisms, which can be summarised by their population dynamics and structure. Population dynamics are the rates of birth, growth, reproduction and survival of individuals, which determine the number of individuals in different stage classes, or, the population structure. Understanding the demography of organisms is particularly difficult, and requires the application of different methods of investigation. A number of methods of investigation are required because each can only investigate particular aspects of population demography, and is subject to problems of inaccuracy and bias. Thus, population demography is best understood by simultaneously collecting complimentary data using different methods of analysis to quantify how and why the dynamics and structure of populations change with their environment. The need for complimentary methods of investigation to understand population demography is even greater for organisms that have complex life histories, such as corals. The life histories of corals are particularly complex because they display a variety of modes of sexual and asexual reproduction, and their rates of growth, reproduction and survival are strongly influenced by the size of the individual, which can change rapidly. Additionally, the rates of sexual recruitment to populations of corals are notoriously variable in space and time. Thus, corals are an ideal model to investigate the usefulness of combining methods of analysis to better understand the demography of organisms. In this thesis I demonstrate that combining data from a number of methods of analysis provides a much better understanding of the demography of populations of the mushroom coral Fungia fungites that were exposed to contrasting regimes of disturbance. I used methods of analysis that produce unique and complimentary results, in the form of genetic, size-structure, life history and experimental data. Genetic data confirmed that I was correctly identifying life history stages of polyps, indicated the extent to which a stock-recruitment relationship existed between the sexual recruits and the adult polyps at each population, and provided an estimate of the relative contribution of asexual recruitment to population maintenance. Changes in the size-structures of populations were quantified to determine the effects of different regimes of disturbance, and these changes in population structure were explained by quantifying the life history traits of polyps and their rates of transition through their life cycle. An experimental manipulation was conducted to specifically quantify the effects of sedimentation on the rates of survival and asexual recruitment of polyps, because sedimentation was an important disturbance whose effects were confounded by additional variables. Together, the set of complimentary data provided a good understanding of population demography in the context of varying levels of exposure to certain disturbances, and enabled the best possible predictions about the future of each population under a variety of conditions

Made available in DSpace on 2016-06-02T19:29:57Z (GMT). No. of bitstreams: 1 5173.pdf: 2537344 bytes, checksum: 189e9f5fe14dc2946f00d093ab36ee54 (MD5) Previous issue date: 2013-04-12 / Universidade Federal de Sao Carlos / The aims of this thesis were: to investigate the heterogeneity of soils and to evaluate the contribution of clonal growth to the population dynamics of woody species under contrasting soil conditions. In the Cerrado-Pantanal ecotone (15º43 S; 56º04 W), the soil possesses typical features from both the soil of Cerrado in the Central Plateau and the soil of the Pantanal Plain. Given the variability of soil and pluviosity, our hypothesis is that clonal growth is the demographic parameter that most contributes to the population growth rate () of five woody species in this ecotone area. This study was carried out in five Cerrado areas. We identified four soil categories; three of them typic Litoplinthic Petric Plintosol, typic Dystrophic Yellow Latosol and Dystrophic Yellow Latosol with plinthite had not yet been described for the study site. The chemical attributes aluminum saturation, magnesium, calcium, pH and manganese explained 38.7% of the variability of the soils in the study sites. The spatial distribution of the edaphic attributes was heterogeneous. These attributes differed between the soils in the Cerrado Pantanal ecotone and the soils of Cerrado in the Central Plateau and of the Pantanal Plain. The population dynamics of Curatella americana, a dominant species with wide geographical distribution, was evaluated in Petric Plintosol (P) and in Yellow Latosol (L), in the dry and rainy seasons. Sexual reproduction and seedling growth were higher in P, whereas clonal growth was higher in L. The population growth rate (λ) was higher in P. The adult survival rate exerted the strongest effect on λ for the two soil categories and the dry and rainy seasons. The interaction between soil category and pluviosity exerted the strongest effect on λ. We also analyzed the dynamics of species with small populations (Bowdichia virgilioides and Roupala montana) and large populations (Curatella americana and Caryocar brasiliense), to determine which demographic parameters characterize woody clonal species with different sizes. For the species with small populations, the rate of sexual reproduction and the density of all life stages were smaller, whereas the clonal growth was higher. Small populations were more susceptible to variations in soil and pluviosity. The rates that most contributed to the λ of Bowdichia virgilioides and Roupala montana in L and P were: respectively, the survival of young ramets and immatures, in the dry season; and for both species, the survival of adults, in the rainy season. For C. americana and C.brasiliense, the survival of adults was the parameter that most contributed to λ, independently of soil category and season. The study indicated an heterogeneity of the superficial layer of the soil and of the soil categories that occur in this area. Soils with contrasting attributes exerted an important effect on the dynamics of woody clonal species. The survival of young and immature individuals, originated from clonal growth, was the rate that most contributed to the λ of the small-sized populations of woody clonal species B. virgilioides and R. montana , whereas the survival of adults was the rate that most contributed to the λ of the large-sized populations C. americana and C. brasiliense. / Os objetivos desta tese foram: investigar a heterogeneidade de solos e verificar qual a contribuição do crescimento clonal para a dinâmica de populações de espécies lenhosas sob condições edáficas contrastantes. Em Cerrado, zona de ecótono com o Pantanal (15º43 S; 56º04 W), o solo é constituído por características inerentes tanto ao solo de Cerrado do Planalto Central como da Planície do Pantanal. Diante da variabilidade de características edáficas e da pluviosidade, a hipótese é que o crescimento clonal apresente-se como o parâmetro demográfico que mais contribua para a taxa de crescimento populacional () de cinco espécies lenhosas nesta zona de ecótono. O estudo foi realizado em cinco remanescentes de Cerrado. Identificou-se quatro classes de solo, sendo que três destas Plintossolo Pétrico Litoplíntico típico, Latossolo Amarelo distrófico típico e Latossolo Amarelo distrófico com plintita ainda não haviam sido descritas para a região em estudo. Saturação por alumínio, magnésio, cálcio, pH e manganês explicaram 38,7% da variabilidade de solos. Houve heterogeneidade na distribuição espacial dos atributos edáficos. Tais atributos diferiram entre os remanescentes e entre solos de Cerrado do Planalto e da Planície do Pantanal. A dinâmica populacional de Curatella americana, espécie dominante e de ampla distribuição geográfica, foi avaliada em Plintossolo Pétrico (FF) e em Latossolo Amarelo (LAd), nas estações seca e chuvosa. A reprodução sexuada e o crescimento de plântulas foram maiores em FF, enquanto o crescimento clonal foi maior em LAd. A taxa de sobrevivência de adultos exerceu maior efeito sobre para as duas classes de solo e estações seca e chuvosa. A interação entre classe de solo e pluviosidade exerceu maior efeito sobre Também foi avaliada a dinâmica das populações de Bowdichia virgilioides e Roupala montana, Curatella americana e Caryocar brasiliense, para determinar quais parâmetros demográficos caracterizam espécies lenhosas clonais com diferentes tamanhos populacionais. A taxa de reprodução sexuada e a densidade de todos os estádios foram menores, enquanto o crescimento clonal foi maior para pequenas populações de espécies lenhosas. Pequenas populações foram mais suscetíveis às variações de solo e de pluviosidade. As taxas que mais contribuíram para de B. virgilioides e R. montana em LAd e em FF foram a sobrevivência de rametas jovens; na estação seca foi a sobrevivência de rametas jovens e de imaturos para as respectivas espécies e na estação chuvosa foi a sobrevivência de adultos para ambas as espécies. Para C. americana e C.brasiliense, a sobrevivência de adultos foi o parâmetro que mais contribuiu para independente da classe de solo e da estação do ano. O estudo comprovou a heterogeneidade da camada superficial do solo e das classes de solo que constituem esta área. Solos de características contrastantes exercem importante efeito sobre a dinâmica de espécies lenhosas clonais. A sobrevivência de indivíduos jovens e de imaturos, provenientes do crescimento clonal, foi a taxa que mais contribuiu para de populações pequenas de lenhosas clonais B. virgilioides e R. montana - ao passo que a sobrevivência de adultos contribuiu para de populações grandes C. americana e C.brasiliense.

Entre os grandes desafios da ecologia está a previsão e o entendimento das flutuações que ocorrem nas populações biológicas. Quando agregamos as intervenções humanas a esse entendimento, podemos, ao mesmo tempo em que fazemos previsões acerca do futuro das populações, entender melhor seus processos naturais de regulação e controle. Neste trabalho, realizamos um estudo da ecologia de duas espécies arbóreas tolerantes à sombra (Guapira opposita (Vell.) Reitz, pertencente ao dossel e Rudgea jasminoides (Cham.) Müll.Arg., pertencente ao sub-bosque) a fim de responder à questão de como a estrutura, a dinâmica populacional e o desempenho de plântulas respondem à heterogeneidade ambiental causada por distúrbios relacionados à fragmentação e à sucessão secundária. No Capítulo 1, testamos se a estrutura populacional, medida pela distribuição de tamanhos, está relacionada ao tamanho e/ou ao estádio sucessional do fragmento. A partir das conclusões geradas nesse primeiro estudo, no Capítulo 2 verificamos se parâmetros da dinâmica populacional, como a taxa assintótica de crescimento e as taxas vitais, explicam as variações encontradas na densidade e na estrutura. No Capítulo 3, testamos se o desempenho das plântulas destas espécies é afetado pelos fatores abertura do dossel e profundidade da serapilheira, os quais são reconhecidamente promotores da heterogeneidade ambiental em fragmentos florestais secundários. Para tanto, marcamos e acompanhamos por dois anos (de 2007 a 2009) populações das duas espécies em seis fragmentos de floresta secundária de distintos estádios sucessionais no Planalto Atlântico Paulista. Quanto à estrutura populacional, Guapira mostrou uma variação associada ao grau de estruturação da floresta, enquanto que para Rudgea o tamanho do fragmento foi a variável explicativa mais importante, o que nos levou a hipotetizar que Rudgea corre risco de extinção local nos fragmentos médios. Com relação às análises de dinâmica populacional, Rudgea apresentou um tempo de duplicação da população 4,5 vezes maior nos fragmentos médios do que nos grandes, o que nos leva a concluir que o risco de extinção local pode ser atenuado ou mesmo revertido. Para Guapira, as taxas não diferiram entre os estádios sucessionais, sendo previstas populações estáveis para ambos (955;8776;1). As análises do efeito da abertura do dossel e da profundidade da serapilheira no desempenho das plântulas demonstraram a importância da interação entre os fatores e a relevância de se considerar as diferentes fases ontogenéticas das plântulas. Através de uma abordagem inédita, demonstramos a importância do emprego de enfoques complementares (estrutura e dinâmica) para que se possa de fato entender o que ocorre em nível populacional. Além disso, destacamos a importância dos estudos de dinâmica para se elucidar os mecanismos demográficos atuantes em cada população. Esse conhecimento é uma ferramenta fundamental para o planejamento de ações direcionais de manejo e conservação. / One of the greatest challenges of ecology is to understand and predict the fluctuations in the biological populations. When we consider the human intervention in this understanding, we can, at the same time, make predictions about the future of the populations and understand better their natural processes of regulation and control. In this work we carried out a study about the ecology of populations of two shade tolerant tree species (Guapira opposita (Vell.) Reitz, that occupies the canopy and Rudgea jasminoides (Cham.) Müll.Arg., that occupies the understory) to answer the question of how the structure, population dynamics and seedling performance respond to environmental heterogeneity caused by disturbances related with fragmentation and secondary succession. In the first chapter we tested if the population structure, measured by size distributions, is related to the size and/or to the successional stage of the fragment. From the conclusions generated in this first study, in the second chapter we evaluated if population dynamics parameters, such as asymptotic growth rate and vital rates explain the variation in density and structure. In the third chapter we tested if the seedling performance of these species is affected by canopy openness and litter depth, which are known promoters of environmental heterogeneity in secondary forest fragments. In order to achieve this goal we marked and followed by two years (2007- 2009) populations of both species in six secondary forest fragments of different successional stages in the Southeast Atlantic Plateau. Regarding the population structure, Guapira showed variation associated to the degree of forest structuring, while for Rudgea the fragment size was the most important explanatory variable, what lead us to hypothesize that Rudgea is in risk of local extinction in the medium fragments. Regarding the analysis of population dynamics, Rudgea showed a population doubling time in the medium fragments 4.5 times greater than in the large ones, what lead us to conclude that this risk of local extinction can be mitigated or even reversed. For Guapira, the rates did not differ between the successional stages, being predicted stable populations in both (955;8776;1). The analysis of the effect of the canopy openness and the litter depth in the seedling performance showed the importance of the interaction between these factors and the relevance of considering the different seedling ontogenetic stages. Through a new approach, we demonstrated how important is to consider complementary studies (structure and dynamics) in order to really understand what happens at population level. Besides, we highlight the importance of studies with dynamics in order to elucidate the demographic mechanisms that occur in each population. This knowledge is a fundamental tool for planning more directional management and conservation actions.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O extrativismo de Produtos Florestais não Madeireiros - PFNMs surge com a prerrogativa do uso dos produtos da biodiversidade com a manutenção da floresta em pé. No entanto, a coleta desses produtos pode afetar, em diferentes níveis, as taxas vitais dos indivíduos da espécie explorada e comprometer os parâmetros demográficos e genéticos das populações. Para garantir que o extrativismo seja uma atividade sustentável e gere desenvolvimento socioeconômico para os povos da floresta, é imprescindível agregar ao conhecimento tradicional informações sobre a ecologia das espécies e técnicas de manejo adequadas. A andiroba (Carapa guianesis Aublet.) é uma árvore de uso múltiplo, fornecendo um dos óleos medicinais mais utilizados na Amazônia e madeira de alta qualidade. Apesar do crescente interesse de mercado, informações sobre a biologia da espécie, aspectos econômicos e ecológicos da coleta das sementes para a extração do óleo e o processamento deste ainda são limitados. Nesse sentido, este estudo teve por objetivo avaliar o potencial produtivo de sementes e a dinâmica populacional de C. guianesis em uma população natural, no sul do Estado de Roraima. Em uma parcela de nove hectares, todos os indivíduos de C. guianensis com DAP &#8805; 10 cm foram classificados em estádios de desenvolvimento e tiveram a produção de sementes estimada durante um intervalo de seis anos. Indivíduos com DAP < 10 cm, denominados regenerantes, foram monitorados em 27 subparcelas de 25 x 25m (1,7 ha). Através de modelos matriciais foram avaliados a estrutura da população, estimada a taxa finita de crescimento populacional (&#955;) e identificado o parâmetro demográfico prioritário para a persistência populacional. Para propor recomendações de manejo, simulações de diferentes regimes de exploração da espécie foram realizadas. O diâmetro do caule das andirobeiras pode ser utilizado para identificar indivíduos produtivos, sendo as árvores com diâmetro acima de 35 cm as mais produtivas. A população estudada encontra-se estável e, se mantidas as condições atuais, não apresenta risco de declínio populacional. O parâmetro demográfico prioritário para a manutenção da população de C. guianensis foi a sobrevivência dos adultos II (DAP &#8805; 35 cm). Em simulações de manejo a remoção ou incremento de regenerantes, ou a retirada de adultos I (10 < DAP < 35 cm), não afetou a taxa de crescimento populacional. No entanto, a remoção de apenas dois adultos II, isto é, interferência inferior a 3% na sobrevivência dos indivíduos com maior potencial reprodutivo da população, a taxa de crescimento populacional decaiu (&#955; < 1). Assim, mantendo-se as condições atuais, a manutenção da taxa de crescimento da população estudada é compatível com o extrativismo de sementes. O manejo de uso múltiplo da espécie (madeira e sementes) deve evitar o corte de árvore nas classes mais produtivas (DAP &#8805; 35 cm) para não comprometer o equilíbrio populacional. / The harvest of non-timber forest products is a way of boosting income for poor people in the tropics, and encouraging forest conservation. However, harvesting of such products may changes ecological processes of the exploited species at different levels. These changes could affect vital rates of individuals and demographic and genetic parameters of the populations. To ensure that the harvesting is a sustainable activity and generate socio-economic development for the people of the forest, it is essential to add to traditional knowledge information about the ecology of the species and appropriate management techniques. The Andiroba (Carapa guianesis Aublet.) is a multipurpose tree, providing one of the most widely used medicinal oils in the Amazon, and high-quality wood. Despite markets growing interest, information on the species biology, economic, and ecological aspects of the seed harvesting and oil processing remain limited. This study aimed to evaluate seed production and population dynamics of C. guianesis in a natural stand in southern Roraima. In a nine-hectare plot, all individuals of C. guianensis with DBH &#8805; 10 cm were classified into developmental stages and tree individual seed production was estimated during a period of six years. Individuals with DBH <10 cm, named recruits, were monitored in 27 subplots of 25 x 25 m (1.7 ha). Matrix models were used to (i) evaluate population structure, (ii) estimate finite rate of increase (&#955;) and (iii) identify priority demographic parameter for population persistence. We used simulations to test different kinds of harvesting intensity to suggest management recommendations. Tree diameter can be used to identify productive individuals; trees with a diameter above 35 cm are the most productive. According to our simulations, the studied population is stable and, if current conditions remain unchanged, there is no risk of population decline. The main population parameter for the maintenance of C. guianensis population was the survival of adult II (DBH&#8805; 35 cm). Simulations of harvesting intensity, regeneration increase, or withdrawal of adults I (10 <DBH <35 cm), did not affect the rate of population growth. However, removal of only two adult II (or less than a 3% interference in the survival of individuals with higher reproductive potential of the population) results in a decrease of population growth rate (&#955; <1). Thus, maintaining current conditions, the growth rate of the population studied is consistent with seeds harvesting. Multiple uses of this species (wood and seeds) should avoid cutting trees in the most productive classes (DBH &#8805; 35 cm) to avoid compromising population growth.

Three separate studies of the effect of plant pathogen on population biology of its host are presented in the thesis. Two are using field data about a widespread system of plant Falcaria vulgaris and its systemic rust fungus Puccinia sii-falcariae. The first study shows, that the disease prevalences in 40 populations of the plant were correlated with the moisture, the soil reaction and the cover of the herb layer at the localities. This was probably a result of the interaction of the life history of the plant and different effect the disease has at various localities. Similar pattern was found in the second study that aimed to determine long-term effect of the disease at the population level at four chosen localities over 4-5 years. Population growth rates were only rarely predicted to be higher for the healthy part of the population when compared with the whole population. Other analyses have however found big differences among years and localities. The locality type (slope vs. field populations) was important factor influencing population stage composition and importance of life cycle transitions for the growth of the population. Finally, the last study explores the possibility that systemic infection in clonal plants might be able to select against clonality. The result of the modelling showed that more...

Diese Arbeit befasst sich mit Quantenfeldtheorien auf nicht-kommutativen Räumen. Solche Modelle treten im Zusammenhang mit der Stringtheorie und mit der Quantengravitation auf. Ihre nicht-störungstheoretische Behandlung ist üblicherweise schwierig. Hier untersuchen wir jedoch drei nicht-kommutative Quantenfeldtheorien nicht-perturbativ, indem wir die Wirkungsfunktionale in eine äquivalente Matrixformulierung übersetzen. In der Matrixdarstellung kann die jeweilige Theorie dann numerisch behandelt werden. Als erstes betrachten wir ein regularisiertes skalares Modell auf der nicht-kommutativen Ebene und untersuchen den Kontinuumslimes bei festgehaltener Nicht-Kommutativität. Dies wird auch als Doppelskalierungslimes bezeichnet. Insbesondere untersuchen wir das Verhalten der gestreiften Phase. Wir finden keinerlei Hinweise auf die Existenz dieser Phase im Doppelskalierungslimes. Im Anschluss daran betrachten wir eine vier-dimensionale U(1) Eichtheorie. Hierbei sind zwei der räumlichen Richtungen nicht-kommutativ. Wir untersuchen sowohl die Phasenstruktur als auch den Doppelskalierungslimes. Es stellt sich heraus, dass neben den Phasen starker und schwacher Kopplung eine weitere Phase existiert, die gebrochene Phase. Dann bestätigen wir die Existenz eines endlichen Doppelskalierungslimes, und damit die Renormierbarkeit der Theorie. Weiterhin untersuchen wir die Dispersionsrelation des Photons. In der Phase mit schwacher Kopplung stimmen unsere Ergebnisse mit störungstheoretischen Berechnungen überein, die eine Infrarot-Instabilität vorhersagen. Andererseits finden wir in der gebrochenen Phase die Dispersionsrelation, die einem masselosen Teilchen entspricht. Als dritte Theorie betrachten wir ein einfaches, in seiner Kontinuumsform supersymmetrisches Modell, welches auf der "Fuzzy Sphere" formuliert wird. Hier wechselwirken neutrale skalare Bosonen mit Majorana-Fermionen. Wir untersuchen die Phasenstruktur dieses Modells, wobei wir drei unterschiedliche Phasen finden. / This work deals with three quantum field theories on spaces with noncommuting position operators. Noncommutative models occur in the study of string theories and quantum gravity. They usually elude treatment beyond the perturbative level. Due to the technique of dimensional reduction, however, we are able to investigate these theories nonperturbatively. This entails translating the action functionals into a matrix language, which is suitable for numerical simulations. First we explore a scalar model on a noncommutative plane. We investigate the continuum limit at fixed noncommutativity, which is known as the double scaling limit. Here we focus especially on the fate of the striped phase, a phase peculiar to the noncommutative version of the regularized scalar model. We find no evidence for its existence in the double scaling limit. Next we examine the U(1) gauge theory on a four-dimensional spacetime, where two spatial directions are noncommutative. We examine the phase structure and find a new phase with a spontaneously broken translation symmetry. In addition we demonstrate the existence of a finite double scaling limit which confirms the renormalizability of the theory. Furthermore we investigate the dispersion relation of the photon. In the weak coupling phase our results are consistent with an infrared instability predicted by perturbation theory. If the translational symmetry is broken, however, we find a dispersion relation corresponding to a massless particle. Finally, we investigate a supersymmetric theory on the fuzzy sphere, which features scalar neutral bosons and Majorana fermions. The supersymmetry is exact in the limit of infinitely large matrices. We investigate the phase structure of the model and find three distinct phases. Summarizing, we study noncommutative field theories beyond perturbation theory. Moreover, we simulate a supersymmetric theory on the fuzzy sphere, which might provide an alternative to attempted lattice formulations.

Dans cette thèse, j’étudie trois sujets reliés : les modèles de tenseurs, les nombres de Hurwitz et les polynômes de Macdonald-Koornwinder. Les modèles de tenseurs généralisent les modèles de matrices en tant qu’une approche à la gravité quantique en dimension arbitraire (les modèles de matrices donnent une version bidimensionnelle). J’étudie un modèle particulier qui s’appelle le modèle quartique mélonique. Sa spécialité est qu’il s’écrit en termes d’un modèle de matrices qui est lui-même aussi intéressant. En utilisant les outils bien établis, je calcule les deux premiers ordres de leur 1=N expansion. Parmi plusieurs interprétations, les nombres de Hurwitz comptent le nombre de revêtements ramifiés de surfaces de Riemann. Ils sont connectés avec de nombreux sujets en mathématiques contemporaines telles que les modèles de matrices, les équations intégrables et les espaces de modules. Ma contribution principale est une formule explicite pour les nombres doubles avec 3-cycles complétées d’une part. Cette formule me permet de prouver plusieurs propriétés intéressantes de ces nombres. Le dernier sujet de mon étude est les polynôme de Macdonald et Koornwinder, plus précisément les identités de Littlewood. Ces polynômes forment les bases importantes de l’algèbre des polynômes symétriques. Un des problèmes intrinsèques dans la théorie des fonctions symétriques est la décomposition d’un polynôme symétrique dans la base de Macdonald. La décomposition obtenue (notamment si les coefficients sont raisonnablement explicites et compacts) est nommée une identité de Littlewood. Dans cette thèse, j’étudie les identités démontrées récemment par Rains et Warnaar. Mes contributions incluent une preuve d’une extension d’une telle identité et quelques progrès partiels vers la généralisation d’une autre. / In this thesis, I study three related subjects: tensor models, Hurwitz numbers and Macdonald-Koornwinder polynomials. Tensor models are generalizations of matrix models as an approach to quantum gravity in arbitrary dimensions (matrix models give a 2D version). I study a specific model called the quartic melonic tensor model. Its specialty is that it can be transformed into a multi-matrix model which is very interesting by itself. With the help of well-established tools, I am able to compute the first two leading orders of their 1=N expansion. Among many interpretations, Hurwitz numbers count the number of weighted ramified coverings of Riemann surfaces. They are connected to many subjects of contemporary mathematics such as matrix models, integrable equations and moduli spaces of complex curves. My main contribution is an explicit formula for one-part double Hurwitz numbers with completed 3-cycles. This explicit formula also allows me to prove many interesting properties of these numbers. The final subject of my study is Macdonald-Koornwinder polynomials, in particular their Littlewood identities. These polynomials form important bases of the algebra of symmetric polynomials. One of the most important problems in symmetric function theory is to decompose a symmetric polynomial into the Macdonald basis. The obtained decomposition (in particular, if the coefficients are explicit and reasonably compact) is called a Littlewood identity. In this thesis, I study many recent Littlewood identities of Rains and Warnaar. My own contributions include a proof of an extension of one of their identities and partial progress towards generalization of one another.