A passagem progressiva de estados de Landau bidimensional (2D) para estados tridimensional (3D) foi estudada em Poços Quânticos Parabólicos (PQW) largos (W = 1000 6000 Å). Utilizou-se como técnica de transporte medidas da magnetoresistência em campo magnético intenso (B = 0 15 T) e inclinado ( = 0 90°; perpendicular paralelo), a baixas temperaturas (T = 50 mK). Observou-se, através da dependência angular das oscilações de Shubnikov de Haas ( = 0 90°), em PQWs cheios, várias sub-bandas ocupadas (5 a 8), a coexistência de estados de Landau 2D e 3D, sendo o gás 3D formado pelo colapso das sub-bandas elevadas, e o gás 2D pertencendo à primeira sub-banda. Através de cálculos do alargamento dos níveis de Landau devido ao espalhamento elástico ( = /2 , onde é o tempo quântico) e de cálculos auto-consistentes da energia de separação entre sub-bandas do PQW (ij = Ej Ei; e 12=12/2), obtiveram-se as condições 2 j-1,j para as sub-bandas elevadas j = 3,4,..., corroborando com as observações experimentais da coexistência de estados de Landau 2D e 3D no poço. Em PQWs parcialmente cheios, com apenas 2 sub-bandas ocupadas, observou-se, através do efeito do anticruzamento de níveis de Landau, de medidas da dependência angular da energia de ativação no regime de efeito Hall quântico, e de comparações com resultados de cálculos da estrutura eletrônica de PQWs em campo magnético inclinado, a coexistência de estados de Landau 2D e 3D, ocorrendo somente em campos intensos e com inclinação acentuada ( = 80 90°). Esta coexistência é diferente da mencionada anteriormente, quando od estados de Landau 3D são observados já em campo perpendicular. / The gradual progress, or evolution, of the two-dimensional (2D) toward three-dimensional (3D) Landau states was studied in wide parabolic quantum Wells (W = 1000 6000 Å). As transport technique, we used measurements of the magnetoresistence in intense (B = 0 15 T) and tilted ( = 0 90°; perpendicular parallel) magnetic Field at low temperature (T = 50 Mk). We observed in PQWs with Five to eight sub-bands occupied full well the coexistence of the 2D and 3D Landau states, through the angular dependence of the Shubnikov de Hass oscillation ( = 0 90°), where the 2D states belong to the lowest sub-band and the 3D states are formed by overlap of the other sub-bands. We calculated the level broadening due to the elastic scattering rate ( = /2 , where is the quantum time), and the energy separation between sub-bands (ij = Ej Ei; e 12=12/2). We obtained 2 j-1,j to j=3,4,... . This confirms the experimental observations of the coexistence of the 2D and 3D states in the well. We also measured PQWs partially full 2 sub-bands occupied. Experiments revel anticrossing of the Landau level (LL) belonging to the lowest sub-band and the last LL belonging to the second sub-band. Such antisrossuing occurs due to a decrease of the energy of the LL with tilt angle. This observation was supported by measurements of the angular dependence of the activation energy in the quantum Hall regime. In these measurements, we also observed the coexistence of the 2D and 3D Landau states. However, the coexistence only occurs at large tilt angles ( = 80 90°). Thus, it is different from the coexistence above mentioned, when 3D Landau states are observed already in the perpendicular magnetic field.

Η παρούσα διπλωματική εργασία πραγματοποιήθηκε στο τμήμα Σωματιδιακής Φυσικής του Πανεπιστημίου Santiago de Compostela της Ισπανίας και αποτελεί τη μελέτη της υπερσυμμετρίας στις τρεις χωροχρονικές διαστάσεις. Έμφαση δίνεται σε θεωρίες που περιέχουν τον όρο Chern-Simons που παιζεί συμαντικό ρόλο στους τομείς έρευνας της θεωρητικής φυσικής. Αρχικά, εισάγουμε τις εισαγωγικές ένοιες της υπερσυμμετρίας στις τρεις διαστάσεις και ακολούθως μελέτουμε την Ν=1 ελάχιστη θεωρία με διάφορες φυσικές ποσότες που περιέχουν τον όρο Chern-Simons. Στην συνέχεια, μελετάμε τις ABJM θεωρίες και αποδεικνύουμε ότι είναι αναλλοίωτες κάτω από μετασχηματισμούς βαθμίδας. Τέλος υπολογίζουμε τις κβαντικές διορθώσεις στην διαταρακτική θεωρία Chern-Simons. / The present thesis took part in Department of Particle Physics of University of Santiago de Compostela, Spain, and is the study of supersymmetry in three spacetime dimensions. Emphasis is given to theories containing the Chern-Simons term that plays an important role in the research areas of theoretical physics. First, we introduce the notion of supersymmetry in three dimensions and then we study the N = 1 minimal theory with various physical quantitative containing the term Chern-Simons. Then, we study the ABJM theories and prove that they are invariant under gauge transformations. Finally we calculate the quantum corrections to the perturbative Chern-Simons theory.

Submitted by Vasti Diniz (vastijpa@hotmail.com) on 2017-09-13T14:26:30Z No. of bitstreams: 1 arquivototal.pdf: 2274734 bytes, checksum: 20f744b4f2279525b9a574a0a0de7838 (MD5) / Made available in DSpace on 2017-09-13T14:26:30Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 2274734 bytes, checksum: 20f744b4f2279525b9a574a0a0de7838 (MD5) Previous issue date: 2017-02-22 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this thesis we deal with G odel-type Universes in the context of modi ed gravity, in particular, Chern-Simons modi ed gravity and Brans-Dicke theory with cosmological constant (BD- ). The G odel-type metrics have been intensively discussed in the General Relativity (GR) over years. It is known that such a metrics present Closed Time-like Curves (CTC's), in other words, the G odel-type metrics are themselves an example of the causality violation. Our goal is verify the consistency of the G odel-type metrics within the Chern-Simons modi ed gravity in both: non-dynamical and dynamical formulations. In the non-dynamical framework, we show that is possible a vacuum solution in contrast to GR. Another essentially new result that we get is the presence of causal solutions for a well-motivated matter source, in general, the solutions have no analogue in GR. Moreover, the vacuum solution represents the limiting case separating the completely causal and non-causal regions, such a property re ects the topological features of the Chern-Simons theory. The primordial distinguishing feature between the Chern-Simons modi ed gravity and GR solutions is the presence of the breaking Lorentz symmetry. It turns out this breaking opens up a range of new solutions. We show that the non-trivial Chern-Simons solutions in the non-dynamical framework is accompanied by rst-order corrections of the Lorentz-violating parameter. Furthermore, in the dynamical framework the geometric parameters are also a ected by second-order corrections of the Lorentzviolating parameter. We also investigated the G odel-type metrics in BD- model. We obtain a vacuum solution which is completely causal, m2 = 4!2, where for ~! ! 1 one recovers the GR with a scalar eld and cosmological constant. It is worth calling attention to the role of the cosmological constant that is fundamental in this context. / Nesta tese tratamos os Universos tipo Code' no contexto da gravidade modificada, em particular, na gravidade modificada de Chern-Simons e na teoria de Brans-Dicke com constante cosmolOgica (BD-A). As metricas tipo Code' vem sendo intensamente discuti­das na Relatividade Geral (RG) ao longo dos anon. Sabe-se que tail metricas apresentam Curvas tipo Tempo Fechadas (CTC's), ou seja, as prOprias metricas tipo Code' sao um exemplo da violagao de causalidade. Nosso objetivo é verificar a consistencia das metricas do tipo Code' dentro da gravidade modificada de Chern-Simons em ambas as formula-goes: nao dinamica e dinamica. Na formulagao nao-dinamica, mostramos que é possivel uma solugao de vacuo diferentemente da RG. Outro resultado essencialmente novo que obtemos é a presenga de solugoes causais para uma fonte de materia bem motivada, em geral, as solugoes nao tem analog° na RG. Alem disso, a solugao de vacuo representa o caso limite que separa as regioes completamente causal e nao causal, tal propriedade re­flete as caracteristicas topolOgicas da teoria de Chern-Simons. A caracteristica que difere fundamentalmente as solugoes da gravidade modificada de Chern-Simons e as da RG é a presenga da quebra da simetria de Lorentz. Acontece que essa quebra abre um leque de novas solugoes. Mostramos que as solugoes nao triviais de Chern-Simons na formulagao nao dinamica sao acompanhadas por corregoes de primeira ordem do parametro de viola­gao de Lorentz. Alem disso, na formulagao dinamica os parametros geometricos tambem sao afetados por corregoes de segunda ordem do parametro de violagao de Lorentz. Investigamos tambem as metricas tipo Code' no modelo BD-A. Obtemos uma solugao de vacuo completamente causal, m2 = 4w2, onde para cD -+ oo recupera-se a GR com um campo escalar e constante cosmolOgica. Vale a pena chamar a atengao para o papel da constante cosmolOgica que é fundamental neste contexto.

Maxim Kontsevich a défini un invariant $Z$ des sphères d'homologie rationnelle orientées de dimension $3$ en 1992, en poursuivant l'étude initiée par Edward Witten du développement perturbatif de la théorie de Chern-Simons.L'invariant $Z$ de Kontsevich est gradué. Il s'écrit $Z=(Z_n)_{nin NN }$, où $Z_n$ prend ses valeurs dans un espace $CA_n$ engendré par des diagrammes trivalents à $2n$ sommets appelésdiagrammes de Feynman-Jacobi de degré $n$.L'invariant $Z$ apparait d'abord comme un invariant $Z(M,tau)$ des sphères d'homologie rationnelle $M$ de dimension $3$ munies d'une parallélisation $tau$.Il est l'exponentielle d'un invariant $z(M,tau)=(z_n(M,tau))_{nin NN }$dont la partie de degré $n$ compte algébriquement les plongements des diagrammes de Feynman-Jacobi connexes à $2n$ sommets assujettis à vérifier certaines conditions.On peut associer un invariant homotopique entier $p_1(tau)$ aux parallélisations $tau$ des variétés orientées de dimension $3$, et il existe un élément $beta=(beta_n)_{nin NN}$ de $CA_n$ appelé anomalie tel que$$z_n(M,tau)-p_1(tau)beta_n$$ soit indépendant de $tau$ et noté $z_n(M)$.$$Z(M)=expleft((z_n(M))_{nin NN}right).$$On sait depuis l'introduction de cette constante par Greg Kuperberg et Dylan Thurston en 1999 que $beta_n=0$ si $n$ est pair et que $beta_1 neq 0$.Cette thèse porte sur le calcul de la première valeur inconnue $beta_3$. Elle en présente des expressions très simplifiées et implémentables sur ordinateur. / The Kontsevich invariant $Z$ of rational homology $3-$ sphere was constructed by Maxim Kontsevich in 1992 using configuration space integrals.This invariant is graduated. It can be written as $Z=(Z_n)_{nin NN}$, where $Z_n$ values in the space $mathcal{A}_n$ of jacobi diagram with order $n$. A Jacobi diagram with order $n$ is a trivalent graph with $2n$ vertices. At a first point, we can see $Z$ as an invariant $Z(M,tau)$ of rational homology $3-$spheres equipped with a trivialisation $tau$ so that $Z$ is the exponential of an invariant $z(M,tau)=(z_n(M,tau))_{ninNN}$. In fact, we can say that $z_n(M,tau)$ counts the number of embeddings of connected jacobi diagrams with order $n$ with some additionnal conditions. We can associate an homotopic integer invariant $p_1(tau)$ to each trivialisation $tau$ of oriented $3-$manifolds and it exists $beta=(beta_n)_{ninNN}$, where $beta_ninmathcal{A}_n$ that is called anomaly so that $$z_n(M,tau) - p_1(tay)$$ is independant of $tau$. We name it $z_n(M)$ and $$Z(M)=exp((z_n(M)_{nin NN})).$$Greg Kuperberg and Dylan Thurston introduced this constant in 1999. We already know that $beta_n=0$ if $n$ is even and $beta_1neq 0$. This thesis is about the computation of $beta_3$. It describes simplified expressions of $beta_3$, and this expressions can be compute with a computer.

A passagem progressiva de estados de Landau bidimensional (2D) para estados tridimensional (3D) foi estudada em Poços Quânticos Parabólicos (PQW) largos (W = 1000 6000 Å). Utilizou-se como técnica de transporte medidas da magnetoresistência em campo magnético intenso (B = 0 15 T) e inclinado ( = 0 90°; perpendicular paralelo), a baixas temperaturas (T = 50 mK). Observou-se, através da dependência angular das oscilações de Shubnikov de Haas ( = 0 90°), em PQWs cheios, várias sub-bandas ocupadas (5 a 8), a coexistência de estados de Landau 2D e 3D, sendo o gás 3D formado pelo colapso das sub-bandas elevadas, e o gás 2D pertencendo à primeira sub-banda. Através de cálculos do alargamento dos níveis de Landau devido ao espalhamento elástico ( = /2 , onde é o tempo quântico) e de cálculos auto-consistentes da energia de separação entre sub-bandas do PQW (ij = Ej Ei; e 12=12/2), obtiveram-se as condições 2 j-1,j para as sub-bandas elevadas j = 3,4,..., corroborando com as observações experimentais da coexistência de estados de Landau 2D e 3D no poço. Em PQWs parcialmente cheios, com apenas 2 sub-bandas ocupadas, observou-se, através do efeito do anticruzamento de níveis de Landau, de medidas da dependência angular da energia de ativação no regime de efeito Hall quântico, e de comparações com resultados de cálculos da estrutura eletrônica de PQWs em campo magnético inclinado, a coexistência de estados de Landau 2D e 3D, ocorrendo somente em campos intensos e com inclinação acentuada ( = 80 90°). Esta coexistência é diferente da mencionada anteriormente, quando od estados de Landau 3D são observados já em campo perpendicular. / The gradual progress, or evolution, of the two-dimensional (2D) toward three-dimensional (3D) Landau states was studied in wide parabolic quantum Wells (W = 1000 6000 Å). As transport technique, we used measurements of the magnetoresistence in intense (B = 0 15 T) and tilted ( = 0 90°; perpendicular parallel) magnetic Field at low temperature (T = 50 Mk). We observed in PQWs with Five to eight sub-bands occupied full well the coexistence of the 2D and 3D Landau states, through the angular dependence of the Shubnikov de Hass oscillation ( = 0 90°), where the 2D states belong to the lowest sub-band and the 3D states are formed by overlap of the other sub-bands. We calculated the level broadening due to the elastic scattering rate ( = /2 , where is the quantum time), and the energy separation between sub-bands (ij = Ej Ei; e 12=12/2). We obtained 2 j-1,j to j=3,4,... . This confirms the experimental observations of the coexistence of the 2D and 3D states in the well. We also measured PQWs partially full 2 sub-bands occupied. Experiments revel anticrossing of the Landau level (LL) belonging to the lowest sub-band and the last LL belonging to the second sub-band. Such antisrossuing occurs due to a decrease of the energy of the LL with tilt angle. This observation was supported by measurements of the angular dependence of the activation energy in the quantum Hall regime. In these measurements, we also observed the coexistence of the 2D and 3D Landau states. However, the coexistence only occurs at large tilt angles ( = 80 90°). Thus, it is different from the coexistence above mentioned, when 3D Landau states are observed already in the perpendicular magnetic field.

Orientadores: Francisco Odair Vieira de Paiva, Augusto César Ponce / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T10:14:29Z (GMT). No. of bitstreams: 1 Presoto_AdilsonEduardo_D.pdf: 2067267 bytes, checksum: 79c3ffe06a88b7cba190920dcf512036 (MD5) Previous issue date: 2011 / Resumo: No trabalho precursor de Brezis, Marcus e Ponce [15], estudou-se problemas semilineares elípticos com uma não linearidade não decrescente, contínua e dependendo apenas da variável dependente e com medidas como dados. Os autores estavam particularmente interessados no caso em que a equação não possuía solução. Numa das técnicas estudadas, eles aproximaram a medida por funções suaves através da convolução e, sob a condição adicional de convexidade da não linearidade, mostraram que as soluções correspondentes convergiam para a solução do mesmo problema com a maior medida menor do que ou igual a medida inicial tal que o problema tinha solução. O nosso objetivo é explorar profundamente este método. Ao invés de lidar com a convolução, consideramos sequências de medidas de Radon que convergem na topologia fraca-estrela e tais que o problema tem solução para cada termo. A pergunta que se põe é: as soluções convergem? Se sim, temos que o limite satisfaz a mesma equação com uma medida, em geral, distinta do limite-fraco, logo desejamos também determinar esta medida. Quando temos uma não linearidade, como descrita no parágrafo acima, as respostas têm um alto grau de variação, conforme os exemplos dados nos trabalhos de Ponce, e são inconclusivas. A proposta da tese é estudar a convergência dessas soluções para equações e sistemas semilineares elípticos com a não linearidade sendo do tipo exponencial. No caso em que temos a equação semilinear no plano, as soluções convergem para a solução do mesmo problema com uma medida que depende apenas do limite-fraco da sequência. Também, vemos que em dimensões superiores essas asserções não se verificam mais. Por fim, o sistema que aplicamos a técnica acima é o Sistema de Chern-Simons, surgido na física teórica e que representa o modelo de Chern-Simons Abeliano relativístico envolvendo duas partículas Higgs e dois campos calibrados / Abstract: In the pioneering work of Brezis, Marcus and Ponce [15], the authors studied elliptic semilinear problems with a continuous nondecreasing nonlinearity which vanishes at origin and depends only on dependent variable, and with measures as inicial data. They were particularly interested in the case which the equation does not have a solution. One of the techniques discussed was the approach of the measure by smooth functions via convolution. Under the additional condition of convexity, they showed that the corresponding solutions converge to the solution for the same problem with the largest measure less than inicial datum such that the problem admits a solution. Our aim is to explore deeply this method. Instead of dealing with the convolution, we consider sequences of Radon measures which converge in weak-star topology and such that the problem has solution for each term. The question posted is: the solutions converge? If yes, the limit solves the same problem with, in general distinct from the weak limit, another measure, thus, we also wish to determine this measure. The purpose of the thesis is to study the convergence of solutions for equations and systems with exponential nonlinearity. If we have the equation semilinear on the plane, the solutions converge to a solution for the same problem with a measure which depends only on weak limit of the sequence. We also see that in upper dimensions the results are no longer assured. In the end, the system concerned is the Chern-Simons System that comes from theoretical physics and it represents a relativistic Abelian Chern- Simons model with two Higgs particles and two gauge fields / Doutorado / Matematica / Doutor em Matemática

Cette thèse étudie la proposition 100% monnaie, telle qu’elle fut formulée aux États-Unis dans les années 1930 par Henry Simons (l’auteur principal du « Plan de Chicago »), Lauchlin Currie et Irving Fisher notamment. L’essence de cette proposition est de divorcer la création de monnaie des prêts de monnaie : les dépôts servant de moyens de paiement seraient soumis à 100% de réserve en monnaie légale, conférant à l’État un monopole de la création monétaire. Cette idée de réforme étant régulièrement sujette à confusion, nous entreprenons de clarifier son concept et d’étudier ses principaux arguments. Au chapitre 1, nous montrons que le 100% monnaie ne saurait être considéré comme un simple avatar des idées de la « Currency School » : contrairement à l’Acte de Peel de 1844, il ne contient en soi aucune règle d’émission, laissant ouvert le débat « règle ou discrétion ». Au chapitre 2, distinguant entre deux grandes approches du 100% monnaie, nous montrons que celui-ci n’implique nullement d’abolir l’intermédiation bancaire basée sur les dépôts d’épargne. Au chapitre 3, nous analysons, à travers les travaux de Fisher, l’objectif principal du 100% monnaie : celui de mettre fin au comportement procyclique du volume de monnaie, causé par le lien de dépendance entre création monétaire et prêts bancaires. Au chapitre 4, nous étudions un autre argument du 100% monnaie : celui de permettre une réduction de la dette publique, en rendant à l’État l’intégralité du seigneuriage – argument souvent critiqué, dont nous montrons qu’il n’est pourtant pas infondé. Alors que le 100% monnaie suscite un regain d’intérêt depuis la crise de 2008, il nous a paru fondamental de clarifier ces questions. / This thesis studies the 100% money proposal, such as it was formulated in the United States in the 1930s by Henry Simons (the main author of the “Chicago Plan”), Lauchlin Currie and Irving Fisher in particular. The essence of this proposal is to divorce the creation of money from the lending of money: deposits serving as means of payment would be subjected to 100% reserves in lawful money, awarding the state a monopoly over money creation. Because this reform idea is regularly subject to confusion, we endeavour to clarify its concept and study its main arguments. In chapter 1, we show that the 100% money proposal ought not to be viewed as a mere avatar of the “Currency School” ideas: contrary to Peel’s Act of 1844, it contains no issuing rule by itself, leaving open the debate “rule or discretion”. In chapter 2, distinguishing between two broad approaches to the 100% money proposal, we show that it does not imply abolishing bank intermediation based on savings deposits at all. In chapter 3, we analyse, through Fisher’s works, the main objective of the 100% money proposal: that of putting an end to the pro-cyclical behaviour of the volume of money, caused by the dependency relationship between money creation and bank loans. In chapter 4, we study another argument of the 100% money proposal: that of allowing a reduction of public debt, by returning the totality of seigniorage back to the state—an oft-criticised argument, which, as we show, is not unfounded however. While the 100% money proposal has been arousing renewed interest since the 2008 crisis, we thought it was fundamental to clarify these issues.

Biomass recalcitrance has been recognized as one of the major barriers that hided the cost-effective conversion of lignocellulosic biomass to bioethanol, therefore the current bioconversion process require an essential step known as pretreatment to increase the cellulose accessibility. This thesis provides information about changes in cellulose accessibility upon different pretreatments, along with how these pretreatments alter the chemical and physical structures of biomass, will be extremely helpful to further optimize the current pretreatment process. Multiple promising analytical techniques including Simons’ stain, NMR cryoporometry, relaxometry, mercury porosimetry was introduced and successfully applied on pretreated biomass samples to characterize the cellulose accessible surface area and biomass porosity. Different pretreatments increase cellulose accessibility through different mechanisms to different extent. Dilute acid pretreatment is more effective than steam explosion in terms of increasing accessible surface area of cellulose as reflected by Simons’ stain and NMR cryoporometry, while NMR relaxometry suggested steam explosion is more effective at pore expansion for the cell wall water pools detected by changes in NMR relaxation time. Alkaline pretreatment decreased cellulose degree of polymerization, cellulose crystallinity, lignin content and subsequently increased cellulose accessibility, with sodium hydroxide pretreatment proved to be much more effective compared lime or soaking in ammonia pretreatment. Delignification through alkaline-based pretreatment is found less effective than removal of hemicellulose using acid in terms of cellulose accessibility increase. Lignin didn’t directly dictate cellulose accessibility but rather restricted xylan accessibility which in turn controls the access of cellulase to cellulose. Pore size distribution analysis based on mercury porosimetry also indicated that the most fundamental barrier in terms of biomass porosity scale for efficient enzymatic hydrolysis is the nano-pore space formed between coated microfibrils, despite some of the porous architecture such as cell lumen and pit could be severely destroyed after pretreatment. The action of cellulase on the characteristics of cellulosic fractions obtained from pretreated biomass was also investigated. Cellulose accessibility was found to increase at the beginning of hydrolysis, and after reaching a maximum value then starting to decrease. Enzymatic hydrolysis resulted in a rapid decrease in the cellulose degree of polymerization then gradually leveled off, suggesting the existence of a synergistic action of endo- and exo-glucanases that contribute to the occurrence of a peeling off type mechanism.

Thesis (Ph.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / We investigate aspects of differential K-theory. In particular, we give a direct proof that the Freed-Lott differential analytic index is well defined, and a short proof of the differential Grothendieck-Riemann-Roch theorem in the setting of Freed-Lott differential K-theory. We also construct explicit ring isomorphisms between Freed-Lott differential K-theory and Simons-Sullivan differential K-theory, define the Simons-Sullivan differential analytic index, and prove the differential Grothendieck-Riemann-Roch theorem in the setting of Simons-Sullivan differential K-theory. / 2031-01-02

It is well-known that in 2 + 1 dimensions the flux attachment transmutes the statistics of a particle.The aim of this master thesis is to study the dualities between bosons and fermions induced by Abeliantopological gauge fields in 2 + 1 dimensions. Chapter 1 and 2 are reviews of known results about thepath integral quantization of Chern-Simons theory and the regularization of the fermionic path integral.In the following chapters, we will derive the statistical transmutation and various Abelian dualities in2 + 1 dimensions.