The cubic focusing nonlinear Schrodinger equation is a canonical model equation that arises in physics and engineering, particularly in nonlinear optics and plasma physics. Cubic NLS is an accessible venue to refine techniques for more general nonlinear partial differential equations. In this thesis, it is shown there exist solutions to the focusing cubic nonlinear Schrodinger equation in three dimensions that blowup on a circle, in the sense of L2-norm concentration on a ring, bounded H1-norm outside any surrounding toroid, and growth of the global H1-norm with the log-log rate. Analogous behaviour occurs in higher dimensions. That is, there exists data for which the corresponding evolution by the cubic nonlinear Schrodinger equation explodes on a set of co-dimension two. To simplify the exposition, the proof is presented in dimension three, with remarks to indicate the adaptations in higher dimension.

The cubic focusing nonlinear Schrodinger equation is a canonical model equation that arises in physics and engineering, particularly in nonlinear optics and plasma physics. Cubic NLS is an accessible venue to refine techniques for more general nonlinear partial differential equations. In this thesis, it is shown there exist solutions to the focusing cubic nonlinear Schrodinger equation in three dimensions that blowup on a circle, in the sense of L2-norm concentration on a ring, bounded H1-norm outside any surrounding toroid, and growth of the global H1-norm with the log-log rate. Analogous behaviour occurs in higher dimensions. That is, there exists data for which the corresponding evolution by the cubic nonlinear Schrodinger equation explodes on a set of co-dimension two. To simplify the exposition, the proof is presented in dimension three, with remarks to indicate the adaptations in higher dimension.

In this thesis we study deformations of functions on singular varieties with a view toward Frobenius manifolds. <br /><br />Chapter 2 is mainly introductory. We prove standard results in deformation theory for which we do not know a suitable reference. We also give a construction of the miniversal deformation of a function on a singular space that to the best of our knowledge does not appear in this form in literature. <br /><br />In Chapter 3 we find a sufficient condition for the dimension of the base space of the miniversal deformation to be equal to the number of critical points into which the original singularity splits. We show that it holds for functions on smoothable and unobstructed curves and for function on isolated complete intersections singularities, unifying under the same argument previously known results. <br /><br />In Chapter 4 we use the previous results to construct a multiplicative structure known as F -manifold on the base space of the miniversal deformation. We relate our construction to the theory of Frobenius manifolds by means of an example: mirrors of weighted projective lines.<br /><br />The appendix is joint work with D. Mond. We study unfolding of composed functions under a suitable deformation category. It also yields an F-manifold structure on the base space, which we use to answer some questions raised by V. Goryunov and V. Zakalyukin on the discriminant on matrix deformations.

Previous studies have suggested that the material graphene might transition into an electron-electron interaction driven, unconventional, time-reversal-symmetry-breaking, (d+id)wave superconducting state upon either significant electron or hole doping, and, in particular, upon doping to the Van Hove singularity. As defects are likely to be introduced in the doping process, we are, in this text, concerned with the effects of defects on this superconducting state near the Van Hove singularity doping. To investigate the effects we use a mean-field treatment of a phenomenological resonant-valence-bond model. We find that the resonant-valence-bond amplitudes, which in the defect free graphene sheet are proportional to the superconducting pairing-potential, are suppressed near the defects, and that the recovery is well described by an exponential, yet anisotropic, recovery. In general, we find that the (d+id)-wave, superconducting state is quite resilient, and that even for strong defects, such as a vacancy, the recovery length is of the order of one lattice constant when extrapolated to weak pairing-potentials; this is compared to a conventional superconducting state of an attractive Hubbard model for which the same decay length is found to be of the order of a half lattice constant. For the defect free graphene sheet the (d+id)-wave state is a completely gapped state. The introduction of vacancies is, however, found to be accompanied by the appearance of midgap states. These states are shown to be localized around the vacancies. In accordance with the nature of this text, we will, for the benefit of students and non-experts, include an introductory section on the fundamental methods and concepts used. It gives a short and hopefully pedagogical introduction to the rudimentary concepts of solid state theory and the microscopic BCS theory of superconductivity.

Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro / Este trabalho buscou compartilhar o debate sobre as metamorfoses e singularidades das rádios comunitárias e a utilização das Tecnologias de Informação e Comunicação (TICs) como linha de fuga capaz de desterritorializar a comunicação comunitária. Guattari, Deleuze e Negri foram os principais referenciais teóricos com que trabalhamos. Fizemos entrevistas com personagens vinculados ao serviço de radiodifusão comunitária no Rio de Janeiro. O cruzamento destes autores com as entrevistas e apreciação de outros materiais teóricos nos permitiu pensar as implicações do uso da internet como mais um meio para o serviço de radiodifusão ser transmitido. Sendo capaz de ampliar as idéias e experiências, podendo fazer parte da rede de produção colaborativa promovida pelo ciberespaço. O esboço deste estudo nos mostrou que a comunicação popular tende a ser encarada erradamente na esfera da ilegalidade, mas os benefícios e a aceitação da comunidade constituem a sua legitimidade. Os dados empíricos revelam que a Lei 9612/98 apesar de legalizar as rádios comunitárias tende exercer um papel castrador. Esta imagem nos leva a pensar o uso das ferramentas da internet como ação política capaz de contribuir para ser mais um espaço de atuação das rádios comunitárias, colaborando para democratização da comunicação. / This study aimed to share the discussion of singularities and metamorphoses of community radios and the use of Information and Communication Technologies (ICTs) as a leak to deterritorialize community communication. Guattari, Deleuze and Negri have been our main theoretical references. We also interviewed largely people linked to community broadcasting service in Rio de Janeiro. The intersection of these authors theories with the interviews and other field work on community radios allowed us to consider the theoretical implications of using Internet as a medium for the broadcasting service. Being able to extend ideas and experiences is part of the network of collaborative production promoted by cyberspace. The outline of this study showed us that popular communication tends to be wrongly seen in the sphere of illegality, but the benefits and community acceptance are its legitimacy. Empirical data show that Law 9612/98 despite having given legalized community radio, plays a castrating role in its development. This image helps thinking the use of internet as a political action in the practice of community radio stations, thus contributing to democratize media in our society.

Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro / Este trabalho buscou compartilhar o debate sobre as metamorfoses e singularidades das rádios comunitárias e a utilização das Tecnologias de Informação e Comunicação (TICs) como linha de fuga capaz de desterritorializar a comunicação comunitária. Guattari, Deleuze e Negri foram os principais referenciais teóricos com que trabalhamos. Fizemos entrevistas com personagens vinculados ao serviço de radiodifusão comunitária no Rio de Janeiro. O cruzamento destes autores com as entrevistas e apreciação de outros materiais teóricos nos permitiu pensar as implicações do uso da internet como mais um meio para o serviço de radiodifusão ser transmitido. Sendo capaz de ampliar as idéias e experiências, podendo fazer parte da rede de produção colaborativa promovida pelo ciberespaço. O esboço deste estudo nos mostrou que a comunicação popular tende a ser encarada erradamente na esfera da ilegalidade, mas os benefícios e a aceitação da comunidade constituem a sua legitimidade. Os dados empíricos revelam que a Lei 9612/98 apesar de legalizar as rádios comunitárias tende exercer um papel castrador. Esta imagem nos leva a pensar o uso das ferramentas da internet como ação política capaz de contribuir para ser mais um espaço de atuação das rádios comunitárias, colaborando para democratização da comunicação. / This study aimed to share the discussion of singularities and metamorphoses of community radios and the use of Information and Communication Technologies (ICTs) as a leak to deterritorialize community communication. Guattari, Deleuze and Negri have been our main theoretical references. We also interviewed largely people linked to community broadcasting service in Rio de Janeiro. The intersection of these authors theories with the interviews and other field work on community radios allowed us to consider the theoretical implications of using Internet as a medium for the broadcasting service. Being able to extend ideas and experiences is part of the network of collaborative production promoted by cyberspace. The outline of this study showed us that popular communication tends to be wrongly seen in the sphere of illegality, but the benefits and community acceptance are its legitimacy. Empirical data show that Law 9612/98 despite having given legalized community radio, plays a castrating role in its development. This image helps thinking the use of internet as a political action in the practice of community radio stations, thus contributing to democratize media in our society.

The Orbifold Landau-Ginzburg Mirror Symmetry Conjecture states that for a quasihomogeneous singularity W and a group G of symmetries of W, there is a dual singularity WT and dual group GT such that the orbifold A-model of W/G is isomorphic to the orbifold B-model of WT/GT. The Landau-Ginzburg A-model is the Frobenius algebra HW,G constructed by Fan, Jarvis, and Ruan, and the B-model is the Orbifold Milnor ring of WT . The unorbifolded conjecture has been verified for Arnol'd's list of simple, unimodal and bimodal quasi-homogeneous singularities with G the maximal diagonal symmetry group by Priddis, Krawitz, Bergin, Acosta, et al. [9], and by Fan-Shen [4] and Acosta [1] for all two dimensional invertible singularities and by Krawitz for all invertible singularities of 3 dimensions and greater in [8]. Based on this Krawitz posed the Orbifold Landau-Ginzburg Mirror Symmetry Conjecture, where the A-model is still the Frobenius algebra HW,G constructed by Fan, Jarvis, and Ruan but constructed with respect to a proper subgroup G of the maximal group of symmetries GW and the B-model is the orbifold Milnor ring of WT orbifolded with respect to a non-trivial group K in SLn of order [GW :J]. I verify this Orbifold Landau-Ginzburg Mirror Symmetry Conjecture for all unimodal and bimodal quasi-homogeneous singularities in Arnol'd's list with G = J < GW, being the minimal admissible diagonal symmetry group. I also discuss some axioms and properties of these singularities.

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