New four dimensional N = 2 field theories can be engineered from compactifying six dimensional (2, 0) superconformal field theory on a punctured Riemann surface. Hitchin’s equation is defined on this Riemann surface and the fields in Hitchin’s equation are singular at the punctures. Four dimensional theory is entirely determined by the data at the punctures. Theory without lagrangian description can also be constructed in this way. We first construct new four dimensional generalized superconformal quiver gauge theory by putting regular singularity at the puncture. The algorithm of calculating weakly coupled gauge group in any duality frame is developed. The asymptotical free theory and Argyres-Douglas field theory can also be constructed using six dimensional method. This requires introducing irregular singularity of Hithcin’s equation. Compactify four dimensional theory down to three dimensions, the corresponding N = 4 theory has the interesting mirror symmetry. The mirror theory for the generalized superconformal quiver gauge theory can be derived using the data at the puncture too. Motivated by this construction, we study other three dimensional theories deformed from the above theory and find their mirrors. The surprising relation of above four dimensional gauge theory and two dimensional conformal field theory may have some deep implications. The S-duality of four dimensional theory and the crossing symmetry and modular invariance of two dimensional theory are naturally related.

We investigate the problem of minimizing a nonconvex function with respect to convex constraints, and we study different techniques to compute a lower bound on the optimal value: The method of using convex envelope functions on one hand, and the method of exploiting nonconvex duality on the other hand. We investigate which technique gives the better bound and develop conditions under which the dual bound is strictly better than the convex envelope bound. As a byproduct, we derive some interesting results on nonconvex duality. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

Through his work in ludics and Geometry of Interaction, Jean-Yves Girard invites us to a change of paradigm in the study of logic: the quest for a transcendental syntax, some kind of idealized language that emerges from the rules of logic. Amongst these rules, "testing" plays a leading role in defining a duality for the interpretation of negation. The present work focuses on a notion of polarity which is a central technique used throughout Girard's work to express linear negation. We describe some properties and illustrate them with examples with the purpose of getting acquainted with the technique. We also highlight how the classical connectives (conjunction and disjunction) arise from an interpretation based on testing. In a sense, this work is intended to provide an alternative introduction to Girard's ideas and we hope it can have some pedagogical value.

This thesis is concerned with so called chiral gauge theories, also known as self dual gauge theories. In particular, the aim of this thesis to investigate the role that chiral gauge theories play in duality symmetries in lower dimensions through dimensional reduction. Chapter one serves as an introduction to the notions of duality in field and string theory. The problems of formulating well defined actions for self-dual gauge theories are introduced as well as a brief presentation of the different approaches used to over come these problems. Chapter two introduces dimensional reduction and demonstrates how duality symmetries arise from the dimensional reduction of self-dual theories in a variety of dimensions and on different compact spaces. Examples are presented where the couplings of the resulting theories are calculated explicitly in terms of the geometrical data of the compact space. The duality generators acting on these couplings are also calculated explicitly and related to the geometry/topology of the compact space. Chapter three deals with the idea of duality manifest actions and their relation to the self-dual theories in higher dimensions. Non-linear Born-Infeld type actions are introduced and again dimensional reduction is shown to play a role in the duality of the Born-Infeld action. This leads to a duality manifest version of the Born-Infeld action. Chapter four describes perhaps the main application of this thesis. The effective action of the M-theory five brane wrapped on a torus is identified with the effective action of the IIB D-3 brane dimensionally reduced on a circle (after some appropriate world volume dualizations). The IIB S-duality then arises as a result of the modular symmetry of the torus. The final chapter contains a brief summary and a hint of further directions for research that were outside the scope of this thesis.

Thesis (Ph. D.)--State University of New York at Binghamton, Department of Mathematical Sciences, 2008. / Includes bibliographical references.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Includes bibliographical references (p. 83-85).

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2001. / Includes bibliographical references. Also available on the Internet.

Proton cyclotron echoes are phenomena related to the proton cyclotron frequency discovered On topside sounder swept-frequency ionograms from the Canadian satellite Alouette II in 1969. Subsequent studies were also limited to the use of the swept-frequency ionograms and devoted only to these so called regular proton cyclotron echoes. The regular proton cyclotron echoes occur on the swept-frequency ionograms at constant apparent ranges predominantly at frequencies below the electron plasma frequency fN and slightly above the electron cyclotron frequency fH. In this dissertation over 2000 topside sounder ionograms (both swept-frequency mode and fixed-frequency mode) obtained from the Alouette II, ISIS I and II satellites of the Alouette-ISIS program are used to investigate proton cyclotron echoes in detail. Examination of the combined swept-frequency and fixed ionograms indicates that the proton cyclotron echoes are also observed on the fixed-frequency ionograms, In addition to some features, such as constant apparent ranges and higher order multiple echoes, which have been already observed on the swept frequency ionograms and also occur on the fixed-frequency ionograms, under some Conditions non-constant apparent ranges and modulations in intensity of the proton cyclotron echoes on the fixed-frequency ionograms are observed. Usually the proton cyclotron echoes on the fixed-frequency ionogram can be observed for a much longer time than on the swept-frequency ionograms due to the fixed sounding frequencies. A proton cyclotron echo can be under observation for several spin periods of the satellite if the plasma parameters encountered by the sounder are appropriate, The modulation in intensity of the proton cyclotron echo by antenna orientation is evident. In terms of analysis of the combined swept-frequency and fixed-frequency ionograms, and the satellite orbital parameters and spin axis attitude, effects on proton cyclotron echoes of antenna orientation with respect to the earth’s magnetic field are examined. It is found that higher intensity and higher harmonics of proton cyclotron echoes occur when the sounding antenna is parallel to the earth’s magnetic field. A new class of proton cyclotron echoes was discovered, which occur on electron plasma resonances. The proton cyclotron echoes on the fH, nfH (n = 2, 3, 4), fQ3 and fD have been observed- The first three are checked in more detail. The proton cyclotron echoes observed on the fH, 4fH, fQ3 and fD resonances exhibit doublet, on the 2fH resonance triplex and on 3fH resonance single while the regular proton cyclotron echoes are always single. A frequency difference of about 5-10 Hz exists between subechoes in a doublet or triplex. The regular proton cyclotron echo seems to correspond to the first echo of the double or triple proton cyclotron echoes. No echo minus exists and most proton cyclotron echoes on the electron plasm a resonances are observed at dip angles whose magnitudes are less than 8°. This new class of proton cyclotron echoes is attributed to the results of nonlinear interactions of ion and electron Bernstein waves or ion Bernstein waves and DKO mode electromagnetic waves (for the fH resonance). Absorption phenomena on the 3fH, 4fH and fQ3 resonance spikes near the proton cyclotron period on swept-frequency ionograms are observed occasionally, but not yet understood. A theory based on nonlinear interaction of two waves is suggested to interpret proton cyclotron echoes. Many observational features of proton cyclotron echoes can be interpreted by this nonlinear interaction model of two waves. / Graduate

Orientador: Maria Gorete Carreira Andrade / Banca: Pedro Luiz Queiroz Pergher / Banca: Ermínia de Lourdes Campello Fanti / Resumo: Baseado na teoria de cohomologia de grupos, Andrade e Fanti definiram um invariante algébrico, denotado por E(G;S;M), onde G é um grupo, S é uma família de subgrupos de G de índice finito e Mé um Z 2G-módulo. O objetivo deste trabalho é definir um invariante dual a E(G;S;M), que denotaremos por E (G;S;M), utilizando a homologia de grupos em vez da cohomologia. Com este invariante, obtemos diversos resultados e aplicações, principalmente nas teorias de grupos e pares de dualidade e de decomposição de grupos. Estes resultados fornecem uma maneira alternativa de obter aplicações e propriedades nestas teorias. E, para desenvolver este trabalho, estudamos as teorias de (co)homologia absoluta e relativa de grupos, bem como suas interpretações topológicas, e a teoria de grupos e pares de dualidade / Abstract: Based on the cohomology theory of groups, Andrade and Fanti defined an algebraic invariant, denoted by E(G;S;M), where G is a group, S is a family of subgroups of G with nite index and M is a Z 2G-module. The purpose of this work is to define a dual invariant of E(G;S;M), which we denote by E (G;S;M), using the homology of groups instead of cohomology. With this invariant, we obtain many results and applications, especially in the duality and splitting theories of groups. These results provide an alternative way to get applications and properties in these theories. And to develop this work, we studied the absolute and relative (co)homology theories of groups, as well as their topological interpretations, and the theories of duality groups and pairs / Mestre

Through his work in ludics and Geometry of Interaction, Jean-Yves Girard invites us to a change of paradigm in the study of logic: the quest for a transcendental syntax, some kind of idealized language that emerges from the rules of logic. Amongst these rules, "testing" plays a leading role in defining a duality for the interpretation of negation. The present work focuses on a notion of polarity which is a central technique used throughout Girard's work to express linear negation. We describe some properties and illustrate them with examples with the purpose of getting acquainted with the technique. We also highlight how the classical connectives (conjunction and disjunction) arise from an interpretation based on testing. In a sense, this work is intended to provide an alternative introduction to Girard's ideas and we hope it can have some pedagogical value.