Let M#k(64) be the space of weakly holomorphic modular forms in level 64 and weight k which can have poles only at infinity, and let S#k(64) be the subspace of M#k(64) consisting of forms which vanish at all cusps other than infinity. We explicitly construct canonical bases for these spaces and show that the coefficients of these basis elements satisfy Zagier duality. We also compute the generating function for the canonical basis.

Let N ∈ {8,9,16,25} and let M#0(N) be the space of level N weakly holomorphic modular functions with poles only at the cusp at infinity. We explicitly construct a canonical basis for M#0(N) indexed by the order of the pole at infinity and show that many of the coefficients of the elements of these bases are divisible by high powers of the prime dividing the level N. Additionally, we show that these basis elements satisfy an interesting duality property. We also give an argument that extends level 1 results on congruences from Griffin to levels 2, 3, 4, 5, 7, 8, 9, 16, and 25.

Matroidal networks are useful tools in furthering research in network coding. They have been used to show the limitations of linear coding solutions. In this paper we examine the basic information on network coding and matroid theory. We then go over the method of creating matroidal networks. Finally we construct matroidal networks from the dual of the fano matroid and the dual of the non-fano matroid, and breifly discuss some coding solutions.

Let M#k(52) be the space of weight k level 52 weakly holomorphic modular forms with poles only at infinity, and S#k(52) the subspace of forms which vanish at all cusps other than infinity. For these spaces we construct canonical bases, indexed by the order of vanishing at infinity. We prove that the coefficients of the canonical basis elements satisfy a duality property. Further, we give closed forms for the generating functions of these basis elements.

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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.

Gauge-string duality has provided a powerful framework for the study of strongly coupled gauge theories and non-perturbative string models. This thesis analyzes the holographic description of non-local gauge theory operators and some aspects of the Bagger-Lambert theory. The latter, as a proposal for a multiple M2-branes effective theory, is conjectured to be the holographic dual of a compactification of M-theory. We show that all half-BPS Wilson loop operators in N=4 SYM - which are labeled by Young tableaus - have a gravitational dual description in terms of D5-branes or alternatively in terms of D3-branes in AdS5xS5. We prove that the insertion of a half-BPS Wilson loop operator in the N=4 SYM path integral is achieved by integrating out the degrees of freedom on a configuration of bulk D5-branes or alternatively on a configuration of bulk D3-branes. We construct a new class of supersymmetric surface operators in N=4 SYM and find the corresponding dual supergravity solutions. Consistency requires constructing N=4 SYM in the D7 supergravity background and not in flat space. This enlarges the class of holographic gauge theories dual to string theory backgrounds to gauge theories in non-trivial supergravity backgrounds. We write down a maximally supersymmetric one parameter deformation of the field theory action of Bagger and Lambert and we show that this theory on RxT2 is invariant under the superalgebra of the maximally supersymmetric Type IIB plane wave. It is argued that this theory holographically describes the Type IIB plane wave in the discrete light-cone quantization (DLCQ). Finally, we show by explicit computation that the Bagger-Lambert Lagrangian realizes the M2-brane superalgebra, including also two p-form central charges that encode the M-theory intersections involving M2-branes.

Gauge-string duality has provided a powerful framework for the study of strongly coupled gauge theories and non-perturbative string models. This thesis analyzes the holographic description of non-local gauge theory operators and some aspects of the Bagger-Lambert theory. The latter, as a proposal for a multiple M2-branes effective theory, is conjectured to be the holographic dual of a compactification of M-theory. We show that all half-BPS Wilson loop operators in N=4 SYM - which are labeled by Young tableaus - have a gravitational dual description in terms of D5-branes or alternatively in terms of D3-branes in AdS5xS5. We prove that the insertion of a half-BPS Wilson loop operator in the N=4 SYM path integral is achieved by integrating out the degrees of freedom on a configuration of bulk D5-branes or alternatively on a configuration of bulk D3-branes. We construct a new class of supersymmetric surface operators in N=4 SYM and find the corresponding dual supergravity solutions. Consistency requires constructing N=4 SYM in the D7 supergravity background and not in flat space. This enlarges the class of holographic gauge theories dual to string theory backgrounds to gauge theories in non-trivial supergravity backgrounds. We write down a maximally supersymmetric one parameter deformation of the field theory action of Bagger and Lambert and we show that this theory on RxT2 is invariant under the superalgebra of the maximally supersymmetric Type IIB plane wave. It is argued that this theory holographically describes the Type IIB plane wave in the discrete light-cone quantization (DLCQ). Finally, we show by explicit computation that the Bagger-Lambert Lagrangian realizes the M2-brane superalgebra, including also two p-form central charges that encode the M-theory intersections involving M2-branes.

In this thesis, we propose and evaluate algorithms for solving the worst-case-coverage deployment problem in ad-hoc wireless sensor networks. The worst-case-coverage deployment problem is to deploy additional sensors in the wireless sensor field to optimize the worst-case coverage. We derive a duality theorem that reveals the close relation between the maximum breach path and the minimum Delaunay cut. The duality theorem is similar to the well-known max-flow-min-cut theorem in the field of network optimization. The major difference lies in the fact that the object function we study in this paper is nonlinear rather than linear. Based on the duality theorem, we propose an efficient dual algorithm to solve the worst-case-coverage deployment problem. In addition, we propose a genetic algorithm for deploying a number of additional sensors simultaneously. We use analytical proofs and simulation results to justify the usage of the proposed approaches.

We explore supersymmetric quantum field theories in three and four dimensions via an analysis of their BPS spectrum. In four dimensions, we develop the theory of BPS quivers which provides a simple picture of BPS states in terms of a set of building block atomic particles, and basic quantum mechanical interactions. We develop efficient techniques, rooted in an understanding of quantum-mechanical dualities, for determining the spectrum of bound states, and apply these techniques to calculate the spectrum in a wide class of field theories including ADE gauge theories with matter, and Argyres-Douglas type theories. Next, we explore the geometric content of quivers in the case when the four-dimensional field theory can be constructed from the six-dimensional (2; 0) superconformal field theory compactified on a Riemann surface. We find that the quiver and its superpotential are determined by an ideal triangulation of the associated Riemann surface. The significance of this triangulation is that it encodes the data of geodesics on the surface which in turn are the geometric realization of supersymmetric particles. Finally we describe a class of three-dimensional theories which are realized as supersymmetric domain walls in the previously studied four-dimensional theories. This leads to an understanding of quantum field theories constructed from the six-dimensional (2; 0) superconformal field theory compactified on a three-manifold, and we develop the associated geometric dictionary. We find that the structure of the field theory is determined by a decomposition of the three-manifold into tetrahedra and a braid which species the relationship between ultraviolet and infrared geometries. The phenomenon of BPS wall-crossing in four dimensions is then seen in these domain walls to be responsible for three-dimensional mirror symmetries. / Physics