This thesis is devoted to the study of two important applications of gauge-gravity duality: the cosmological singularity problem and conformal fluid dynamics. Gauge-gravity duality is a concrete dual relationship between a gauge theory (such as electromagnetism, the theories of weak and strong interactions), and a theory of strings which contains gravity. The most concrete application of this duality is the AdS/CFT correspondence, where the theory containing gravity lives in the bulk of an asymptotically anti-de-Sitter space-time, while the dual gauge theory is a deformation of a conformal field theory which lives on the boundary of anti-de-Sitter space-time(AdS). Our first application of gauge-gravity duality is to the cosmological singularity problem in string gravity. A cosmological singularity is defined as a spacelike region of space-time which is highly curved so that Einstein’s gravity theory can be no longer applied. In our setup the bulk space-time has low curvature in the far past and the physics is well described by supergravity (which is an extension of standard Einstein gravity). The cosmological singularity is driven by a time dependent string coupling in the bulk theory. The rate of change of the coupling is slow, but the net change of the coupling can be large. The dual description of this is a time dependent coupling of the boundary gauge theory. The coupling has a profile which is a constant in the far past and future and attains a small but finite value at intermediate times. We construct the supergravity solution, with the initial condition that the bulk space-time is pure AdS in the far past and show that the solution remains smooth in a derivative expansion without formation of black holes. However when the intermediate value of the string coupling becomes weak enough, space-time becomes highly curved and the supergravity approximation breaks down, mimicking a spacelike singularity. The resulting dynamics is analyzed in the dual gauge theory with a time dependent coupling constant which varies slowly. We develop an appropriate adiabatic expansion in the gauge theory in terms of coherent states and show that the time evolution continues to be smooth. We cannot, however, arrive at a definitive conclusion about the fate of the system at very late times when the coupling has again risen and supergravity again applies. One possibility is that the energy which has been supplied to the universe is simply extracted out and the space-time goes back to its initial state. This could provide a model for a bouncing cosmology. A second possibility is that dissipation leads to a thermal state at late time. If this possibility holds, we show that such a thermal state will be described either by a gas of strings or by a small black hole, but not by a big black hole. This means that in either case, the future space-time is close to AdS. We then apply gauge-gravity duality to conformal fluid dynamics. The long wavelength behavior of any strongly coupled system with a finite mean free path is described by an appropriate fluid dynamics. The bulk dual of a fluid flow in the boundary theory is a black hole with a slowly varying horizon. In this work we consider certain fluid flows which become supersonic in some regions. It is well known that such flows present acoustic analogs of ergoregions and horizons, where acoustic waves cannot propagate in certain directions. Such acoustic horizons are expected to exhibit thermal radiation of acoustic waves with temperature essentially given by the gradient of the velocity at the acoustic horizon. We find acoustic analogs of black holes in charged conformal fluids and use gauge-gravity duality to construct dual gravity solutions. A certain class of gravitational quasinormal wave modes around these gravitational backgrounds perceives a horizon. Upon quantization, this implies that these gravitational modes should have a thermal spectrum. The final issue that we study is fluid-gravity duality at zero temperature. The usual way of constructing gravity duals of fluid flows is by means of a small derivative expansion, in which the derivatives are much smaller than the temperature of the background black hole. Recently, it has been reported that for charged fluids, this procedure breaks down in the zero temperature limit. More precisely, corrections to the small derivative expansion in the dual gravity of charged fluid at zero temperature have singularities at the black hole horizon. In this case, fluid-gravity duality is not understood precisely. We explore this problem for a zero temperature charged fluid driven by a low frequency, small amplitude and spatially homogeneous external force. In the gravity dual, this force corresponds to a time dependent boundary value of the dilaton field. We calculate the bulk solution for the dilaton and the leading backreaction using a modified low frequency expansion. The resulting solutions are regular everywhere, establishing fluid-gravity duality to this order.

Serre classes of modules over a ring R are important because they describe relationships between certain classes of modules and sets of ideals of R. We characterize the Serre classes of three different types of modules. First we characterize all Serre classes of noetherian modules over a commutative noetherian ring. By relating noetherian modules to artinian modules via Matlis duality, we characterize the Serre classes of artinian modules. A module M is reflexive with respect to E if the natural evaluation map from M to its bidual is an isomorphism. When R is complete local and noetherian, take E as the injective envelope of the residue field of R. The main result provides a characterization of the Serre classes of reflexive modules over a complete local noetherian ring. This characterization depends on an ability to “construct” reflexive modules from noetherian modules and artinian modules. We find that Serre classes of reflexive modules over a complete local noetherian ring are in one-to-one correspondence with pairs of collections of prime ideals which are closed under specialization.

Following [10] we continue the study of the problem of expected utility maximization in incomplete markets. Our goal is to find minimal conditions on a model and a utility function for the validity of several key assertions of the theory to hold true. In [10] we proved that a minimal condition on the utility function alone, i.e. a minimal market independent condition, is that the asymptotic elasticity of the utility function is strictly less than 1. In this paper we show that a necessary and sufficient condition on both, the utility function and the model, is that the value function of the dual problem is finite. (authors' abstract) / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

This thesis encompasses a collection of four pieces of work on wave-particle duality, weak-value-assisted tomography, high-dimensional quantum key distribution, and phase-only holograms. In the work on duality, we derive a novel duality relation, and we sketch a thought experiment that leads to an apparent violation of the duality principle. In the project on tomography, we perform a state determination procedure with weak values, and we study the accuracy of the method. In the quantum cryptography project, we optimize an experimental implementation of a quantum cryptography system where two parties share information with the orbital angular momentum degree of freedom of entangled photon pairs. Finally, in the work on holography, we establish the exact solution to the encryption of a phase-only hologram, and experimentally demonstrate its application to spatial light modulators. The four projects provide improvements on measurement procedures in applied and fundamental quantum mechanics.

It is impossible to overstate the importance of symmetry in physics and mathematics. Symmetry arguments play a central role in a broad range of problems from simplifying a system of linear equations to a deep role in organizing the fundamental principles of physics. They are used, for instance, in Noether’s theorem to find the consequences of symmetry of a dynamics. For many systems of interest, the dynamics are sufficiently complicated that one cannot hope to characterize their evolution completely, whereas by appealing to the symmetries of the dynamical laws one can easily infer many useful results. In part I of this thesis we study the problem of finding the consequences of symmetry of a (possibly open) dynamics from an information-theoretic perspective. The study of this problem naturally leads us to the notion of asymmetry of quantum states. The asymmetry of a state relative to some symmetry group specifies how and to what extent the given symmetry is broken by the state. Characterizing these is found to be surprisingly useful to constrain which final states of the system can be reached from a given initial state. Another motivation for the study of asymmetry comes from the field of quantum metrology and relatedly the field of quantum reference frames. It turns out that the degree of success one can achieve in many metrological tasks depends only on the asymmetry properties of the state used for metrology. We show that some ideas and tools developed in the field of quantum information theory are extremely useful to study the notion of asymmetry of states and therefore to find the consequences of symmetry of an open or closed system dynamics. In part II of this thesis we present a novel application of symmetry arguments in the field of quantum estimation theory. We consider a family of multi-copy estimation problems wherein one is given n copies of an unknown quantum state according to some prior distribution and the goal is to estimate certain parameters of the given state. In particular, we are interested to know whether collective measurements are useful and if so to find an upper bound on the amount of entanglement which is required to achieve the optimal estimation. We introduce a new approach to this problem by considering the symmetries of the prior and the symmetries of the parameters to be estimated. We show that based on these symmetries one can find strong constraints on the amount of entanglement required to implement the optimal measurement. In order to infer properties of the optimal estimation procedure from the symmetries of the parameters and the prior we come up with a generalization of Schur-Weyl duality. Just as Schur-Weyl duality has many applications to quantum information theory and quantum algorithms so too does this generalization. In this thesis we explore some of these applications.

This thesis is aimed at making an inquiry into the theoretical roots of a dualistic approach used in the anaysis of state/market relation and at showing in what ways this approach dominates the current debates on globalization and the state. For the theoretical discussion, the works of Hayek, Weber and Polanyi / for the actual discussions, selected methodologies from the international political economy literature are examined. The study argues that two dominant approaches emerging in the political economy literature, which can be identified as &ldquo / state-centric&rdquo / and &ldquo / market forces-centric&rdquo / , do reproduce state/market duality and that an alternative perspective which may overcome this duality can only be elaborated within the framework of the form analysis.

Due to various corporate scandals and failures, there has been a renewed interest on the role of boards in the performance of firms. This thesis examines the relationship between the key board characteristics and firm performance. Unlike most studies on boards which predominantly use only financial variables affecting governance, I take a different approach by combining them with non-financial variables. This combined set of variables is used for theoretical and empirical modelling. Based on the extant literature, I develop a conceptual framework and a set of hypotheses to examine the relationship between board characteristics and firm performance. Board characteristics considered in this research include board size, director ownership, CEO duality, gender diversity, educational qualification of board members and number of board meetings. Additionally, I use board size as a moderating variable to examine how the effect of other board characteristics is contingent on board size. Firm performance is measured by return on assets. I test my hypotheses on a longitudinal sample of 156 firms over a four year period from 2004 to 2007. My sample includes all firms listed on New Zealand stock exchange as on November 2007. Empirical analysis is undertaken using Generalised Least Squares analyses. The findings of the study show that board characteristics such as board size, CEO duality and gender diversity were positively related with firm performance, where as director ownership, board meetings and the number of board members with PhD level education was found to be negatively related. Board size was found to be moderating some of these relationships, indicating the critical role being played by board size in the design and role of corporate boards. The findings also provide partial evidence to different governance theories, further indicating the need for theoretical pluralism to gain insights into boards’ functioning. The study contributes to the understanding of board-performance link by examining both the traditional variables such as board size, CEO duality, and number of board meetings as well as other organisational attributes such as gender diversity and competence variables represented by women and PhD holders, respectively. The theoretical framework and the findings of my thesis are expected to stimulate scholars for further research to identify the contingency conditions upon which the board characteristics and firm performance may be dependent.

[Truncated abstract. Formulae and special characters can only be approximated. See PDF version for accurate reproduction.] Theories of self-dual supersymmetric nonlinear electrodynamics are generalized to a curved superspace of 4D N = 1 supergravity, for both the old-minimal and the newminimal versions of N = 1 supergravity. We derive the self-duality equation, which has to be satisfied by the action functional of any U(1) duality invariant model of a massless vector multiplet, and show that such models are invariant under a superfield Legendre transformation. We construct a family of self-dual nonlinear models, which includes a minimal curved superspace extension of the N = 1 supersymmetric Born- Infeld action. The supercurrent and supertrace of such models are explicitly derived and proved to be duality invariant. The requirement of nonlinear self-duality turns out to yield nontrivial couplings of the vector multiplet to Kähler sigma models. We explicitly construct such couplings in the case when the matter chiral multiplets are inert under the duality rotations, and more specifically to the dilaton-axion chiral multiplet when the group of duality rotations is enhanced to SL(2,R). The component structure of the nonlinear dynamical systems introduced proves to be more complicated, especially in the presence of supergravity, as compared with well-studied effective supersymmetric theories containing at most two derivatives (including nonlinear Kähler sigma-models). As a result, when deriving their canonically normalized component actions, the traditional approach becomes impractical and cumbersome. We find it more efficient to follow the Kugo-Uehara scheme which consists of (i) extending the superfield theory to a super-Weyl invariant system; and then (ii) applying a plain component reduction along with imposing a suitable super-Weyl gauge condition. This scheme is implemented in order to derive the bosonic action of the SL(2,R) duality invariant coupling to the dilaton-axion chiral multiplet and a Kähler sigma-model.

Thesis (Ph. D.)--State University of New York at Binghamton, of Department of Mathematical Sciences, 2007. / Includes bibliographical references.

Thesis (M. S.)--Mathematics, Georgia Institute of Technology, 2009. / Committee Chair: Evans Harrell; Committee Member: Guillermo Goldsztein; Committee Member: Mohammad Ghomi