We construct the moduli spaces of stable maps, Mg,n(pr, d), via geometric invariant theory (GIT). This construction is only valid over Spec C, but a special case is a GIT presentation of Mg,n, which is valid over any algebraically closed field. Our method follows that of Gieseker in [G] and Swinarski in [Sw2], though our proof that the semistable set is nonempty is different.

The differences between classical and quantum mechanics were highlighted early in the development of quantum mechanics when Schrodinger proposed the thought experiment of a cat in a superposition of alive and dead. In this thesis I try to understand these differences by considering superpositions of large objects at a single particle level. Research in the field of superconductors has provided evidence for macroscopic quantum superpositions (or cat states) of currents in superconducting loops. Bose-Einstein condensates of ultracold atoms provide another promising system for experimentally producing similar results. I begin by describing two straightforward schemes that make macroscopic superpositions of superfluid flow states of Bose-Einstein condensates trapped in optical lattice rings. The first scheme achieves a superposition of three flow states by nonadiabatically evolving the barrier heights between the sites. The second scheme produces a superposition of two flow states by applying a 7f phase around the ring. This could be experimentally achieved by physically rotating the sites or imparting angular momentum from two co-propagating lasers. The next part of the thesis investigates why it is difficult to produce macroscopic superpositions. By treating the interaction strength between the atoms as a perturbation I show three reasons, other than decoherence, why macroscopic superpositions are hard to make. Firstly, the energy of the two distinct flow states must be sufficiently close. Secondly, coupling between the two states must be sufficiently strong, and thirdly, other states must be well separated from those two flow states. To make larger superpositions I look at a Josephson junction coupled to a superfluid loop. This shows that making superpositions depends on the number of atoms in the junction rather than the whole system. Finally I propose ways of developing the work. This concentrates on how the systems could experimentally create macroscopic superpositions and how we could measure signatures of these states. I then suggest ways of using the systems, such as quantum information and precision measurement schemes.

In this thesis, we explore two approaches to string phenomenology. In the first half of the work, we investigate compactifications of M-theory on spaces with co-dimension four, orbifold singularities. We construct M-theory on <e2j71N by coupling ll-dimensional supergravity to a seven-dimensional Yang-Mills theory located on the orbifold fixed-plane. The resulting action is supersymmetric to leading non-trivial order in the ll-dimensional Newton constant. Vle thereby reduce M-theory on a G2 orbifold with <e2 j71N singularities, explicitly incorporating the additional gauge fields ?-t the singularities. We derive the Kahler potential, gauge-kinetic function and super:potential for the resulting N = 1 four-dimensional theory. Blowing-up of the orbifold is described by a Higgs effect induced by continuation along D-flat directions. Using this interpretation, we show our results are consistent with the corresponding ones obtained for smooth G2 spaces. Finally, we consider switching on flux and Wilson lines on singular loci of the G2 space, and discuss the relation to N = 4 gYM theory. In the second half, we develop an algorithmic framework for Es x Es heterotic compactifications with monad bundles, including new and efficient techniques for proving stability and calculating particle spectra. We begin by considering cyclic Calabi-Yau manifolds where we classify positive monad bundles, prove stability, and compute the complete particle spectrum for all bundles. Next, we generalize the construction to bundles on complete intersection Calabi-Yau manifolds. \Ve show that the class of positive monad bundles, subject to the heterotic anomaly condition, is finite (rv 7000 models). We compute the particle spectrum for these models and develop new techniques for computing the cohomology of line bundles on CICYs. There are no anti-generations of particles and the spectrum is manifestly moduli-dependent. VVe further investigate the slope-stability of positive monad bundles and develop a new method for proving stability of SU(n) vector bundles on CICYs

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The main aim of this thesis is to investigate non-Einsteinian interactions in a scalartensor theory and a tensor-tensor theory of gravity with torsion. We first explore perfect fluid and spinning particle dynamics in a scalar-tensor theory of gravity with scalar field interactions, and derive equations of motion for a charged perfect fluid both from gauge identities and a variational principle in background nonRiemannian spacetime (metric compatible connection with torsion), a scalar field and an electrom(l.gnetic field. For a spinning particle, we use gauge identities with given source currents to obtain its equations of motion with scalar field interactions, and solve its equations of motion in two different backgrounds: one is a BransDicke torsion field and the other is a constant pseudo-Riemannian curvature with constant scalar field and zero torsion. Moreover, we calculate the precession of a gyroscope moving along abound orbit in the weak limit of a vacuum Kerr-BransDicke solution with torsion. In Chapter 4, the equations of motion for massive spinless particles in a tensor-tensor theory of gravity with torsion are investigated. We first apply the perturbation scheme to the system of field equations and discover a perturbed torsion wave solution. Furthermore, we obtain gauge transformations of perturbed field variables and examine the polarizations of this torsion wave solution from autoparallel deviation. The longitudinal modes of the torsion wave polarization has been found. The polarizations (both transverse and longitudinal modes) of the torsion wave are quite different from the gravitational wave in the linearized GR.

In 1964, John Bell showed that a local hidden variables model cannot accommodate all the statistical predictions of quantum theory. This incompatibility was demonstrated by way of the negation of an inequality which put bounds on the expectation values of experimental outcomes. This inequality is known as Bell's inequality, its generalizations are known as Bell-type inequalities, and a Bell or Belltype experiment is a quantum experiment that tests for the violation of such an inequality. . According to the cl)lssical theory of stochastic input-output systems in spacetime, a quantum experiment may be considered as a black-box that has input and output ports distributed along its spacetime boundary and which is at rest in the laboratory frame. For each run of an experiment there are classical input events, i, and classical output events, j, which represent the settings of the devices that determine what quantum variable is to be measured and the outcomes of a quantum measurement respectively. In this thesis, quantum experiments of the Bell-type are considered within this framework. For such stochastic input-output systems, the relation between the inputs and outputs is defined in terms of the measurable conditional probabilities !P(jli), which depend on the probabilities !PCF) of local transfer functions F We define weak nonlocality, or nonlocality in the sense of Bell, as astatistical property of the classical inputs and outputs for which it is impossible to express the conditional probabilities in terms of local transfer functions alone. At least one nonlocal transfer function may have to have nonzero probability. We show that for a given Bell-type experiment, the corresponding Bell-type inequalities are in fact the facet-defining inequalities of a polytope which is given as the convex hull of a finite set of local deterministic systems. This is the Bell-polytope, which, in general, has many facets. Using BellTest, a computer program which we designed for the analysis of such experiments, we find the Bell-type inequalities for ideal and realistic Bell-type experiments which have outputs excluded in the ideal setup.

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