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This thesis project investigates two of the mechanisms that have been postulated to explain the origin of the molecular homochirality found in living systems. These mechanisms are i. the Yamagata-Rein Hypothesis, which predicts that the weak neutral currents between electrons and nucleons result in a Parity-Violating Energy Difference (PVED) between a pair of molecular enantiomers; and ii. the Vester-Ulbricht Hypothesis, which predicts that the β-particles produced during weak force mediated nuclear β-decay interact differently with each of a pair of molecular enantiomers. Both hypotheses predict a small enantiomeric excess and theorise that such an excess might have been the prebiotic chiral influence that led to homochirality, which is generally accepted to be a prerequisite for biology. It is found that those PVED computations carried out the coupled-perturbed level of theory are approximately one order of magnitude larger than those previously computed at the entry-level uncoupled-perturbed level of theory. The results computed as part of this thesis project include the first to apply Density-Functional theory to the computation of the PVED. The coupled-perturbed Hartree-Fock and Kohn-Sham computations of the PVED are found to be in good agreement with recent related computational studies. The hypersensitivity of the PVED to small conformational changes, coupled to the difficulties in knowing the aqueous phase structure of even small chiral biomolecules such as α-alanine, means that it is difficult to make any unambiguous conclusions regarding which of a pair of biomolecular enantiomers might have been favoured by the PVED in a probiotic regime. These results do indicate that the largest β-asymmetries are generated in the forward scattering regimes and that the sign of the asymmetries may change suddenly as the energy of the incoming β-particle increases so as to allow the removal of electrons from successive molecular orbitals during the impact ionisation of the molecule.

In the introductory chapters I review this problem and recall how in IIB compactifications the dilation and complex structure moduli can be stabilised by 3-form fluxes. There exist very many possible discrete flux choices which motivates the use of statistical techniques to analyse this discretuum of choices. Such approaches generate formulae predicting the distribution of vacua and I describe numerical tests of these formulae on the Calabi-Yau ℙ⁴_[1,1,2,2,6]. Stabilising the Kähler moduli requires nonperturbative superpotential effects. I review the KKLT construction and explain why this must in general be supplemented with perturbative Kähler corrections. I show how the incorporation of such corrections generically leads to non-supersymmetric minima at exponentially large volumes, giving a detailed account of the α’ expansion and its relation to Kähler corrections. I illustrate this with explicit computations for the Calabi-Yau ℙ⁴_[1,1,1,6,9]. The next part of the thesis examines phenomenological applications of this construction. I first describe how the magnitude of the soft supersymmetry parameters may be computed. In the large-volume models the gravitino mass and soft terms are volume-suppressed. As we naturally have <i>V</i> >>1, this gives a dynamical solution of the hierarchy problem. I also demonstrate the existence of a fine structure in the soft terms, with gaugino masses naturally lighter than the gravitino mass by a factor 1n. A second chapter gives a detailed analysis of the relationship of moduli stabilisation to the QCD axions relevant to the strong CP problem, proving a no-go theorem on the compatibility of a QCD axion with supersymmetric moduli stabilisation. I describe how QCD axions can coexist with nonsupersymmetric perturbative stabilisation and how the large-volume models naturally contain axions with decay constants that are phenomenologically allowed and satisfy the appealing relationship <i>f</i>²<i>_a</i>~ <i>M_PM_susy. </i>A further chapter describe how a simple and predictive inflationary model can be built in the context of the above large-volume construction, using the no-scale Kähler potential to avoid the <i>η</i> problem.

It is a common conjecture that all convex polyhedra must be edge-unfoldable but to date a valid proof of this has escaped discovery. This dissertation presents several new directions in the quest for the proof. Also discussed is a method which may lead to a counterexample to the conjecture through the construction of ‘hard to unfold’ polyhedra. Algorithmic solutions are discussed for the task of determining the specific set of edges which must be cut in order that an unfolding not self-intersect. A series of <i>Unfolder </i>algorithms are explored and compared, in terms of both algorithmic design and empirical performance on test data. No surface of uniformly negative internal curvature with fewer than two border curves is unfoldable. The <i>coolinoids </i>are a class of non-convex polyhedra having exactly two border curves and negative curvature at every internal vertex, which may be constructed so as to be unfoldable without overlap. The fascinating interaction between construction and overlap in coolinoids is modelled and explored.

Bayesian inference offers us a powerful tool with which to tackle the problem of data modelling. However, the performance of Bayesian methods is crucially dependent on being able to find good models for our data. The principal focus of this thesis is the development of models based on Gaussian process priors. Such models, which can be thought of as the infinite extension of several existing finite models, have the flexibility to model complex phenomena while being mathematically simple. In this thesis, I present a review of the theory of Gaussian processes and their covariance functions and demonstrate how they fit into the Bayesian framework. The efficient implementation of a Gaussian process is discussed with particular reference to approximate methods for matrix inversion based on the work of Skilling (1993). Several regression problems are examined. Non-stationary covariance functions are developed for the regression of neuron spike data and the use of Gaussian processes to model the potential energy surfaces of weakly bound molecules is discussed. Classification methods based on Gaussian processes are implemented using variational methods. Existing bounds (Jaakkola and Jordan 1996) for the sigmoid function are used to tackle binary problems and multi-dimensional bounds on the softmax function are presented for the multiple class case. The performance of the variational classifier is compared with that of other methods using the CRABS and PIMA datasets (Ripley 1996) and the problem of predicting the cracking of welds based on their chemical composition is also investigated. The theoretical calculation of the density of states of crystal structures is discussed in detail. Three possible approaches to the problem are described based on free energy minimization, Gaussian processes and the theory of random matrices. Results from these approaches are compared with the state-of-the-art techniques (Pickard 1997).

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In this thesis we consider some aspects of two dimensional Conformal Field Theory and their connection to String Theory. We have also studied some aspects of supersymmetry of M-Theory on Ricci-flat seven manifolds with 4-form fluxes. We concentrate mainly on certain supersymmetric extensions of the coset models due to Goddard, Kent and Olive (GKO). These models are known as the Kazama-Suzuki (KS) models and they are characterized by their <i>N</i> = 2 superconformal symmetry. Two series of the KS models enjoy a duality called level-rank duality which can be described roughly as duality between the dimension of the target space and the level of coset. We believe that the path-integral approach is the closest in spirit to string theory. Therefore, we formulate the level-rank duality of KS models in the path-integral approach by using the realization of GKO cosets as gauged Wess-Zumino-Novikov-Witten (gauged-WZNW) models. We derive, for a class of KS models, an expression for the partition function which is symmetric in the parameters of the level-rank duality. We compute the central charge of the models from this expression which matches that of Kazama and Suzuki in the operator approach. We then work out the target space metric and the dilation of the gauged-WZNW model based on the GKO coset SU(3)/(SU(2) x U(1)). It turns out to be quite a complicated metric with a non-trivial dilation. We verify, as a consistency check, that they satisfy the appropriate string theory effective equations of motion. We then argue that this background can arise naturally in type II string theory compactified down to <i>AdS₃</i> space. We then turn to Eleven Dimensional Supergravity which is the low energy limit of M-theory. We adopt a metric ansatz which is a warped product of four dimensional Minkowski space and a (non-compact) seven manifold with 4-form fluxes turned on it. We derive the condition for unbroken supersymmetry with fluxes and non-trivial warp-factor. We show that the same condition implies that the seven manifold is conformal to a Ricci-flat manifold. We also point out the limitation of some naive ansatze about the structure of the Killing spinor. At this stage we are unable to give an explicit solution to the supersymmetry condition.

After reviewing previously published techniques, a new algorithm is presented for optimising variable parameters in explicitly correlated many-body trial wavefunctions for use in variational quantum Monte Carlo (VMC) and diffusion quantum Monte Carlo (DMC) calculations. The method optimises the parameters with respect to the VMC energy by extending a low-noise VMC implementation of diagonalisation to include parameters which affect the wavefunction to higher than first-order. Similarly to minimising the variance of the local energy by fixed-sampling, accurate results are achieved using a relatively small number of VMC configurations because the optimisation is based on a least-squares fitting procedure. The method is tested by optimising six small examples intended to broadly cover the range of systems and wavefunctions typically treated using VMC and DMC, including atoms, molecules, and extended systems. Least-squares energy minimisation is found to be stable, fast enough to be practical, and capable of achieving lower VMC energies than minimisation of the filtered underweighted variance of the local energy (and the underweighted mean absolute deviation from the median local energy) by fixed-sampling. Least-squares energy minimisation is used to optimise four different wavefunctions for each of the all-electron first row atoms, from lithium to neon: single-determinant Slater-Jastrow wavefunctions with and without backflow transformations, and multi-determinant Slater-Jastrow wavefunctions with and without backflow transformations. The optimisations are more stable and successful than some previous variance minimisations using similar wavefunctions. The DMC energies of the energy-optimised wavefunctions for the atoms from boron to neon are significantly lower than previously published results, and, using the multi-determinant Slater-Jastrow wavefunctions with backflow, the calculations recover at least 90% of the correlation energies for lithium, beryllium, boron, carbon, nitrogen and neon, 97% for oxygen, and 98% for fluorine.

In the first part of this thesis, a new method is proposed to circumvent the relaxation bottleneck that prevents high-temperature Bose-Einstein condensation of microcavity polaritons. This is achieved by relaxing polaritons with a coherent beam of phonons that is pumped in the growth direction of the quantum well. By tuning the frequency of these phonons to the energy difference between the bottleneck distribution and the ground state, resonant scattering relaxes the exciton reservoir very efficiently. Within a simple rate equation model, it is shown that above a threshold phonon intensity, an unstable and faster-than-exponential instability of the ground state population occurs, with macroscopic occupation possible on sub-picosecond timescales. Numerical results for GaAs and CdTe are presented, confirming that this method could be highly effective in alleviating the bottleneck effect. The second part addresses the problem of decoherence in the sub-Ohmic spin boson model. This previously unimportant model has recently attracted attention due to the discovery of several new physical realisations. Using Silbey and Harris’ variational method, a zero-temperature phase transition between coherent and incoherent ground states is found. The critical spin-bath coupling is extracted, and found to agree well with numerical calculations. Calculations at finite temperatures also yield transitions between coherent and incoherent spin dynamics at a critical temperature. Considering the dynamics of the variational Hamiltonian, it is shown that fluctuations of the bath lead to damping of the coherent spin-dynamics. Approaching the incoherence transition, these dynamics become highly non-Markovian. This chapter also contains a formal demonstration that an alternative variational state, the displaced-squeezed state (DSS), leads to incorrect results in the thermodynamic limit.