Abstract:
We propose a geometric method to parameterize inequivalent vacua by dynamical data. Introducing quantum Clifford algebras with arbitrary bilinear forms we distinguish isomorphic algebras --as Clifford algebras-- by different filtrations resp. induced gradings. The idea of a vacuum is introduced as the unique algebraic projection on the base field embedded in the Clifford algebra, which is however equivalent to the term vacuum in axiomatic quantum field theory and the GNS construction in C^*-algebras. This approach is shown to be equivalent to the usual picture which fixes one product but employs a variety of GNS states. The most striking novelty of the geometric approach is the fact that dynamical data fix uniquely the vacuum and that positivity is not required. The usual concept of a statistical quantum state can be generalized to geometric meaningful but non-statistical, non-definite, situations. Furthermore, an algebraization of states takes place. An application to physics is provided by an U(2)-symmetry producing a gap-equation which governs a phase transition. The parameterization of all vacua is explicitly calculated from propagator matrix elements. A discussion of the relation to BCS theory and Bogoliubov-Valatin transformations is given.

Abstract:
Some examples of Type-I vacua related to non geometric orbifolds are shown. In particular, the open descendants of the diagonal $Z_3$ orbifold are compared with the geometric ones. Although not chiral, these models exhibit some interesting properties, like twisted sectors in the open-string spectra and the presence of ``quantized'' geometric moduli, a key ingredient to ensure their perturbative consistency and to explain the rank reduction of their Chan-Paton groups.

Abstract:
Region covariance descriptor recently proposed has been approved robust and elegant to describe a region of interest, which has been applied to visual tracking. We develop a geometric method for visual tracking, in which region covariance is used to model objects appearance; then tracking is led by implementing the particle filter with the constraint that the system state lies in a low dimensional manifold: affine Lie group. The sequential Bayesian updating consists of drawing state samples while moving on the manifold geodesics; the region covariance is updated using a novel approach in a Riemannian space. Our main contribution is developing a general particle filtering-based racking algorithm that explicitly take the geometry of affine Lie groups into consideration in deriving the state equation on Lie groups. Theoretic analysis and experimental evaluations demonstrate the promise and effectiveness of the proposed tracking method.

Abstract:
Region covariance descriptor recently proposed has been approved robust and elegant to describe a region of interest, which has been applied to visual tracking. We develop a geometric method for visual tracking, in which region covariance is used to model objects appearance; then tracking is led by implementing the particle filter with the constraint that the system state lies in a low dimensional manifold: affine Lie group. The sequential Bayesian updating consists of drawing state samples while moving on the manifold geodesics; the region covariance is updated using a novel approach in a Riemannian space. Our main contribution is developing a general particle filtering-based racking algorithm that explicitly take the geometry of affine Lie groups into consideration in deriving the state equation on Lie groups. Theoretic analysis and experimental evaluations demonstrate the promise and effectiveness of the proposed tracking method.

Abstract:
This work summarises some of the attempts to explain the phenomenon of dark energy as an effective description of complex gravitational physics and the proper interpretation of observations. Cosmological backreaction has been shown to be relevant for observational (precision) cosmology, nevertheless no convincing explanation of dark energy by means of backreaction has been given so far.

Abstract:
The Tomita-Takesaki modular groups and conjugations for the observable algebras of space-like wedges and the vacuum state are computed for translationally covariant, but possibly not Lorentz covariant, generalized free quantum fields in arbitrary space-time dimension d. It is shown that for $d\geq 4$ the condition of geometric modular action (CGMA) of Buchholz, Dreyer, Florig and Summers \cite{BDFS}, Lorentz covariance and wedge duality are all equivalent in these models. The same holds for d=3 if there is a mass gap. For massless fields in d=3, and for d=2 and arbitrary mass, CGMA does not imply Lorentz covariance of the field itself, but only of the maximal local net generated by the field.

Abstract:
This thesis is a review of backreaction, which is a possible alternative to dark energy and modified gravity. We describe in detail the 3+1 covariant formalism and Frobenius' theorem. We present the averaging procedure developed by Buchert, the Buchert equations and the generalization of these equations to the case of general matter. We then generalize to arbitrary number of spatial dimensions. We focus on the case of 2+1 dimensions, where the relation between the topology and the geometry of a surface imposes a global constraint on backreaction.

Abstract:
We present the Bayesian analysis of four different types of backreation models, which are based on the Buchert equations. In this approach, one considers a solution to the Einstein equations for a general matter distribution and then an average of various observable quantities is taken. Such an approach became of considerable interest when it was shown that it could lead to agreement with observations without resorting to dark energy. In this paper we compare the LambdaCDM model and the backreation models with SNIa, BAO, and CMB data, and find that the former is favoured. However, the tested models were based on some particular assumptions about the relation between the average spatial curvature and the backreaction, as well as the relation between the curvature and curvature index. In this paper we modified the latter assumption, leaving the former unchanged. We find that, by varying the relation between the curvature and curvature index, we can obtain a better fit. Therefore, some further work is still needed -- in particular the relation between the backreaction and the curvature should be revisited in order to fully determine the feasibility of the backreaction models to mimic dark energy.

Abstract:
We consider a Metropolis-Hastings method with proposal kernel $\mathcal{N}(x,hG^{-1}(x))$, where $x$ is the current state. After discussing specific cases from the literature, we analyse the ergodicity properties of the resulting Markov chains. In one dimension we find that suitable choice of $G^{-1}(x)$ can change the ergodicity properties compared to the Random Walk Metropolis case $\mathcal{N}(x,h\Sigma)$, either for the better or worse. In higher dimensions we use a specific example to show that judicious choice of $G^{-1}(x)$ can produce a chain which will converge at a geometric rate to its limiting distribution when probability concentrates on an ever narrower ridge as $|x|$ grows, something which is not true for the Random Walk Metropolis.

Abstract:
We investigate consequences of allowing the Hilbert space of a quantum system to have a time-dependent metric. For a given possibly nonstationary quantum system, we show that the requirement of having a unitary Schreodinger time-evolution identifies the metric with a positive-definite (Ermakov-Lewis) dynamical invariant of the system. Therefore the geometric phases are determined by the metric. We construct a unitary map relating a given time-independent Hilbert space to the time-dependent Hilbert space defined by a positive-definite dynamical invariant. This map defines a transformation that changes the metric of the Hilbert space but leaves the Hamiltonian of the system invariant. We propose to identify this phenomenon with a quantum mechanical analogue of the principle of general covariance of General Relativity. We comment on the implications of this principle for geometrically equivalent quantum systems and investigate the underlying symmetry group.