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A classical Hamiltonian system can be reduced to a subsystem of "relevant observables" using the pull-back under a Poisson embedding of the "relevant phase space" into the "full phase space". Since a quantum theory can be thought of a noncommutative phase space, one encounters the problem of the embedding of noncommutative spaces, when one tries to extend the reduction via a pull-back to a quantum theory. This problem can be solved for a class of physically interesting quantum systems and embeddings using an analogy to finding the base space of an embedded fibre bundle via the projection in the full fibre bundle. The resulting construction is then applied to Loop Quantum Gravity to extract a cosmological sector. This sector turns out to be similar but not equivalent to Loop Quantum Cosmology.

The description of noncommutative space will be given. I will show the relation between field theory on kappa-Minkowski space and the one in Minkowski. This construction leads to deformed energy momentum conservation law for energies close to the Planck scale.

At large scales the CMB spectrum measured by WMAP appears to have an anomalously low power of the quadrupole and an asymmetry of power at l < 40. We show that with an initial stage of fast roll of the inflation and a gradient in the initial conditions a simple chaotic inflation model may be capable of explaining both anomalies.

I consider a six dimensional space-time, in which two of the dimensions are compactified by a flux. Matter can be localized on a codimension one brane coupled to the bulk gauge field and wrapped around an axis of symmetry of the internal space. By studying the linear perturbations around this background, I show that the gravitational interaction between sources on the brane is described by Einstein 4d gravity at large distances. This is one of the first complete study of gravity in a realistic brane model with two extra dimensions, in which the mechanism of stabilization of the extra space is fully taken into account.

I discuss two instances in which nonlinear perturbations in cosmological models are important. First, in de Sitter space-time, the bare necessity that the perturbations should be part of a consistent Taylor expansion of the field equations leads to the requirement, using the 'linearization stability' arguments of the '70's, that the quantum field theory of a scalar field on de Sitter space-time is manifestly de Sitter invariant (not covariant). Second, the concern that in slow-roll inflation the effect of second order perturbations on the long wavelength (super Hubble) perturbations could be much stronger than that of the first order perturbations, for a wide range of slow-roll conditions, is explored in the context of a linear inflation potential and chaotic inflation.

We argue that four-dimensional quantum gravity may be essentially renormalizable provided one relaxes the assumption of metricity of the theory. We work with Plebanski formulation of general relativity in which the metric (tetrad), the connection as well as the curvature are all independent variables and the usual relations among these quantities are only on-shell. One of the Euler-Lagrange equations of this theory guarantees its metricity. We show that quantum corrections generate a counterterm that destroys this metricity property, and that there are no other counterterms, at least at the one-loop level. There is a new coupling constant that controls the non-metric character of the theory. Its beta-function can be computed and is negative, which shows that the non-metricity becomes important in the infra red. The new IR-relevant term in the action is akin to a curvature dependent cosmological ``constant\'\' and may provide a mechanism for naturally small ``dark energy\'\'.

We explore the role of rotational symmetry of quantum key distribution (QKD) protocols in their security. Specifically, in the first part of the talk, we consider a generalized QKD protocol with discrete rotational symmetry. Note that, before our work, each QKD protocol seems to have a different security proof. Given that the techniques of those proofs are similar, it will be interesting to have a unified proof for QKD protocols with symmetry (e.g., the BB84 protocol and the SARG04 protocol). This is exactly what we achieve in our work. We show that rotational symmetry plays an important role in the unified security proof of QKD protocols with symmetry, leading to simple and structural security relations. In the second part, we consider a QKD protocol that does not possess rotational symmetry and analyze its security. Interestingly, even without any rotational symmetry, this protocol can still be proven secure. However, the security relation is not as simple as those in the first part, due to the lack of symmetry. Therefore, although rotational symmetry is not required in a QKD protocol to ensure its security, rotational symmetry does provide significant simplification in the security analysis, leading to simple security relations.

A nonrotating black hole placed in a tidal environment (that is, subjected to the gravitational interactions produced by other nearby bodies) is not described by the Schwarzschild solution to the Einstein field equations. Instead, its metric is given by a perturbed version of this exact solution, and the spacetime is no longer stationary nor spherically symmetric. After reviewing the situation in Newtonian theory, I shall describe how the metric of a tidally distorted black hole is calculated. Special attention will be placed on the general description of the tidal environment, the choice of a good coordinate system to describe the perturbed black hole, and the consequences on the structure of the event horizon