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For quantum gravity, the requirement of metric positivity suggests the use of noncanonical, affine kinematical field operators. In view of gravity\'s set of open classical first class constraints, quantization before reduction is appropriate, leading to affine commutation relations and affine coherent states. The anomaly in the quantized constraints may be accommodated within the projection operator approach, which treats first and second class quantum constraints in an equal fashion. Functional integral representations are derived for expressions both with and without constraint imposition. As with all coherent state formulations, close contact between the classical and quantum theories is maintained throughout. Perturbative nonrenormalizability is understood as a partial hard-core behavior of the interaction, which as soluble models suggest, leads to a perturbative formulation, not about the traditional free theory, but rather about a suitable pseudofree theory that properly incorporates the essence of the hard core.

In an effort to better understand the class of operations on a bipartite system which preserve positivity of partial transpose (PPT operations), we have investigated the (non-asymptotic) transformation of pure states to pure states by operations in this class. Under local operations and classical communication (LOCC) Nielsen\'s majorization criterion provides a necessary and sufficient condition for such a transformation. This can be used to show that under LOCC a phenomenon called catalysis can occur, where an otherwise impossible transformation can be made possible by the provision of an entangled catalyst state, which must be recovered unchanged after the transformation (hence the name). I will present some recent work where we have found a necessary condition for obtaining a given pure state from a maximally entangled state via PPT operations. This condition is conjectured to be sufficient also, and we can prove this for the case where the goal state has Schmidt rank three. We have also shown that catalysis occurs under PPT operations, and have derived a necessary and sufficient condition for PPT pure state transformations where both the initial state and the catalyst are maximally entangled.

Recent developments in the field of Numerical Relativity have not only provided key insights of binary black hole systems but also began influencing its future role. Undoubtedly one of the most important future drivers in the near future of the field will be its role as another element within the study of spectacular astrophysical phenomena involving strongly gravitation scenarios. Connecting (yet to be observed) gravitational waves with observations within the electromagnetic spectra will be one ultimate goal of this enterprise. This talk will illustrate this interplay within the study of magnetized binary neutron star systems as a stepping stone towards more ambitious goals.

The rich network of string dualities provides powerful constraints in the structure of the theory. The connection of ten-dimensional type II theories to eleven-dimensional supergravity compactified on a circle and on a torus allows one to compute many perturbative high genus terms as well as the complete sum of non-perturbative contributions to a given higher derivative coupling of the string effective action. The same duality connection leads to a series of surprising non-renormalization theorems. These control the UV behavior of maximal supergravity in lower dimensions, and indicate that N=8 supergravity in four dimensions might be a finite quantum gravity theory.

I will review relativistic quantum theory that is based on Wigner\'s unitary representations of the Poincare group, Dirac\'s forms of dynamics, and Newton-Wigner\'s definition of the position operator. Formulas will be derived that transform particle observables between different inertial reference frames. In the absence of interactions, these formulas coincide with Lorentz transformations from special relativity. However, when interaction is turned on, some deviations appear. The relationship of this result to the Currie-Jordan-Sudarshan \'no-interaction\' theorem will be mentioned, and the status of Lorentz transformations within quantum and classical theories of interacting particles will be discussed.

This talk presents some recent results in renormalizable noncommutative quantum field theory. After introducing the renormalization group approach in the commutative setting I will procede to its generalization to the simplest noncommutative model, $phi_4^{star 4}$ on the Moyal space. The well known phenomenon of ultraviolet/infrared mixing is cured by adding a harmonic potential term to the free action. Under the new renormalization group, adapted to the noncommutative geometry, this model turns out to be renormalizable to all orders in perturbation theory. Moreover it is {f asymptotically safe} at all orders in perturbation theory. The consequences of this results are discussed.

What does a fractional quantum Hall liquid and Kitaev\'s proposals for topological quantum computation have in common? It turns out that they are physical systems that exhibit degenerate ground states with properties seemingly different than ordinary (Landau-type) vacua, such as the ground states of a Heisenberg magnet. For example, those (topologically quantum ordered)states cannot be characterized by (local) order parameters such as magnetization. How does one characterize this new order? I will present a unifying framework which will allow us to engineer physical systems displaying topological quantum order. What are the physical properties of these new orders? How robust are they to temperature effects? What are they useful for? Topologically quantum ordered states of matter seem to be ideal physical systems to store and manipulate quantum information since they are believed to be robust against decoherence with an environment, and thus appropriate for building a quantum computer and quantum memories. I will discuss the role of temperature in the protection of quantum information. Have we finally found a technological application for quantum Hall liquids?