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According to general relativity, space-time ends at singularities and classical physics just stops. In particular, the big bang is regarded as The Beginning. However, general relativity is incomplete because it ignores quantum effects. Through simple models, I will illustrate how the quantum nature of space-time geometry resolves the big bang singularity. Quantum physics does not stop there. Indeed, quantum space-times can be vastly larger than what general relativity had us believe, with unforeseen physical effects in the deep Planck regime.

Quantum entanglement has two remarkable properties. First, according to Bell\'s theorem, the statistical correlations between entangled quantum systems are inconsistent with any theory of local hidden variables. Second, entanglement is monogamous -- that is, to the degree that A and B are entangled with each other, they cannot be entangled with any other systems. It turns out that these properties are intimately related.

Understanding dynamics of strongly coupled quantum field theories is an important problem in both condensed matter physics and high energy physics. In condensed matter systems, interacting quantum field theories can arise either at a critical point, or in a finite region of a parameter space. In the former case, massless modes arise as a result of fine tuning of external parameters, while, in the latter case, massless modes are protected by topology and/or symmetry. In this talk, I will discuss two examples in 2+1 dimensions (one for each case) where one can understand strong coupling physics nonperturbatively. In the first example, a lattice model which describes a superconducting phase transition will be discussed. In this model, a superconformal symmetry dynamically emerges at the quantum critical point and one can predict non-trivial critical exponents using the enlarged symmetry, even though there is no underlying supersymmetry in the microscopic model. In the second example, I will discuss about a 2+1 dimensional non-relativistic quantum field theory which is dual to a gravitational theory in the AdS4 background with a charged black hole. The spectral function of a fermion field exhibits an interesting non-Fermi liquid behavior, that is, all momentum points inside the Fermi surface are critical and the gapless modes are defined in a critical Fermi ball in the momentum space.

We all know that the EPR argument fails, and we can all provide proofs of one sort or another that it can\'t work. But in spite of this, there\'s something curiously tempting about the reasoning, and the temptation sometimes leads to needless perplexity about other issues. This paper will do two things. It will offer a diagnosis of where the EPR argument goes wrong that shows why we should be suspicious long before we get to Bell-type results, and then use the thought behind this diagnosis to suggest an orientation toward thinking about quantum states. The proposal for understanding states will have some things in common with Bayesian approaches, but will part company with them on some crucial points.

I will review recent progress in testing with cosmological data the inflationary hypothesis for describing the very early universe. I will present snapshots of different aspects of confronting the theory with data, including a \'bottom-up\' approach: the latest results from a systematic reconstruction of the inflationary dynamics; and a \'top- down\' approach: testing specific string theoretic constructions that attempt to implement inflation, while predicting distinctive observables not found in simple field-theory models. I will discuss the ambiguities inherent in attempting to quantify generic predictions of the inflationary \'paradigm\' (as opposed to the predictions of specific models). Finally, I will discuss (in a manner accessible to theoreticians) the astrophysical complexities underlying an observational program to look for primordial tensor modes that will discriminate between inflation and alternative theories.

Recent PAMELA and ATIC results may represent a breakthrough in dark matter searches beyond its gravitational imprint. After briefly reviewing the possible (classes of) explanations for the observed excesses in positron and electron cosmic ray fluxes I will focus on a two component dark matter model that may provide an explanation for large boost factors needed in the dark matter annihilation interpretation of the signals.

It has long been thought that theories based on equations of motion possessing derivatives of order higher than second are not unitary. Specifically, they are thought to possess unphysical ghost states with negative norm. However, it turns out that the appropriate Hilbert space for such theories had not been correctly constructed, and when the theory is formulated properly [Bender and Mannheim, PRL 100, 110402 (2008). (arXiv:0706.0207 [hep-th]] there are no ghost states at all and time evolution is fully unitary. Unitarity can be established for theories based on both second and fourth order derivatives, and for theories based on fourth order derivatives alone. In this latter case the Hamiltonian is a non-diagonalizable, Jordan-block operator which possesses fewer eigenstates than eigenvalues. Despite the lack of completeness of the energy eigenstates, a consistent, unitary quantum mechanics for the theory can still be formulated [Bender and Mannheim, PRD 78, 025022 (2008). (arXiv:0807.2607 [hep-th]).] The implications of these results for the construction of a consistent theory of quantum gravity in four spacetime dimensions will be briefly discussed.

As well known, cosmic ray experiments can put strong constraints on possible Lorentz Invariance Violations. In particular, the presence of the so called GZK \'cut-off\' may indicate that protons do propagate in the Universe as expected from relativistic invariance. The presence of this feature in the spectrum has been convincingly indicated by the HiRes and Auger experiments, while the Auger Observatory has given indication on the correlation of Ultra High Energy Cosmic particles with nearby sources, as predicted by the GZK feature. I review the experimental results and discuss in particular both the theoretical and experimental intricacies and the limits on LIV parameters that can be deduced.

This course is aimed at advanced undergraduate and beginning graduate students, and is inspired by a book by the same title, written by Padmanabhan. Each session consists of solving one or two pre-determined problems, which is done by a randomly picked student. While the problems introduce various subjects in Astrophysics and Cosmology, they do not serve as replacement for standard courses in these subjects, and are rather aimed at educating students with hands-on analytic/numerical skills to attack new problems.

This course is aimed at advanced undergraduate and beginning graduate students, and is inspired by a book by the same title, written by Padmanabhan. Each session consists of solving one or two pre-determined problems, which is done by a randomly picked student. While the problems introduce various subjects in Astrophysics and Cosmology, they do not serve as replacement for standard courses in these subjects, and are rather aimed at educating students with hands-on analytic/numerical skills to attack new problems.