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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.

A quantum channel models a physical process in which noise is added to a quantum system via interaction with its environment. Protecting quantum systems from such noise can be viewed as an extension of the classical communication problem introduced by Shannon sixty years ago. A fundamental quantity of interest is the quantum capacity of a given channel, which measures the amount of quantum information which can be protected, in the limit of many transmissions over the channel. In this talk, I will show that certain pairs of channels, each with a capacity of zero, can have a strictly positive capacity when used together, implying that the quantum capacity does not completely characterize a channel\'s ability to transmit quantum information. As a corollary, I will show that a commonly used lower bound on the quantum capacity - the coherent information, or hashing bound - is an overly pessimistic benchmark against which to measure the performance of quantum error correction because the gap between this bound and the capacity can be arbitrarily large.

According to the second law of thermodynamics the entropy of a system cannot decrease by adiabatic state transformations. In quantum mechanics, the \'degree of entanglement\' of a state cannot increase under state transformations of a certain kind (local operations assisted by classical communication) In this talk I will explore the significance of the analogy between these two statements.

After using the complex Hilbert space formalism for quantum theory for so long, it is very easy to begin to take for granted features like projection operators and the projection postulate, the algebra of observables, symmetric transition probabilities, linear evolution, etc.... Over the past 50 years there have been many attempts to gain a better understanding of this formalism by reconstructing it from different kinds of (sometimes) physically motivated assumptions. By looking at how the above features are motivated and used in different reconstructions, it becomes clear just how special and restrictive many of them are. The question is then what a theory which does not have some of these features looks like. Another interesting question is whether there are any reasons to be suspicious of postulating them in reconstructions or when trying to generalize or apply the quantum formalism to untested situations.

The QUaD experiment has recently released CMB polarization results at el>200 which are the most sensitive to date. The predicted series of peaks in the EE spectrum are shown to be present for the first time while BB remains undetectable. After briefly reviewing the motivation for polarization measurements I will move on to the experiment, observations, analysis technique and the final results. Finally I will mention on-going efforts to detect gravitational wave B modes.

There are many results showing that the probability of entanglement is high for large dimensions. Recently, Arveson showed that the probability of entanglement is zero when the rank of a bipartite state is no larger than half the dimension of the smaller space. We show that that the probability of entanglement is zero when the rank of a bipartite state is no larger than half the maximum of the rank of its two reduced density matrices. Our approach is quite different from that of Arveson and uses a different measure. But both approaches show that the separable states lie in a lower dimensional manifold given a reasonable parameterization of the separable states. This is joint work with Elisabeth Werner, using on characterizations of the extreme points of qubit channels given by Ruskai, Szarek and Werner and the extreme points of entanglement breaking channels given by M. Horodecki, Shor and Ruskai.

During the last two decades Alain Connes developed Noncommutative Geometry, which allows to unify two of the basic theories of modern physics: General Relativity and the Standard Model of Particle Physics. In the noncommutative framework the Higgs boson, which had previously to be put in by hand, and many of the ad hoc features of the standard model, appear in a natural way. The aim of my talk is to motivate this unification from basic physical principles and to give a flavour of its derivation. I will give an overview of the basic tools such as almost-commutative spectral triples and the spectral action principle. The latter allows to derive the Einstein-Hilbert Lagrangian and the Standard Model Lagrangian together with a set of relations among the Standard Model parameters.

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.