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Low-temperature phases of strongly-interacting quantum many-body systems can exhibit a range of exotic quantum phenomena, from superconductivity to fractionalized particles. One exciting prospect is that the ground or low-temperature thermal state of an engineered quantum system can function as a quantum computer. The output of the computation can be viewed as a response, or 'susceptibility', to an applied input (say in the form of a magnetic field). For this idea to be sensible, the usefulness of a ground or low-temperature thermal state for quantum computation cannot be critically dependent on the details of the system's Hamiltonian; if so, engineering such systems would be difficult or even impossible. A much more powerful result would be the existence of a robust ordered phase which is characterised by its ability to perform quantum computation. I'll discuss some recent results on the existence of such a quantum computational phase of matter. I'll outline some positive results on a phase of a toy model that allows for quantum computation, including a recent result that provides sufficient conditions for fault-tolerance. I'll also introduce a more realistic model of antiferromagnetic spins, and demonstrate the existence of a quantum computational phase in a two-dimensional system. Together, these results reveal that the characterisation of quantum computational matter has a rich and complex structure, with connections to renormalisation and recently-proposed concepts of 'symmetry-protected topological order'.

Classical constraints come in various forms: first and second class, irreducible and reducible, regular and irregular, all of which will be illustrated. They can lead to severe complications when classical constraints are quantized. An additional complication involves whether one should quantize first and reduce second or vice versa, which may conflict with the axiom that canonical quantization requires Cartesian coordinates. Most constraint quantization procedures (e.g., Dirac, BRST, Faddeev) run into difficulties with some of these issues and may lead to erroneous results. The Projection Operator Method involves no gauge fixing, no auxiliary variables of any kind, and can treat simultaneously any and all kinds of constraints. It also admits a phase space path integral formulation with similar features.

I will discuss three ways in which (the string landscape and) eternal inflation is fun: (1) because it motivates revisiting some beautiful, classic calculations; (2) because its global description requires asking novel questions with possible broad ramifications; and (3) because it leads to experimental predictions.

The simplest technicolor model contains would-be Goldstone bosons to provide masses for the observed W and Z particles, replacing the standard Higgs mechanism. Perhaps surprisingly, it also contains an additional Goldstone boson that is a natural dark matter candidate. A recent lattice simulation has confirmed the symmetry-breaking pattern, explored the mass spectrum of the lightest technihadrons, and established an effective field theory.

A recently discovered class of active galactic nuclei, TeV luminous blazars, constitute a small fraction of the power output of black holes. Nevertheless, there are suggestions that unlike the UV and X-ray luminosity of quasars, the very-high energy gamma-ray emission from the TeV blazars can be thermalized on cosmological scales with order unity efficiency, resulting in a potentially dramatic heating of the low-density intergalactic medium. The way in which this occurs, however, imparts a variety of peculiar properties to this novel heating source, resulting in a number of robust cosmological consequences. I will discuss the process by which TeV blazars heat the Universe, the strange properties that this heating has, and the variety of signatures that it has left behind, many of which have already been observed!