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Computational complexity theory is a branch of computer science dedicated to classifying computational problems in terms of their difficulty. While computability theory tells us what we can compute in principle, complexity theory informs us regarding what is feasible. In this chapter I argue that the science of quantum computing illuminates the foundations of complexity theory by emphasising that its fundamental concepts are not model-independent. However this does not, as some have suggested, force us to radically revise the foundations of the theory. For model-independence never has been essential to those foundations. The fundamental aim of complexity theory is to describe what is achievable in practice under various models of computation for our various practical purposes. Reflecting on quantum computing illuminates complexity theory by reminding us of this, too often under-emphasised, fact.

Recently a boundary energy-momentum tensor Tzz has been constructed from the soft graviton operator for any 4D quantum theory of gravity in asymptotically flat space. Up to an "anomaly" which is one-loop exact, Tzz generates a Virasoro action on the 2D celestial sphere at null infinity. Here we show by explicit construction that the effects of the IR divergent part of the anomaly can be eliminated by a one-loop renormalization that shifts Tzz.

Utilizing the Fermi measurement of the gamma-ray spectrum toward the Galactic Center, we derive some of the strongest constraints to date on the dark matter (DM) lifetime in the mass range from hundreds of MeV to above an EeV. Our profile-likelihood based analysis relies on 413 weeks of Fermi Pass 8 data from 200 MeV to 2 TeV, along with up-to-date models for diffuse gamma-ray emission within the Milky Way. We model Galactic and extragalactic DM decay and include contributions to the DM-induced gamma-ray flux resulting from both primary emission and inverse-Compton scattering of primary electrons and positrons. For the extragalactic flux, we also calculate the spectrum associated with cascades of high-energy gamma-rays scattering off of the cosmic background radiation. We argue that a decaying DM interpretation for the 10 TeV-1 PeV neutrino flux observed by IceCube is disfavored by our constraints. We interpret the results in terms of individual final states and in the context of simplified scenarios such as a hidden-sector glueball model.

Cosmology has seen great progress thanks to precision measurements and is bound to greatly benefit from upcoming Large Scale Structure and Cosmic Microwave Background data. I will point out a number of interesting directions. In particular, I discuss how the microphysics of inflation may be tested in galaxy surveys through “fossil” signatures originating from squeezed primordial correlations. I further elaborate on the constraining power of CMB spectral distortions on small-scale cosmological fluctuations and on particle decays in the very early Universe in relation to reheating. I also describe some of the possible constraints on inflation and reheating from future B-mode observations.

A central theme of modern condensed matter physics is the study of topological quantum matter enabled by quantum mechanics, which provides a further "topological" twist to the classical theory of ordered phases. These quantum-entangled phases of matter such as fractional quantum Hall phases, spin liquids, and some non-Fermi liquids, are typically strongly-correlated and thus cannot be studied within conventional perturbative approaches. Because of the spectacular emergent phenomena as well as their potential for realistic applications, there has been much recent interest in exploring the physics of these exotic phases. In this talk, I show that the powerful methods of quantum field theory, namely quantum anomaly and duality, can expose the global landscape in parameter space of these gapped and gapless topological quantum phases and lead to several novel insights on these phases. As a demonstration of this principle, we study clean fractional quantum Hall transitions, composite Fermi liquids, and the surface of fractional topological insulators. Despite long and storied histories, these three systems are at the frontier of our knowledge of two and three dimensional topological phases. I show that the non-perturbative approach for these systems, i.e., the duality, sheds some new light on these systems and allows us to resolve some longstanding puzzles, which have not been clear previously. Furthermore, it uncovers novel physics of these intrinsically strongly-correlated phases of matter.

The parametrized particle: notion of Dirac observables - Canonical analysis of the first order action for 3D gravity - Constraints of 3D gravity (1st order formulation): Gauss constraints and Flatness constraints.