Colloquium

Many systems take the form of networks: the Internet, the World Wide Web, social networks, distribution networks, citation networks, food webs, and neural networks are just a few examples. I will show some recent empirical results on the structure of these and other networks, particularly emphasizing degree sequences, clustering, and vertex-vertex correlations. I will also discuss some graph theoretical models of networks that incorporate these features, and give examples of how both empirical measurements and models can lead to interesting and useful predictions about the real world.

Noncommutative geometry is a more general formulation of geometry that does not require coordinates to commute. As such it unifies quantum theory and geometry and should appear in any effective theory of quantum gravity. In this general talk we present quantum groups as a microcosm of this unification in the same way that Lie groups are a microcosm of usual geometry, and give a flavour of some of the deeper insights they provide. One of them is the ability to interchange the roles of quantum theory and gravity by `arrow reversal'. Another is that noncommutative spaces typically carry a canonical 1-parameter evolution or intrinsic time created from the fundamental conflict between noncommuting coordinates and differential calculus. In physical terms one could say that quantising space typically has an anomaly for the spatial translation group and this forces the system to evolve. We give an example where we derive Schroedinger's equation in this way.

The causal set -- mathematically a finitary partial order -- is a candidate discrete substratum for spacetime. I will introduce this idea and describe some aspects of causal set kinematics, dynamics, and phenomenology, including, as time permits, a notion of fractal dimension, a (classical) dynamics of stochastic growth, and an idea for explaining some of the puzzling large numbers of cosmology. I will also mention some general insights that have emerged from the study of causal sets, the most recent one concerning the role of intermediate length-scales in discrete spacetime theories.

Superconducting circuits based on Josephson junctions are promising candidates for the implementation of solid-state qubits. In most of the recent experiments on these circuits, the qubits are controlled by a classical field containing a large number of photons. The possibility of coherently coupling these systems to a single photon has been recently suggested, opening the possibility to study analogs of quantum optics in condensed matter systems. I will review one of these proposals based on a superconducting charge qubit fabricated inside a high quality transmission line resonator and will describe its recent experimental realization. When the qubit is brought into resonance with the resonator, vacuum Rabi splitting is observed indicating that the regime of strong coupling has been reached. When the qubit is detuned from the cavity, I will explain how quantum non-demolition measurement can be realized. I will discuss how the measurement process can be quantitatively understood in this regime allowing us to explore the effect of measurement back-action on the qubit and to extract, for the first time in superconducting qubits, large visibility in Rabi oscillations.

A key issue in the context of (compact) extra dimensions is the one of their stability. Any stabilization mechanism is effective only up to some given energy scale; if they can approach this energy, 4$d observers can excite the fluctuations of the internal space, and probe its existence. Stabilization mechanisms introduce fields in the internal space; perturbations of these fields are mixed with perturbations of the metric, so that their study requires a complete GR treatment. After presenting the general framework, I will then discuss some relevant applications. I will present the exact coupling of the radion to codimension one branes, extending the regime of validity of the results in the literature. I will then focus on de Sitter compactifications, showing that the cosmological expansion has typically the effect of destabilizing the internal space. The final part of the talk will be devoted to related work in progress in less conventional areas of brane models: the localization of gravity towards the IR brane (corresponding to a dual description of emerging gravity from the CFT), and the inclusion of ghost fields in the bulk.

Since the seminal discovery of the neutrino by Cowan and Reines in the late 1950's, intense experimental and theoretical effort has focused on the elucidation of neutrino properties and the role they play in elementary particle physics, astrophysics, and cosmology. Neutrinos are born in the fusion reactions powering our Sun and are thought to be the driving mechanism for supernova explosions. Neutrinos exist in copious amounts as the primordial afterglow of the Big Bang and, if massive, would play a role in the evolution and ultimate fate of the Universe. Central to many of the key issues in neutrino physics is the question of whether neutrinos possess non-zero rest mass. If neutrinos are massive, then one expects flavor mixing to occur in the neutrino sector which could lead to the phenomena of neutrino oscillations and the possibility of CP violation in the neutrino sector. A detailed understanding of the microscopic properties of neutrinos can serve to pave the way to a unified description of the fundamental forces of Nature.

In this talk we assume that Quantum Einstein Gravity (QEG) is the correct theory of gravity on all length scales. We use both analytical results from nonperturbative renormalization group (RG) equations and experimental input in order to describe the special RG trajectory of QEG which is realized in Nature. We identify a regime of scales where gravitational physics is well described by classical General Relativity. Strong renormalization effects occur at both larger and smaller momentum scales. The former are related to the (conjectured) nonperturbative renormalizability of QEG. The latter lead to a growth of Newton's constant at large distances. We argue that this effect becomes visible at the scale of galaxies and could provide a solution to the astrophysical missing mass problem which does not require dark matter. A possible resolution of the cosmological constant problem is proposed by noting that all RG trajectories admitting a long classical regime automatically imply a small cosmological constant.