This paper first describes, from a high level viewpoint, the main challenges that had to be solved in order to develop a theory of spin glasses in the last fifty years. It then explains how important inference problems, notably those occurring in machine learning, can be formulated as problems in statistical physics of disordered systems. However, the main questions that we face in the analysis of deep networks require to develop a new chapter of spin glass theory, which will address the challenge of structured data.

Hard spheres are ubiquitous in condensed matter: they have been used as models for liquids, crystals, colloidal systems, granular systems, and powders. Packings of hard spheres are of even wider interest, as they are related to important problems in information theory, such as digitalization of signals, error correcting codes, and optimization problems. In three dimensions the densest packing of identical hard spheres has been proven to be the FCC lattice, and it is conjectured that the closest packing is ordered (a regular lattice, e.g, a crystal) in low enough dimension. Still, amorphous packings have attracted a lot of interest, because for polydisperse colloids and granular materials the crystalline state is not obtained in experiments for kinetic reasons. We review here a theory of amorphous packings, and more generally glassy states, of hard spheres that is based on the replica method: this theory gives predictions on the structure and thermodynamics of these states. In dimensions between two and six these predictions can be successfully compared with numerical simulations. We will also discuss the limit of large dimension where an exact solution is possible. Some of the results we present here have been already published, but others are original: in particular we improved the discussion of the large dimension limit and we obtained new results on the correlation function and the contact force distribution in three dimensions. We also try here to clarify the main assumptions that are beyond our theory and in particular the relation between our static computation and the dynamical procedures used to construct amorphous packings. There remain many weak points in our theory that should be better investigated. We hope that this paper can be useful to present the state of the art of the method and to stimulate new research in this field

From bird flocks to fish schools, animal groups often seem to react to environmental perturbations as if of one mind. Most studies in collective animal behavior have aimed to understand how a globally ordered state may emerge from simple behavioral rules. Less effort has been devoted to understanding the origin of collective response, namely the way the group as a whole reacts to its environment. Yet, in the presence of strong predatory pressure on the group, collective response may yield a significant adaptive advantage. Here we suggest that collective response in animal groups may be achieved through scale-free behavioral correlations. By reconstructing the 3D position and velocity of individual birds in large flocks of starlings, we measured to what extent the velocity fluctuations of different birds are correlated to each other. We found that the range of such spatial correlation does not have a constant value, but it scales with the linear size of the flock. This result indicates that behavioral correlations are scale free: The change in the behavioral state of one animal affects and is affected by that of all other animals in the group, no matter how large the group is. Scale-free correlations provide each animal with an effective perception range much larger than the direct interindividual interaction range, thus enhancing global response to perturbations. Our results suggest that flocks behave as critical systems, poised to respond maximally to environmental perturbations. O f all distinctive traits of collective animal behavior the most conspicuous is the emergence of global order, namely the fact that all individuals within the group synchronize to some extent their behavioral state (1-3). In many cases global ordering amounts to an alignment of the individual directions of motion, as in bird flocks, fish schools, mammal herds, and in some insect swarms (4-6). Yet, global ordering can affect also other behavioral states, as it happens with the synchronous flashing of tropical fireflies (7) or the synchronous clapping in human crowds (8).The presence of order within an animal group is easy to detect. However, order may have radically different origins, and discovering what is the underlying coordination mechanism is not straightforward. Order can be the effect of a top-down centralized control mechanism (for example, due to the presence of one or more leaders), or it can be a bottom-up self-organized feature emerging from local behavioral rules (9). In reality, the lines are often blurred and hierarchical and distributed control may combine together (10). However, even in the two extreme cases, discriminating between the two types of global ordering is not trivial. In fact, the prominent difference between the centralized and the self-organized paradigm is not order, but response.Collective response is the way a group as a whole reacts to its environment. It is often crucial for a group, or for subsets of it, to respond coherently to perturbations. For gregarious animals under strong predatory pressure,...