We study disordered and frustrated systems such as spin-glasses, structural glasses and random photonics models by means of advanced methods in statistical mechanical of disordered and complex systems, namely Replica Symmetry Breaking theory, Cavity method, Belief and Survey propagation, Supersymmetric path integral formulation of the dynamics (à la Martin-Siggia-Rose), Renormalization group approaches (on hierarchical lattice, on finite dimensional squared and cubic cells, on hierarchical models) and enhanced Monte Carlo methods for the numerical simulation of the dynamics both at and off-equilibrium.

By means of the above mentioned techniques we investigate phase transitions and states organisation in complex systems both at high dimensionality, where the mean-field approximation is exact) and in low dimension, where a phase transition is still present but it belongs to a different universally class. Phenomena studied in the recent years are the spin-glass transition and the low temperature replica symmetry breaking, the structural glass transition and of the organisation of the stable and metastable glassy states below the dynamic arrest transition, the random field Ising model transition, the Anderson localization.

**Glassy and slowly relaxing systems. **A glass can be viewed as a liquid in which a huge slowing down of the diffusive motion of the particles has destroyed its ability to flow on experimental timescales. The slowing down is expressed through the relaxation time, that is, generally speaking, the characteristic time at which the slowest measurable processes relax to equilibrium. Cooling down from the liquid phase, the slow degrees of freedom of the glass former are no longer accessible and the viscosity of the undercooled melt grows several orders of magnitude in a relatively small temperature interval. As a result, in the cooling process, from some point on, the time effectively spent at a certain temperature is not enough to attain equilibrium: the system is said to have fallen out of equilibrium. Nature and characterization of this non-equilibrium glassy regime and of the glass transition are a challenging issue that stimulates deep theoretical work concerning frustrated systems in diverse representations. We work on the theoretical representation of the behavior of viscous liquids, structural glasses and spin-glasses, on the critical slowing down occurring near-by the dynamic arrest, on the aging dynamics, on the extension of glass theories beyond the limit of validity of mean-field approximation.

**Inverse problem in statistical mechanics**. Given a data set and a model with some unknown parameters, the inverse problem aims to find the values of the model parameters that best fit the data. We focus on systems of interacting elements, in which the inverse problem concerns the statistical inference of the underling interaction network and of its coupling coefficients from observed data on the dynamics of the system. Versions of this problem are encountered in physics, biology, social sciences and finance, neuroscience (just to cite a few), and are becoming more and more important due to the increase in the amount of data available from these fields. A standard approach used in statistical inference is to predict the interaction couplings by maximizing the likelihood function. This technique, however, requires the evaluation of the partition function that, in the most general case, concerns a number of computations scaling exponentially with the system size. Boltzmann machine learning approach uses Monte Carlo sampling to compute the gradients of the Log-likelihood looking for stationary points but this method is computationally manageable only for small systems. A series of faster approximations, such as naive mean-field, independent-pair approximation inversion of Thouless-Anderson-Palmer equations, small correlations expansion, adaptive TAP, adaptive cluster expansion or Bethe approximations have been developed in the last 15 years. These techniques take as input means and correlations of observed variables and most of them assume a fully connected graph as underlying connectivity network, or expand around it by perturbative dilution. In most cases, network reconstruction turns out to be not accurate for small data sizes and/or when couplings are strong or, else, if the original interaction network is sparse. A further method, substantially improving performances for small data, is the so-called Pseudo-Likelyhood Method (PLM), implemented with regularization or with decimation. We work on the analysis of the performances of the various inference methods, on their improvement and on their application to new problems.

**Disordered protein states. **The ordered structure of proteins is one of the basic paradigms of classical biology, and it provides an explanation for many aspects of their functioning. Nevertheless, in many cases proteins operate in environments far from equilibrium, or possess labile conformations that convert towards order only under particular conditions. Examples include protein folding/unfolding in the presence of temperature and pressure variations, or configuration reorganizations induced by ligand binding in intrinsically disordered proteins. The statistical properties of these ensembles of structures can be studied with sampling techniques based on classical molecular dynamics simulations.

**Molecular networks.** We are interested in characterizing emergent properties of large networks of interacting molecules of biological significance, e.g. proteins or nucleic acids, using equilibrium and non-equilibrium statistical mechanics methods. Our central goal is to understand what makes these networks optimal and in which precise sense, how the laws of physics limit their performance in such tasks as noise or information processing, and whether they can sustain collective effects similar to those that characterize more traditional systems studied in statistical physics. In turn, our hope is to gain insight about the evolution of the large-scale organization of the known molecular networks that govern cellular and multi-cellular activities.