A beam of light is characterized by the following local properties: intensity, phase, and polarization. In most practical cases, these optical properties are either uniform or varying in space in a smooth, simple fashion. But it is nowadays possible to create strongly space-variant light beams, in which one or more of these properties vary in space in a prescribed way, forming nontrivial geometrical patterns. In other words, it is possible to endow light with a geometrical “structure”. We will not be concerned here so much with the case of patterns of light intensity, which may be considered just as usual optical images. In my talk, I will mainly focus on optical patterns of phase and/or polarization. In contrast to intensity, which is defined as a nonnegative real number, phase and polarization can be represented as points in closed manifolds, e.g. a circle or a sphere. A pattern of these properties may then acquire a rich geometrical structure, including the possible appearance of topological singularities of different kinds, e.g. optical scalar vortices (singularities of phase) and vector-vortices (singularities of polarization), a multiple-helix shape of the optical wavefront, and other rather nontrivial three-dimensional structures of the light field. While conceiving these structures in theory is often very simple, realizing them in the lab is usually not as simple. There are today different tools allowing the experimenter to control the spatial structure of light. In my talk, I will mainly focus on a relatively recent invention for generating phase and polarization vortices and other structures exploiting a singular-patterned liquid crystal cell, commonly named “q-plate”. This name is due to the presence of a topological singularity of given charge “q” in the medium structure, exploiting the long-range orientational order that characterizes liquid crystals. Interestingly, even the working principle allowing this device to control the structure of light is somehow “geometrical” in its nature, being related to the so-called “geometric phase”, an ubiquitous concept crossing many boundaries of physics, ranging from optics to classical mechanics, to quantum mechanics.

Liquid Crystals (LC) are anisotropic fluids characterized by long range orientational order and pair correlations. Mesoscale models, based on the drastic simplification of representing molecules as simple rigid objects such as (LC) theories and computer simulations. While these approaches are still very valuable in obtaining the general properties of complex LC one of the most important current challenges is to relate a realistic molecular structure to spherocylinders or ellipsoids or even spins on a lattice have been the cornerstone of the first generation of liquid crystal physical observables and predict properties such as morphologies, order parameters, and phase-transition temperatures. 
Atomistic molecular dynamics (MD) simulations, consisting in the numerical solution of Newton equations of motion for all the atoms in the system now start to make this possible, also allowing the test of classical theories for bulk LC (e.g. Maier-Saupe or Onsager). However, for most practical applications LC are not used in bulk but in thin films where the LC is aligned with the help of surface interactions, so it is somewhat surprising that surface effects are still described only empirically, while little is known on their molecular origin. In the talk we shall show that computer simulations start to shed some light on the interfacial behavior of liquid crystals and show examples for the prediction of the alignment and anchoring of  LC at the interface with different solid surfaces e.g. silicon or crystalline and glassy silica with different roughness (see figure). Simulations show in various cases that molecular organizations at the interface differ radically from those in the bulk, showing either discontinuities or broad distributions of orientations rather than the simple Dirichlet type boundary conditions assumed by many continuum type theories. In the talk an introduction to these systems and a discussion of some open problems will be presented.

As required by information theory, channel codes with long code words are necessary building blocks in a transmission system to achieve reliable communications with minimal power and maximal throughputs over noisy physical channels.
Nowadays capacity-achieving large random binary codes are actually adopted in most telecommunication standards. The breakthrough that allowed their practical use has been to substitute the optimal maximum likelihood decoding techniques at the receiver with suboptimal iterative techniques based on “belief” propagation.
Belief propagation is a powerful inference technique working on graphs used in many different applications.  Graph nodes are associated to factors or constraints of the model while graph edges are associated to random variables. Belief propagation algorithm proceeds by iteratively updating messages associated to the random variables, according to the constraints imposed by nodes.

In the framework of channel decoding the graph nodes represent the deterministic, usually linear, code constraints. They are either associated to a (binary)  parity check sum, or to a repetition of the variable. Belief propagation is initialized with a set of messages obtained from the noisy channel observations of the transmitted bits and proceeds iteratively, updating the messages according to the code constraints until convergence is reached.
Most practically used codes are linear and binary codes, so that messages propagated in the graph are binary messages usually represented with a single scalar (the Log-Likelihood Ratio).
In order to increase the throughput of communication systems, the use of  non binary codes is an attractive solution as each symbol can carry more information bits. Non binary codes can be constructed over groups,  rings,  or fields and there is a vast literature on the design of capacity achieving non binary codes.
The extension of the application of belief propagation to the non binary codes however poses several complexity problems as both message representation and message updating at check node grows at least linearly with the cardinality of the non binary alphabet and consequently exponentially with the increase of required throughput.

In this talk I will start by recalling the fundamental ideas and terminology beyond binary channel coding constructions and corresponding iterative decoding with belief propagation and other iterative techniques. I will then extend the concepts to non binary codes and summarize the main algorithms and complexity problems related with them. I will then introduce a new algorithm, named Analog Digital Belief Propagation (ADBP) which solves the complexity problems of belief propagation for non binary  codes. I will discuss the main properties of the algorithm, its possible extensions and code design problems related to the adoption of this algorithm.
During the talk I will also try to provide some open problems related to encoding and decoding of non binary codes  that will hopefully  stimulate the discussion and the interaction with the attendees having a mathematical background.

A minimal model for studying the mechanical properties of amorphous solids is a disordered network of point masses connected by springs. At a critical value of its mean connectivity, such a network becomes fragile: it undergoes a rigidity transition signaled by a vanishing shear modulus and transverse sound speed. We first investigate analytically and numerically the linear and non-linear visco-elastic response of these fragile solids by probing how shear fronts propagate through them.

Our approach, that we tentatively label shear front rheology, provides an alternative route to standard oscillatory rheology. In the linear regime, we observe at late times a diffusive broadening of the fronts controlled by an effective shear viscosity that diverges at the critical point. No matter how small the microscopic coefficient of dissipation, strongly disordered networks behave as if they were over-damped because energy is irreversibly leaked into diverging non-affine fluctuations. Close to the transition, the regime of linear response becomes vanishingly small: the tiniest shear strains generate strongly non-linear shear shock waves. The inherent non-linearities trigger an energy cascade from low to high frequency components that keep the network away from attaining the quasi-static limit. This mechanism, reminiscent of acoustic turbulence, causes a super-diffusive broadening of the shock width.

Finally, we show that the mechanism of failure of such networks, consists of meandering cracks whose width diverges at the transition. Thus, upon approaching the critical point, we can effectively zoom inside the fracture process zone.