**ALBERT FERRANDO**University of Valencia

Symmetry is able to produce in us a natural aesthetic attraction related to its capability of ordering our vision of physical phenomena. In my lectures I will motivate the introduction of this fascinating concept in the field of linear and nonlinear waves by briefly reviewing its impact in physics. We will see how symmetry has been a powerful tool to classify and, thus, understand systems with a large variety of diverse and apparently unrelated elements. A perfect example is constituted by linear and nonlinear waves propagating in a discrete symmetry media. The mathematical formalization of symmetry in such systems is provided by discrete group theory. I will review some basic properties of this theory by applying it to the case of linear and nonlinear optical waves in discrete symmetry media and I will show how it can be used to provide an elegant (i.e., aesthetic) way of classifying the enormous amount of possible stationary states that these systems admit. A particular emphasis will be given to the important case of optical vortices, i.e., to solutions exhibiting circulating energy fluxes, discretized angular momentum, and phase singularities. We will learn how discrete rotational symmetry changes our usual understanding of these objects and produce a panoply of fascinating and, sometimes, counter-intuitive phenomena. **ICFO+ Theory Lecture, January 9, 2018, 10:15. Blue Lecture Room**