Little more than one hundred years have gone by since the birth of Ettore Majorana, a highly renowed theoretical physicist. His career was brief and irregular but very intense, and he disappeared in March 1938 in circumstances that still are not completely clear. This volume is a contribution to a better understanding of the scientific, academic and human personality of Ettore Majorana, beyond the layers of legendary aspects which have accumulated over the years. Based on primary sources alone-scientific literature of the period and numerous archival documents-the figure of Ettore Majorana emerges in a completely new light. The young scientist is intensely involved in scientific research, completely independent, always striving to offer innovative contributions of the highest level according to the most advanced international standards, and very determined to make his results known by following a shrewd publication strategy. He is profoundly interested in his academic career, very scrupulous with institutional relationships, and attentive to his students. Moreover, his documented scholarly activity is much wider than previously reported. This historical analysis shows also that Majorana had a very important role in orienting research in Rome, especially in the sector of the statistical model of the atom and in Nuclear Physics. It is also clear that there are aspects of Ettore Majorana's life which transmit a solid historical legacy from the cultural and human points of view, besides the scientific legacy which is universally recognized to him. A rich reproduction of original documents completes the volume.

The main aim is to present at the level of beginners several modern tools of micro-local analysis which are useful for the mathematical study of nonlinear partial differential equations. The core of these notes is devoted to a presentation of the para-differential techniques, which combine a linearization procedure for nonlinear equations, and a symbolic calculus which mimics or extends the classical calculus of Fourier multipliers. These methods apply to many problems in nonlinear PDE's such as elliptic equations, propagation of singularities, boundary value problems, shocks or boundary layers. However, in these introductory notes, we have chosen to illustrate the theory on two selected and relatively simple examples, which allow becoming familiar with the techniques. They concern the well-posedness of the Cauchy problem for systems of nonlinear PDE's, firstly hyperbolic systems and secondly coupled systems of Schrödinger equations which arise in various models of wave propagation.