Complex systems theory : from spin glasses to neural networks

Since the 1970s, the study of spin glasses has been at the heart of disordered systems theory. Initiated by experimental observations, such as magnetic susceptibility anomalies or ageing phenomena, it has given rise to a considerable amount of work by physicists and mathematicians. Numerous concepts, such as the replica method and its interpretation, have been developed in an attempt to solve spin glass models that are sometimes very simple to formulate. These ideas have made it possible to tackle other questions in a priori distant fields (neural networks, protein folding, error-correcting codes, genealogies, optimization...). The aim of this lecture is to outline the main advances made in understanding the spin glass problem, which in many respects is and will remain the archetype of a complex system.