Mathematical Methods in Social and Economical Science

Is modern science able to study social matters like those related to immigration phenomena on solid mathematical grounds? Can we for instance determine cultural robustness and the causes behind abrupt changes from cultural legacies? Can we predict, cause or avoid swings? A novel approach is under investigation using the statistical mechanics formalism devised for the study of phase transitions in physics. Three decades ago, statistical mechanics witnessed the introduction of disordered interactions in spin glass models. In addition to the description of condensed matter alloys, the proposed statistical mechanics formalism has proven to be an excellent method to study problems well beyond the original motivation and has indeed invaded fields as diverse as combinatorial optimization and neural networks. Interesting applications have been found in the biological, economical, and social sciences. Today, the rich mathematical structure of spin glasses is a leading research topic in probability theory. The introduction of random couplings can be seen as a first attempt to introduce random interactions in particle systems. Diluted spin glasses have moved the theory toward adding a new source of randomness in the connectivity property of the interaction network like those of Erdös-Rényi random graphs. Those are also relevant in computer science, since many random optimization problems are mapped in a natural way into the study of ground states of diluted mean-field spin glass models for example, the K-sat model has been solved within the framework of “one-step replica symmetry breaking”