The Mathematical Models in Science course will be devoted to mathematical models to understand complex systems, that is, systems consisting of a huge number of "individuals" that interact with each other, and give rise to emergent phenomena, not only explained by the individual characteristics of each one. In other words systems in which "the whole is much greater than the sum of the parts". Systems of this type are very frequent in Physics, Biology, Sociology, Ecology, Epidemiology and other areas of knowledge.

The aim is therefore to construct a conceptual (and formal) framework to explain how interactions between the (microscopic) elements of a system can lead to cooperative phenomena, and emerging properties of process dynamics. This strategy, which allows us to move from microscopic interaction to emergent collective phenomena characteristic of all Complex Systems, is strongly inspired by the methodology of Statistical Physics. It is seen as a general paradigm of the passage from the site to the large-scale global properties of complex systems, and has served as a motivation for many areas of mathematics (dynamical systems, knot theory, enumerative geometry, and others).

The mathematical models used are vast:. from information theory, entropy, random fields, Gibbs measures, statistical physics models, percolation, cellular automata, agent modeling, and many others, all using "classical" mathematical methods, which will be reviewed during class.

Several applications will be addressed to Mathematics and Natural Sciences described above. The course does not presuppose any background in Physics, Biology or other sciences.

João Nuno Tavares