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