The pluridisciplinarity character of the project makes up its
main originality. We intend to promote the development of nonlinear science
through a focussed training of Young Researchers ("fellows"). These fellows
are expected to play a role as future leaders of the field in Europe.
- The pluridisciplinarity character of the field of nonlinear
physics results essentially from the fact that all scientific disciplines
have to deal with nonlinear ordinary and partial differential equations. (ODEs
and PDEs) Universality of the routes to chaos, for example, can be understood
as generic behaviors of low-dimensional dynamical systems. It is quite instructive
to remark that the understanding of this question progressed a lot in the
years 1970-80 by works implementing simple ideas from phase transitions (renormalisation
group). This progress has been done by physicists (not by mathematicians).
The situation today is similar with respect to spatio-temporal phenomena observed
in many different areas of science (from physics to biology), which seems
also to possess a great degree of universality. The lack of a mathematical
framework for PDEs, similar to the qualitative theory of ODEs (Poincare) makes
the question very difficult. We quote Feynman, Lectures II/2 (41): ìthe next
great era of awakening of human intellect may well produce a method of understanding
the content of equations. Today we cannot. Today we cannot see that the water
flow equations contain such things as the barber pole structure of turbulence
that one sees between rotating cylinders. Today we cannot see whether Schroedingers
equation contains frogs, musical composers, or morality -- or whether not.î.
The development of a qualitative theory of partial differential equation is
the decisive open problem, which probably needs new mathematical concepts.
There is actually a great difficulty for learning the subject,
which is connected with its ambiguous relation to mathematics. The qualitative
theory of ODEs, for example, is not present in the curriculum of most physicists,
because it is typically considered part of mathematics. The situation is quite
serious. Students in physics often start a PhD thesis, without even knowing
the Poincare representation of the dynamics (phase portrait) of the system that
they may consider.