The workshop “Mathematical Models and Methods for Planet Earth,” organized by the Italian National Institute for Advanced Mathematics (INdAM) under the auspices of MPE2013 in Rome, May 27-29, finished a few days ago.
The workshop offered an interesting view of the broad spectrum of ongoing applications of mathematics to life on our planet.
Several presentations focused on the dynamics of planet Earth. Since Earth is a celestial body in the solar system, it is exposed to hazardous impacts with other natural or artificial objects (e.g., meteorites or spatial debris). It is also the basis of spatial missions designed by applying new concepts. Earth provides a delicate environment for millions of different life forms; thus, some talks were devoted the dynamics of the Earth’s interior, its oceans and climate (including its evolution on geological time scales). Moreover, science must consider human beings and their activities; thus, a significant part of the workshop focused on mathematical models of problems in medicine, biology, social prevention, economics, politics, internet diffusion, etc.
All the speakers made a substantial effort to avoid too technical contents, so their presentations were accessible for a highly heterogeneous audience. The slides presented at the workshop are currently being collected and will be made available here.
It is now time for the participants to relax and process at least part of the concepts and ideas presented at the workshop. Synthesizing the new trends in this kind of applied mathematics is a big effort, but it is worth a try.
Let us start with the provocative question raised by one of the speakers (J. Laskar) at the end of the beautiful public lecture by Christiane Rousseau: “Nowadays, what can the Earth do for Mathematics?” At that moment, the question sounded as a bizarre attempt to reverse the problem: this kind of rhetorical argument can be very useful to improve the understanding of a wide problem, but it is not always meaningful. However, the question is extremely natural for people working in celestial mechanics, as was shown with a beautiful example in the last talk: The study of the secular variations of the planetary orbital elements pushed Lagrange and Laplace to introduce solutions of systems of linear differential
equations. It took me a long time to realize that Laskar was probably referring to the changes experienced in the relation between applied mathematics and other fields of science in the past few decades.
Astronomy and physics have been the main (perhaps, the only) sources of problems that enabled the birth of new branches of mathematics until the beginning of the 20th century. In this context, it is important to recall that, in Hilbert’s list of 23 unsolved problems in mathematics introduced at the International Congress of Mathematics in Paris in 1900, just one was concerned with arguments other than the pure mathematical ones: the axiomatization of physics. Moreover, in the past, the genesis of great ideas has often been the result of a beautiful interaction between experimental measures with a few significant digits that was explained by the “first order” of a new theory, the refinement of which was found in agreement with more precise new measures. To mention some examples, we can quote the discovery of the law of gravitation and the introduction of the two-body model or quantum mechanics and the study of light emissions of the hydrogen atom. This beautiful scenario with continuous positive interactions between mathematics, theoretical physics and technological progress involved in the experimental measures seems to have lost its nice simplicity. Nowadays, the amount of data in some scientific problems is often overwhelmingly large (as explained in
Perozzi’s talk). Most of the speakers showed that in the last century mathematics was applied to many fields different from physics (e.g., game theory, dynamics of biological populations, etc.).
The apparent loss of simplicity in the relations between mathematics and its applications is not by chance. The problems arising in sciences studying the Earth and the evolution of life presently point to a common Grand Challenge: complexity. The mathematical approaches shown in the talks at the workshop deal with complexity in various ways. First of all, mathematics is often used to extract meaningful information from the huge amount of data: even literary textual data can be successfully analyzed and classified by applying statistical concepts! Moreover, the role of mathematics is extremely important in providing well defined models to properly study problems. In this context, the approach often defines a sort of “Russian doll” structure of models of increasing difficulty. In particular, this was seen in some talks based on probabilistic techniques: some set of equations depending on many parameters were initially settled; then, some of the parameters were switched off, so to make the behavior of the system (sometimes numerically, other times in easy analytical ways) predictable and comparable with some expected/known phenomenon. Finally, the system was restudied by restoring the dependency on all the parameters previously neglected. Often, in the widest generality, the considered systems provide a lot of open subproblems which are hard to solve from a mathematical point of view. On the whole, mathematics is now relevant not only to provide solutions but also to set up well-posed problems.
As a final comment on the INdAM MPE2013 initiative, let us consider the new trends about people working in mathematics. Everybody can see that complexity often makes the things more complicated and, thus, mathematicians in universities and research centers are more and more specialized in their own field(s) of interest. Who can now master so many mathematical arguments as Poincaré or Hilbert were able to do a century ago? All the speakers of the workshop clearly showed that mathematicians are now challenged also in the opposite direction: more and more research topics require deep mathematical knowledge, often to tackle problems in the context of teams. The abstraction can allow mathematics to still be pure, but mathematicians are more and more requested to be part of the rest of the scientific community.
Univ. of Rome “Tor Vergata”