NONLINEART
Now days if you want to apply for a scientific grant, you have to be proficient in handling the neoteric yet overused concepts of sustainable development, quantum computers, globalization of information, business models and nano-neurotechnology. What made these terms to win over many other coinages?
Was it because of a diversification of disagreements, or because of an increase in cooperation? This is a question belonging to the theory of complex systems. According to Eugene Gendlin, and also in the spirit of Heidegger and Noica, there are three main types of conflicts involving living systems: the battle, or absolute adversity (when one disappears), the contention or the relative adversity (when only one can win), and the debate, or complex adversity (when nobody wins but a new truth reveals). By induction, there are three main procedures of understanding complex systems. One way is to begin from a particular complex system and addresses a variety of questions coming from that particular domain and its points of view. The second approach eliminates the deterministic point of view and uses statistical theories like random matrices, artificial intelligence or super-computer simulations. The third approach cuts across particular domains such as the earth and life sciences.
While the first two approaches lead to domain-specific cross disciplinary fields the third one is responsible for the interdisciplinary type of inquiry. It starts from fundamental general questions relevant to all domains, and searches for rigorous methods to solve the open particular problems. I am trying to develop a different approach for the science of complex systems. Usual models (based on non-linear differential equations, dynamic systems, graph theory, cellular automata, stochastic processes, or information theory) are well adapted to study local problems. However they cannot simultaneously take into account different multiplicities, nor explain how the system can have both robustness and flexibility. See for example the endless set of attempts to predict of both the laminar flow and the turbulence through an understanding of solutions to the Navier-Stokes equations (one of the Clay Millennium Problems). New approaches are welcome and seem necessary, in particular to treat challenging problems like correlation between phenomena at different levels, self-organization, robustness and flexibility of the system, memory evolutionary systems, and growing of supplementary structures.
This topic is wide and I can only do what Tom Thumb was suggested to do when King Thunston asked him to harvest an infinite crop of wheat in one day. Get a sickle and begin harvest one row at a time. At present I work on problems related to free (dynamical) boundaries in the evolution of complex systems. Consequently, I intend to understand more about fluids by doing experiments in our new Wave Lab (LB-173-A), or by understanding mathematically the inherent stochastic structure of Rogue waves.