The mathematical description and understanding of "diffusive processes" is central to many areas of science, from geometry and probability to continuum mechanics, fluid dynamics, population dynamics and game theory to name a few. Ricci flows, the Navier Stokes equation, non linear elasticity, futures options, all carry an element of diffusion, viscosity, uncertainty that correspond to a similar mathematical description. An extensive theory already permeates, under different circumstances and goals, wide areas of analysis, geometry and applied mathematics. The investigators perceive, however, that there are new, emerging areas of added complexity. It is expected that through the collaborative effort, new general methods will emerge providing some sort of unification and cohesion like the one existing today for classical infinitesimal diffusion processes and their role in the sciences (modeling, simulation, prediction).
Problems which are to be studied include reaction diffusion phenomena in random environments, phase transition problems, non local diffusion processes and other applications motivated by questions of population dynamics, congestion issues in transportation, diffusion and segregation phenomena in social sciences, formation and dynamics of hotspots of criminal activity, phase transition problems involving nonlocal (long range) interactions and surface diffusion related to electric fluid droplets and complex fluids in nano technology and biology. The project involves a system of personnel exchanges between the home institutions, designed to provide a rich training experience for students and postdocs. In addition, there will be an emphasis month every year of the project in one of the home universities and a summer school designed to bring members of the group - including graduate students and postdocs - together to stimulate scientific progress. In this way the fruits of the research will be exposed to a broader audience, thereby, helping educate and attract a new generation of researchers to these exciting emerging mathematical challenges and ideas.
