FRG: Collaborative Research: Emerging issues in the sciences involving non standard diffusion

对“扩散过程”的数学描述和理解是许多科学领域的核心，从几何学和概率学到连续介质力学、流体动力学、种群动力学和博弈论等等。Ricci流、纳维斯托克斯方程、非线性弹性、期货期权，都带有扩散、粘度和不确定性的元素，这些元素都对应于类似的数学描述。在不同的情况和目标下，广泛的理论已经渗透到分析、几何和应用数学的广泛领域。然而，调查人员认为，还有一些新的、正在出现的领域变得更加复杂。预计通过合作，新的通用方法将出现，为经典的无穷小扩散过程及其在科学(建模、模拟、预测)中的作用提供某种形式的统一和凝聚力。需要研究的问题包括随机环境中的反应扩散现象、相变问题、非局部扩散过程以及由人口动力学问题引起的其他应用、交通中的拥堵问题、社会科学中的扩散和分离现象、犯罪活动热点的形成和动力学、涉及非局部(远程)相互作用的相变问题以及与纳米技术和生物学中的电液滴和复杂流体有关的表面扩散。该项目涉及国内机构之间的人员交流系统，旨在为学生和博士后提供丰富的培训经验。此外，该项目每年将在一所本土大学和一个暑期学校举行重点活动，旨在将该小组的成员--包括研究生和博士后--聚集在一起，以促进科学进步。通过这种方式，研究成果将向更广泛的受众展示，从而帮助教育和吸引新一代研究人员接受这些令人兴奋的新兴数学挑战和想法。

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.