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，从几何和概率论到连续介质力学、流体动力学、种群动力学和博弈论等等。里奇流、纳维·斯托克斯方程、非线性弹性、期货期权，所有这些都带有扩散、黏性、不确定性的元素，它们对应于类似的数学描述。在不同的环境和目标下，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，研究人员发现，还有新的、正在出现的、更加复杂的领域。预计通过合作的努力，新的通用方法将出现，提供某种统一和凝聚力，就像今天存在的经典无穷小扩散过程及其在科学（建模，模拟，预测）中的作用一样。要研究的问题包括随机环境中的反应扩散现象、相变问题、非局部扩散过程和其他由人口动态问题、交通拥挤问题、社会科学中的扩散和隔离现象、犯罪活动热点的形成和动态等问题引起的应用。纳米技术和生物学中涉及电液滴和复杂流体的非局部（远程）相互作用和表面扩散的相变问题。该项目涉及国内院校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，每年将有一个月的重点月在其中一所国内大学进行，并有一个暑期学校，旨在将小组成员（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，研究成果将接触到更广泛的受众，从而帮助教育和吸引新一代的研究人员来研究这些令人兴奋的新兴数学挑战和思想。

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.