FRG: Collaborative Research : Emerging Issues in the Sciences Involving Non-Standard Diffusion

FRG：合作研究：涉及非标准扩散的科学中的新问题

• 批准号: 1065979
• 负责人: Luis Silvestre
• 金额: $ 38万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1065979&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Emerging; Issues

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.

对“扩散过程”的数学描述和理解是许多科学领域的核心，从几何和概率到连续介质力学、流体动力学、群体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，都带有与类似数学描述相对应的扩散、粘度、不确定性元素。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，调查人员认为，有一些新的、正在出现的领域更加复杂。预计通过协作努力，将出现新的通用方法，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中的作用（建模、模拟、预测）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、非局部扩散过程以及由人口动态、拥塞问题引发的其他应用 社会科学中的运输、扩散和隔离现象问题，犯罪活动热点的形成和动态，涉及非局域（长程）相互作用的相变问题以及与纳米技术和生物学中的电流液滴和复杂流体相关的表面扩散问题。该项目涉及母校之间的人员交流制度，旨在为学生和博士后提供丰富的培训经验。此外，每年都会在一所母校举办该项目的重点月，并举办暑期学校，旨在将小组成员（包括研究生和博士后）聚集在一起，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研究人员面对这些令人兴奋的新兴数学挑战和想法。