Interacting Particle Systems and Mean-field games Workshops

该项目将支持来自美国的研究生、博士后研究人员和早期职业研究人员参加将于2022年3月13日至27日在加拿大蒙特利尔数学研究中心举行的“相互作用粒子系统和流体动力学极限”研讨会或2022年4月10日至17日在加拿大蒙特利尔数学研究中心举行的“平均场游戏”研讨会。这两个研讨会都是2022年1月至7月在CRM举办的一个更大的跨学科主题项目“概率和偏微分方程”的一部分。概率论和偏微分方程理论（PDEs）是数学的重要领域，在方法和目标上有大量的重叠。在这两个领域，主要目标之一是提供工程、物理、化学和生物系统如何随时间变化的精确模型。概率论经常关注在微观水平上随机和/或不可预测的系统如何在宏观水平上变得高度有序。偏微分方程理论经常关注这些宏观系统的时空演化。几十年来，概率和偏微分方程这两个领域之间有着卓有成效的相互作用，每个领域的直觉和数学技术都在另一个领域找到了应用。这个项目着重于这种相互作用的两个方面，这两个方面都与概率粒子系统在充分“缩小”时与偏微分方程的相似程度有关。其中之一，即平均场博弈领域，描述了通过图形结构（通常是完整图形）耦合扩散而进化的策略控制交互代理的缩放限制。第二，相互作用的粒子系统和流体动力学极限，通常侧重于粒子系统在更多几何设置中的偏微分方程近似，例如晶格（在空间和时间上都采取适当的细网格限制）。该项目的目标是支持美国的初级研究人员和来自代表性不足群体的研究人员参加一个关于概率和偏微分方程的主题学期（特别是他们参加两个关于平均场游戏和相互作用粒子系统主题的研讨会），该研讨会将于2022年上半年在加拿大蒙特雷姆萨伊姆的mathacimmatiques研究中心举行。主题学期网站保存在http://www.crm.umontreal.ca/2022/Probab22/index_e.php相互作用粒子系统和流体动力学极限研讨会http://www.crm.umontreal.ca/2022/Particules22/index_e.php和平均场游戏研讨会http://www.crm.umontreal.ca/2022/Games22/index_e.php.This，奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，认为值得支持。

This project will support participation of graduate students, post-doctoral researchers and early career researchers from the United States of America in one of the workshops "Interacting Particle Systems and Hydrodynamic Limits" to be held from March 13-27, 2022, or the "Mean-Field Games" workshop to be held from April 10-17, 2022 at the Centre de Recherches Mathematiques (CRM) in Montreal, Canada. Both workshops are part of a larger interdisciplinary thematic program on "Probabilities and PDEs" held at CRM from January to July 2022. Probability theory and the theory of partial differential equations (PDEs) are important areas of mathematics with substantial overlap in their methods and goals. In both fields, one of the major aims is to provide accurate models of how engineered, physical, chemical and biological systems change over time. Probability frequently focuses on how systems which are random and/or unpredictable at the microscopic level can become highly ordered at the macroscopic level. PDE theory frequently focuses on the spatial and temporal evolution of such macroscopic systems. For decades there has been a fruitful interplay between the two fields probability and PDEs, with both intuitions and mathematical techniques from each area finding application in the other. This project focuses on two aspects of that interplay, which are both related to how probabilistic particle systems resemble PDEs when sufficiently "zoomed out". One of these, the area of mean-field games, describes scaling limits of strategically controlled interacting agents evolving as diffusions coupled via a graph structure (often the complete graph). The second, interacting particle systems and hydrodynamic limits, typically focuses on PDE approximations for particle systems in more geometric settings, such as lattices (on taking an appropriate fine-mesh limit in both space and time). The goal of this project is to support the participation of US-based junior researchers and researchers from underrepresented groups in a thematic semester on Probability and PDEs (and in particular their participation in two workshops, on the subjects of mean-field games and interacting particle systems), taking place in the first half of 2022 at the Centre de Recherches Mathématiques in Montréal, Canada. The thematic semester website is maintained at http://www.crm.umontreal.ca/2022/Probab22/index_e.php the Interacting Particle Systems and Hydrodynamic Limits Workshop at http://www.crm.umontreal.ca/2022/Particules22/index_e.php and the Mean-Field Games Workshop at http://www.crm.umontreal.ca/2022/Games22/index_e.php.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.