CAREER: Renormalization and higher rank parabolic actions

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。这个项目的目的是促进抛物线系统的知识，抛物线系统指的是出现在数学和物理的许多领域的一类复杂系统。有一个重要的教育部分：该项目计划支持实验数学实验室的活动，国际数学协会是该实验室的联合主任，以及女孩谈话数学(GTM)项目，这两个项目都由马里兰大学主办。前者的目的是让本科生参与研究，后者是为了吸引高中生从代表性不足的群体转向STEM学科。抛物型系统是既不混沌(即表现出指数复杂性)也不完全刚性的动力系统类型。抛物型系统通常表现出一种多项式的复杂性。一个这样的系统的例子是粒子在无摩擦、无口袋的非矩形多边形台球桌上的运动，在边界处完全有弹性碰撞。高阶抛物系统出现在非周期瓦片的研究中，与数学物理的应用密切相关。抛物系统可以通过被称为重整化方法的技术来研究。该项目的一个重要组成部分是致力于开发算子代数语言中的重整化工具，即使用来自算子代数的不变量来获得动态不变量。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). This project aims to advance knowledge on parabolic systems, which refer to a class of complex systems that appear in many areas of mathematics as well as physics. There is a significant educational component: this project plans to support the activities of the Laboratory of Experimental Mathematics, of which the PI is a co-director, as well as the Girls Talk Math (GTM) program, both hosted at the University of Maryland. The former activity is aimed at engaging undergraduate students in research, and the latter at attracting high-school students from underrepresented groups towards the STEM-disciplines.Parabolic systems are the types of dynamical systems that are neither chaotic (that is, exhibit exponential complexity), nor completely rigid. Parabolic systems typically exhibit a type of polynomial complexity. An example of one such system is the motion of a particle on a frictionless and pocketless billiard table of non-rectangular polygonal shape with completely elastic collisions at the boundary. Higher rank parabolic systems appear in the study of aperiodic tilings, closely connected to applications to mathematical physics. Parabolic systems can be studied through techniques known as renormalization methods. A significant component of the project is devoted to developing renormalization tools in the language of operator algebras, that is, using invariants coming from operator algebras to obtain dynamical invariants.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.