CAREER: Randomness in Geometry and Dynamics

该项目涉及使用动力系统，概率论和几何的想法来理解各种几何和动力学背景下的随机性。PI正在使用空间统计和间隙分布开发抛物动力系统复杂性的新措施，并使用重整化方法来计算这些不变量。PI和他的合作者还创造了随机几何对象和结构的新概念，通过对几何对象的模空间进行适当的测量，并通过计算各种函数关于这些测度的矩来理解这些随机对象的属性。本项目正在设计复杂性测度，以帮助检测新类型的混沌（“对初始条件的敏感依赖”）物理系统在新情况下的时间演化行为，例如在磁场存在下金属费米表面上电子的运动。PI和一组工程师正在利用对几何对象的几何和模量空间的测量理论的理解来建模和优化锂离子电池的性能。该项目的研究开发和使用了许多计算实验，以及模型的可视化和动画。其中许多是由学生团队准备的，他们在lllingo几何实验室（IGL），一个由PI创立的设施，专注于研究，可视化和社区参与。该奖项的教育部分集中在IGL的发展和扩展上，涉及各种各样的参与者：从学龄前儿童到研究生再到教师和教育工作者。IGL的活动将他们引入数学作为探索性的奋进，并发展数学探索者的社区。

This project involves using ideas from dynamical systems, probability theory and geometry to understand randomness in various geometric and dynamical contexts. The PI is developing new measures of complexity for parabolic dynamical systems using spacing statistics and gap distributions, and using renormalization methods to compute these invariants. The PI and his collaborators are also creating new notions of random geometric objects and structures, by placing appropriate measures on moduli spaces of geometric objects, and understanding properties of these random objects by computing moments of various functions with respect to these measures.The measures of complexity are being designed in this project are constructed to help detect new types of chaotic (`sensitive dependence on initial conditions') behavior of time evolution of physical systems in new situations, such as the motion of an electron on the Fermi surface of a metal in the presence of a magnetic field. Understanding the geometry and measure theory of moduli space of geometric objects is being used by the PI and a team of engineers in modeling and optimizing the performance of a lithium-ion battery. The research in this project develops and uses many computational experiments, as well as visualizations and animations of the models. Many of these are prepared by teams of students in the lllinois Geometry Lab (IGL), a facility founded by the PI, focusing on research, visualization and community engagement. The educational component of this award centers on development and expansion of the IGL, which involves a broad variety of participants: all the way from preschoolers to graduate students to teachers and educators. The activities of the IGL introduce them to mathematics as an exploratory endeavor, and develop a community of mathematical explorers.