CAREER: Stochastic Forward and Inverse Problems Involving Dynamical Systems

动力系统长期以来一直在数学和许多其他学科中作为模型进行研究，如天气预报、地球物理建模、分子生物学、金融数学和生态学。这些系统的数学描述揭示了各种可能的行为，并且众所周知，在应用设置中，这些系统的复杂行为可以显着影响建模结果。这个项目的教育重点包括对从中学到研究生的学生进行概率论、统计学和动力学方面的培训和教育。特别是，PI将开发引人入胜的概率和统计教育活动，并将其提供给北卡罗来纳大学夏洛特分校大学预科课程的初高中学生。这些活动将广泛传播，教师将接受培训。此外，PI将为本科生建立一个概率论的夏季研究项目，研究生将参与该项目的各个方面。该项目侧重于从概率和统计的角度分析动力系统。从概率的角度来看，该项目试图解决“前向问题”，即从系统集合中随机选择一个动力系统，并希望表征其行为。这一系列的研究将阐明在一个典型的系统中，人们可以期望看到什么样的行为。从统计学的角度来看，该项目侧重于“逆问题”，在这个问题中，人们希望从对（可能未知的）动力系统的观察中学习或得出推论。更具体地说，该项目涉及分析统计推理方法在应用于动态系统或群体行为观察时的性能。这个方向的结果将为统计程序何时可以成功地应用于动力系统提供理论指导。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Dynamical systems have long been studied both within mathematics and as models in many other disciplines, such as weather forecasting, geophysical modeling, molecular biology, financial mathematics, and ecology. The mathematical description of these systems has revealed a remarkable variety of possible behavior, and it is known that in applied settings the complex behavior of these systems can significantly impact modeling outcomes. The educational focus of this project involves training and educating students from middle school to graduate school in probability, statistics, and dynamics. In particular, the PI will develop engaging educational activities in probability and statistics and deliver them to middle and high school students in UNC Charlotte's Pre-College Program. These activities will be disseminated widely, and teachers will receive training in their delivery. Additionally, the PI will establish a summer research program in probability for undergraduate students, and graduate students will be involved in all aspects of the project.This project focuses on the analysis of dynamical systems from both probabilistic and statistical points of view. From the probabilistic point of view, the project seeks to address the ``forward problem," in which a dynamical system is chosen at random from a collection of systems and one would like to characterize its behavior. This line of research will shed light on what type of behavior one can expect to see in a typical system. From the statistical perspective, the project focuses on the ``inverse problem," in which one would like to learn or draw inferences from observations of a (possibly unknown) dynamical system. More specifically, the project involves analyzing the performance of statistical inference methods when applied to observations of a dynamical system or group action. Results in this direction will provide theoretical guidance on when statistical procedures may be successfully applied in the context of dynamical systems.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.