CAREER: Stochastic Forward and Inverse Problems Involving Dynamical Systems

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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.