This course will explore applications of mathematical data science to analysis and statistical modeling of dynamical systems. We will focus on operator-theoretic approaches, which characterize dynamical systems through their induced action on linear spaces of observables. Remarkably, this action is through linear evolution operators—the Koopman operators, which act on observables by composition with the dynamical flow map, and the transfer operators, which are the dual operators acting on spaces of measures. Using this framework, we will study how techniques from linear operator theory can be employed in a variety of problems in nonlinear dynamics, such as forecasting and identification of coherent observables. In addition, we will study, both theoretically and numerically, associated data-driven approximation techniques, using samples taken on dynamical trajectories to approximate observables and evolution operators.
A tentative weekly plan for the course can be found here.