From the human brain to the electric power grid, scientists and engineers increasingly depend on large-scale computer models to understand, predict, and even control complex dynamical systems. However, these models often reflect the complexity of the systems they represent, and require supercomputers or large computing clusters to run. For many tasks of modern scientific computing, this can represent an impediment. For example, when estimating the values of model parameters based on physical measurements (a critical step in using a model to make predictions), it is necessary to run a model many times. Even with ever-increasing computational power, this may be time-consuming. This project studies computational and mathematical techniques for constructing simpler models that nevertheless capture key features of more complex models, so that tasks like parameter estimation can be done more efficiently. The algorithms resulting from this research are expected to be applicable to a wide range of scientific and engineering problems. This research project is also expected to be a good training vehicle for young scientists in interdisciplinary research. In more detail, this project concerns a discrete-time approach to model reduction for high-dimensional chaotic / noisy dynamical systems. The primary aims are (i) to develop a general mathematical framework for discrete-time model reduction, focusing on enabling both short- to medium-range forecasting as well as reproducing selected long-time statistics; (ii) adapting the general methods to handle specific types of dynamical systems that arise in practice, e.g., chaotic, noisy, etc.; (iii) apply the methodology to model reduction and data-driven modeling problems in specific biological and physical applications. As many dynamical systems of interest in contemporary science and engineering consist of a large number of degrees of freedom interacting across spatial and temporal scales, detailed first-principles models may not provide a practical basis for tasks like uncertainty quantification, parameter / state estimation, and optimization. Moreover, for many such systems, our knowledge of pertinent facts and principles are incomplete. Reduced models that capture the essential dynamics without resolving all degrees of freedom are thus of practical utility. They are also of great interest in their own right, as good reduced models leave out irrelevant details and often contain useful insights into the phenomenon at hand. This project draws on diverse ideas from applied probability, dynamical systems, and statistical physics, and provide ample opportunities for training mathematical scientists for the mastery of these tools. 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.