Dynamics and stochastics, both interpreted in their broadest sense, are important mathematical areas that have made many contributions to applications. There are many natural links between these two areas that lead to a better appreciation of techniques and methods used in either field. In addition, an approach that combines stochastic modelling with a dynamical-systems analysis often has the best chances of tackling a problem successfully: three examples are nonlinear optics, where noise from various sources plays an important role in laser and fiber dynamics; stochastic networks, where stability of the network can, in many cases, be determined by the analysis of a related fluid limit that is characterized by an ordinary or partial differential equation; and cell physiology, where stochastic models of ion channel gates often provide better agreement with experiments. A similar trend occurs on the theoretical level: probabilistic methods are an important tool that aids in the analysis of PDEs, for instance in the derivation and validation of scaling laws and the dynamics of fronts in heterogeneous media; conversely, dynamical-systems methods provide insight into the behavior of PDEs with noise. The goal of this project is to broaden and enhance the scope and quality of the educational and research training provided to graduate students and postdoctoral fellows by integrating research and education in the fields of dynamics, stochastics, and their applications and to involve more undergraduate students in courses and research experiences in applied mathematics.
Dynamical systems and stochastic processes are highly active and exciting fields of research that make important contributions to many applications in economics and the natural and social sciences, whilst also being of intrinsic mathematical interest. Dynamical-systems theory is concerned with time-dependent processes, while stochastics deals with nondeterministic, random processes. Examples where the interplay between these two fields is important are noise fluctuations in the design of high-power lasers, the long-time behavior of random interacting systems such as consensus formation in social networks, self-assembly of micro- and nanostructures for drug delivery, random fluctuations in biological cell processes, and importance sampling of rare events in finance. Providing more systematic and integrated training in dynamics and stochastics for graduate students and postdoctoral fellows will prepare these groups better for careers in academia and industry. It will also lead to an increase in the number of US mathematicians trained in stochastic systems, and therefore make the US better able to compete with Europe, which has a larger community in this area. The initiatives for undergraduate students will increase the number of students who are exposed to applied mathematics and are engaged in summer research experiences.
