Big Data and the Dynamical Systems Approach: New Directions and Applications (chemistry, geophysical fluid dynamics) in Applied Mathematics

The methodology of applied and computational mathematics provides capability, insight, and predictive possibilities for a variety of fields in engineering and science. Historically, the pursuit of applied mathematics has had a synergistic effect in the sense that it has often led to new developments in mathematics. For example, the development of fluid and solid mechanics in the 19th century played a significant role in the development of the theory of nonlinear partial differential equations in the 20th century.

Over the past thirty years science and engineering have moved towards the consideration of more complex systems; for example, systems modeling particular environmental situations (e.g. Arctic ocean transport, oil spills) or the dynamics of complex molecular systems. Such complex systems do not readily give rise to tractable models through the traditional approach of physics based modeling. Rather, much understanding of such systems comes from new capabilities for acquiring data that describe the behavior of such systems with remarkable spatial and temporal resolution.

In this talk I will show that the framework of dynamical systems theory can be used in conjunction with data to provide new understanding and predictive capability for such complex systems. To do this I will consider three case studies in the geophysical and environmental fluid mechanics setting (i.e. an oil spill, path planning for autonomous underwater vehicles, and Arctic ocean transport phenomena). I will also consider the (surprising) impact that this approach has had in recent years on chemical reaction dynamics. If time permits, I will discuss how this work has motivated new developments in mathematics. Finally, I will discuss implications for future research directions and possibilities, as well as the needs for students in applied mathematics planning to embark on careers in this type of research.