Science in the 21st century will focus on multiscale challenges, such as those arising from materials science, geoscience, life sciences, and social sciences, be guided by data driven models, and will interrogate these systems via large scale computation. Currently, these problems are typically described using phenomenological models with poorly quantified parameters. However, as information technologies continue to improve we expect to see more models based on representations of time series data. The thesis of this course is that we need a novel framework for nonlinear dynamics that is designed to address these types of systems.
With this in mind we will discuss the decomposition of dynamics from the perspectives of lattices (attractors) and posets (gradient-like dynamics) and the reconstruction of dynamics via algebraic topology (Conley-Morse theory). We will also discuss how this framework naturally allows for computer assisted proofs of global dynamics over large sets of parameter space. The particular applications that will be discussed will depend on the interests of the students, but may include topics from ecology, regulatory networks, fluids, and materials science.
course notes will be provided
Math 351, 441 or permission of instructor