This project will address several important questions concerning higher-rank smooth group actions: characterization of actions that cannot be obtained from classical dynamical systems (diffeomorphisms and flows) via classical constructions; the prevalence of chaotic behavior in the space of partially hyperbolic higher-rank actions; the extent to which global hypoellipticity of the sublaplacian for conservative actions gives a general set-up for stability under perturbations; the search for new nonalgebraic examples of weakly rigid higher-rank actions. The creation of a bridge between findings on algebraic higher-rank actions that rely on analytic tools and the more dynamical approach used for general smooth actions is crucial for an improved understanding of rigidity phenomena. For algebraic higher-rank actions, the project will study connections between cohomological obstructions obtained geometrically and those obtained analytically from the induced action on representations. Part of the strategy is to work towards answering these questions by focusing on representative examples. The principal investigator's main interest is in actions by groups that have higher rank but that lack rich geometric or algebraic structure (e.g., abelian groups, nilpotent groups, solvable groups).
Persistence of dynamics under perturbations is an old question in science. We completely understand the future and the past only for sufficiently simple systems, which are merely approximations of observed phenomena. Chaotic behavior was initially considered to be a pathology. However, it turns out to be a source of stability. Studies showing that it is a healthy heartbeat that demonstrates the presence of chaos are not surprising, provided that one accepts the fact that nature prefers stability. For systems that fall under the heading "group actions" the relation between chaos and stability is even more dramatic: weaker chaotic behavior tends to imply stronger stability for the system. Group actions can be thought of as systems with multidimensional time. As such, they are useful models in biology (neural networks), chemistry (quasi-crystals), and computer science (multidimensional data storage). In particular models, chaos appears in different guises. It is a goal of this project to explore conditions under which systems with diverse chaotic behavior preserve their dynamical properties under perturbations. This topic is especially amenable to introducing students to research in the area of dynamical systems. Through the study of simple models, students can develop intuition, learn what the open problems are, and make their own contribution to the actual research by performing specific computations. This is the rationale for the project's outreach component, which is aimed at high-school girls and the goal of which is to introduce mathematical research and insights into academic careers to female students at an early stage in their intellectual development.
