NSF Award
2404309
Dynamical systems and ergodic theory investigate the evolution of physical, biological or mathematical systems over time, such as turbulence in a fluid flow, the motions in planetary systems or the evolution of diseases. Fundamental ideas and concepts such as information, entropy, chaos and fractals have had a profound impact on our understanding of the world. Dynamical systems and ergodic theory have developed superb tools with applications to sciences and engineering. Symbolic dynamics, for example, has been instrumental in developing efficient and safe codes for computer science. Tools and ideas from smooth dynamics are used as far afield as cell biology and meteorology. Geometry is a highly developed, ancient yet superbly active field in mathematics. It studies curves, surfaces and their higher dimensional analogs, their shapes, shortest paths, and maps between such spaces. Surveying the land for his principality, Gauss developed the fundamental notions of geodesics and curvature, laying the groundwork for modern differential geometry. It has close links with physics and applied areas like computer vision or the more current geometric and topological data analysis. Geometry and dynamics are closely connected. Indeed, important dynamical systems such as the geodesic flow come from geometry, and conversely one can use geometric tools to study dynamics. One main goal of this project is to study symmetries of dynamical systems, especially when one system is unaffected by the changes brought on by the other. The quest is to study these systems via unexpected symmetries. Important examples arise from geometry when the space contains many flat subspaces. Under additional assumptions, one can classify such spaces. Finally, group theory is introduced in both dynamics and geometry via the group of symmetries of a geometry or dynamical system. The principal investigator will continue training a new generation of researchers in mathematics, and students at all levels in their mathematical endeavors. This project includes support for research training opportunities for graduate students and summer research experiences for undergraduates.<br/><br/>This project will investigate rigidity phenomena in geometry and dynamics, especially actions of higher rank abelian and semi-simple Lie groups and their lattices. The latter is part of the Zimmer program. Particular emphasis will be put on hyperbolic actions of such groups. As higher-rank semisimple Lie groups and their lattices contain higher-rank abelian groups, the classification and rigidity problems for the abelian and semi-simple cases are closely related, with abundant cross-fertilization. The goal is the classification of such actions. Closely related are the study of automorphism groups of geometric structures. A further goal is to understand topological joinings of lattice actions on Furstenberg boundaries and the related problem of classifying discrete subgroups in semisimple ie groups with higher Prasad-Rapinchuk rank. Investigations in geometry will address higher-rank Riemannian manifolds and their classification, introducing novel methods. The dynamics of geodesic and frame flows will also be studied, with investigations of discrete subgroups of Lie groups for rank rigidity and measure properties. Besides establishing new results, the principal investigator also strives to find and introduce novel methods for investigating these problems which will lend themselves to applications in other areas.<br/><br/>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.
