Dynamical Rigidity through Incidence Geometry

About This Grant

This project will develop analytical methods to approximate the state of certain dynamical systems to within manageable errors, with an eye towards computational feasibility. Dynamical systems provide mathematical models for the long-term trajectories of objects moving according to physical principles. The subject has its foundations in Newtonian mechanics, and features widely in both pure and applied mathematics (e.g., fluid dynamics, airflow dynamics, Hamiltonian mechanics). Hamiltonian mechanics, which provides an alternative formulation to Newtonian mechanics and is particularly useful in classical and quantum mechanics, expresses the time evolution of a system in terms of partial derivatives of a certain energy. The resulting equations, called Hamilton’s equations, describe how the coordinates and momenta of a system evolve over time. The project will seek to further the understanding of quantitative aspects of the analysis of various dynamical systems, under the mathematical rubric of rigidity. The project also provides opportunities for the training and mentoring of early career researchers, especially graduate students. The PI will contribute to the dissemination of mathematical knowledge through the organization of various conferences, workshops, and long research programs. The project resides at the intersection of dynamical systems, ergodic theory, number theory, and geometry. It seeks both to establish new rigidity results and to advance and refine existing results, in the setting of the dynamics of group actions on structured spaces. A major focus is on an enhanced understanding of finitary analysis and effective conclusions. Rigidity phenomena for the dynamics of group actions on homogeneous spaces and moduli spaces have been studied extensively. However, a quantitative understanding of the behavior of orbits has proven to be challenging. This project seeks results in this direction, with an emphasis on systems with polynomial rates, and will also investigate implications of these findings within the domains of number theory and geometry. In addition, recent results of the PI and collaborators regarding finitary analysis on homogeneous spaces will be studied with an eye towards extensions to other settings, including dynamics on the moduli spaces of Riemann surfaces. 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.