Grants and Contributions:

Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023)

This proposal outlines the study of three connected research directions in the qualitative theory of differential equations: (1) the natural generalization of the classical N-body problem of celestial mechanics to spaces of constant curvature, namely spheres and hyperbolic spheres; (2) the extension of these equations to the case when the curvature of the spheres and hyperbolic spheres varies (i.e. they inflate or contract in time); (3) the generalization of the Vlasov-Poisson equations of stellar dynamics to spheres and hyperbolic spheres. Both (2) and (3) are built on (1), but in different directions. While (2) extends to a system of non-autonomous ordinary differential equations, with potential applications in cosmology, (3) uses (1) in the framework of kinetic theory, and may help understand some aspects of galactic dynamics. For (1), the main concept to be explored is that of central configurations, which we recently introduced for this system by exploring its analogue from the Euclidean case. Central configurations allow a unifying treatment of relative equilibria, which are solutions for which the point masses maintain constant mutual distances in time. Their study is fundamental for understanding the differential equations. For this purpose we will use geometric and algebraic methods in the study of dynamical systems, differential and non-Euclidean geometry, topology, geometric mechanics, as well as the theory of Lie groups and algebras. We will be interested in aspects related to the number of central configurations (a generalization of the Wintner-Smale conjecture) as well as in the stability of relative equilibria. For (2), we will rely on a recent result we obtained, which shows that the relative equilibria of (1) can be used to obtain solutions of the non-autonomous equations of motion that correspond to an expanding or contracting curved universe. In particular, we may be able to understand some dynamical aspects related to the current expansion of the universe under Hubble's law and find new connections between classical mechanics and general relativity. Techniques similar to the ones described above can be used in this problem too. For (3), we recently succeeded to prove that some interesting qualitative properties, such as Landau damping, occur in the linearized version of the Vlasov-Poisson system on spheres and hyperbolic spheres. We aim to show that they occur in the general case as well. To achieve this goal we will have to surmount the difficulties of extending certain techniques used in the Euclidean case by Villani, Penrouse, Mouhot, and others, a nontrivial task, given the new setting of the problem. Nevertheless, we already have some indication that this plan can be achieved. This research proposal offers many opportunities to engage undergraduates and graduate students as well as postdoctoral fellows.