Mathematical and numerical methods in celestial mechanics

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Celestial mechanics has historically served as a source of mathematical discovery and new problems.  This is still true today where the contemporary needs of space science bring a new wave of interest in the theoretical and practical aspects of celestial mechanics making its connections with mathematics and applications stronger than ever before. One of the main aspects of celestial mechanics is the study of the Newtonian N-body problem, namely the analysis of the motion of N point particles in a setting where the dynamics are dictated by Newton's law of motion. The study of the N-body problem now includes just about any dynamical system that remotely resembles the Newtonian N-body problem, and beyond the insight it provides in astronomy and space science it  has many applications in science and engineering.My research program has two main directions. The first concerns investigations of a theoretical nature. The second emphasizes applied aspects. From the theoretical point of view I am interested in studying properties of some special solutions of the N-body problem (i.e. homographic motions and relative equilibria) and the special configurations that are allowed in such motions (i.e. central configurations). The study of central configurations is important because they play a ``central role" in understanding N-body behavior: they characterize the dynamical behavior of collisions and expansions and play a key role in the topology of the integral manifolds of the N-body problem. From a more application oriented  point of view my research program is directed towards developing numerical methods that improve computational efficiency and permit new systems to be simulated with greater accuracy. Additionally, I am interested in analyzing several aspects of the full-body problems.  This is a fascinating class of problems that has many open questions and touches on numerous important issues in science and engineering,  as for example binary asteroids, the dynamics of the Earth-Moon system, reaction and ionization of molecules, stability and control of underwater vehicles and  rendezvous  and docking of space vehicles.