Topics in Differential Dynamics and Differential Topology

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The project concerns two topics: the geometry and dynamics of billiard-like dynamical systems, and the topology and geometry of Legendrian knots and curves in contact 3-dimensional manifolds. The former includes the study of the classical Birkhoff billiards, dual (or outer) billiards, and projective billiards. The techniques include symplectic topology, KAM theory, Aubry-Mather theory, integral geometry, and symbolic dynamics. The latter includes applications of "quantum" topology to Legendrian and transverse knots and links, study of the recently introduced contact homology rings and their applications, and applications of Sturm theory and the theory of generating functions to the global geometry of Legendrian curves. The motivation for the study of billiards is two-fold. First of all, mathematical billiards are intimately related to geometrical optics, and progress in the study of billiards may have practical applications in optics. Secondly, billiards provide a very good model in the theory of dynamical systems, and many developments in various areas of dynamical systems have been stimulated by problems from the theory of mathematical billiards. The theory of Legendrian curves belongs to the intersection of two very active research areas: symplectic topology and knot theory. Both have deep connections with theoretical physics: the former, with classical mechanics; the latter, with quantum physics. The Legendrian knot theory provides a good testing ground for symplectic topology and knot theory, and progress in the former will stimulate new developments in these fundamental theories.