Topics in Dynamics, Differential Topology and Differential Geometry

This proposal concerns a variety of concrete research problems from the theory of dynamical systems, differential topology and differential geometry, mostly motivated by physical or engineering applications. They include estimating below the number of periodic motions in systems with elastic collisions (billiards, ideal gas models), rigidity of integrable billiards, motion of an electric charge in magnetic field (in particular, the magnetic analog of Hilbert's fourth problem), tire track geometry and flotation problems, bihamiltonian approach to the filament equation in hydrodynamics and its discrete versions, complexity of the motion planning problem in topological robotics, non-degenerate immersions and embeddings of manifolds (the latter are closely related with H-principle of differential topology and with the topology of configuration spaces of smooth manifolds).

There are two themes, unifying the diverse problems considered in the proposal. The first is the interplay between continuous and discrete (for example, evolution of curves describing vortices in ideal fluid versus similar evolution of polygons). The second unifying theme is the interplay between rigidity and flexibility of remarkable dynamical or geometrical phenomena (for example, is a round cylinder the only uniform cylindrical solid - a log - that floats in equilibrium in all positions? The answer depends on the density of the log.)