Geometric and Topological Study of Systems with Impact and Related Topics

The proposed research concerns the geometry and topology of dynamical systems with impact: classical billiards, billiards in magnetic fields, outer billiards and other systems defined by variational principles or geometrical constructions. The research focuses on the number and structure of periodic orbits, stability properties of these systems, rigidity of integrable dynamics and related problems of geometry and topology. The techniques include variational principles (in particular, Morse theory) and related methods of algebraic topology and geometry, in particular, topological study of various configuration spaces. Applications to other areas are concerned, in particular, topological issues of the motion planning problems in robotics.

The main emphasis of the proposed research is in the interrelations between seemingly distant ideas and approaches from geometry and topology, on the one hand, and dynamical systems, mechanics, electrodynamics and robotics, on the other. The proposed research will make an impact on undergraduate research in mathematics via the Mathematics Advance Studies Semesters program (MASS) at Penn State of which the PI is the Director.