Abstract Kononenko The main emphasis in our a program of research is on the singular geometry and its applications to billiards systems. The research focuses on geometric methods that allow one to restate various problems in the theory of semi-dispersing billiards in terms of geodesics in some Alexandrov spaces (typically of non-positive curvature). The problems that can be attacked with our methods have to do with counting the number of collisions in billiards, counting the number of periodic orbits, calculating entropy and other invariants of billiards systems. The most important open problems that we plan to work on are the construction of universal model spaces for billiards in non-positive curvature, and proving the finiteness of the topological entropy for the first-return map for billiards in non-positive curvature. In addition, hope to construct examples of billiards with areas of positive curvature, which have infinite entropy. We also plan to continue our work in the areas of differentiable and cohomological rigidity of lattice actions, and partially hyperbolic dynamical systems. The theory of semi-dispersing billiards lies at the heart of the foundations of the statistical physics. Much of our proposed research focuses on new geometric methods in the theory of semi-dispersing billiards that we have recently developed. Our new methods have already allowed us to solve several old problems that had been open for at least 15 years. In particular we have established the finiteness of entropy (measure of chaos in the system) for broad classes of billiards. Among other things we plan to use our approach to study various other statistical properties of hard-ball systems (the main model of statistical physics). In addition to our research on billiards we plan to work in the areas of rigidity and hyperbolic dynamics. In particular, we are interested in various questions related to the notion of stable ergodicity (a property of a s ystem to remain chaotic even after arbitrary, but "small" perturbations).
