The theory of dynamical systems studies the time-evolution of complicated, multi-component systems, like particle systems in statistical physics, reaction kinetics from chemistry, the behavior of the atmosphere (hence the relevance in weather forecasting), population dynamics, developments on the stock market, etc. By nature, this theory is closely related to - and is partly arising from - the theory of differential equations and stochastic processes. An interesting feature of the theory of dynamical systems is that it helps us better understand such crucial phenomena in the time evolution of the above mentioned systems, as the high sensitivity of the solution to initial conditions, sometimes referred to as chaos, or chaotic behavior. My present proposal targets the investigation and better understanding of a popular and important class of mostly chaotically behaving mathematical models, namely the so called mathematical billiards. They got their name after the fact that they model the physical motion of ball shaped particles interacting with each other via elastic collisions.
Due to the above said, this project is primarily devoted to a special class of non-uniformly hyperbolic dynamical systems with singularities, namely hyperbolic, or at least partially hyperbolic mathematical billiards. Such systems have been playing an important role in the rigorous mathematical foundation of statistical physics, so that studying them and establishing their ergodic and statistical properties is getting more and more physical relevance. The main part of the proposal focuses on a the final steps in proving a fundamental conjecture regarding hard ball systems, namely the so called "Boltzmann-Sinai Ergodic Hypothesis", which states that any finite system of (totally elastic) hard spheres moving on a flat torus is fully hyperbolic and ergodic, of course, on the level set of its trivial first integrals. The proof of this conjecture (in its full generality) up until now has been notoriously withstanding any attack against it. The hardest part in presenting an inductive proof (induction on the number of interacting spheres) has been so far proving the so called "Ansatz", a global dynamical-geometric condition, claiming the almost sure hyperbolicity of singular orbits inside the realm of all singular trajectories. Since by now I got extremely close to make the closing steps in this big project that spanned over a couple decades of hard research work of several people, I can now confidently include finishing the proof of this hypothesis as the first major part of the recent proposal. Furthermore, this plan also contains my promise to join efforts with my colleague N. I. Chernov to publish a comprehensive book on the complete proof of the Boltzmann-Sinai Ergodic Hypothesis. The goal of publishing such a volume is two-fold: First, it would provide a comprehensive exposition of this theory with a unified system of notations, references, etc. for easier reading. Secondly, it would more thoroughly and clearly explain many involved technicalities of the theory, in order to make it accessible for a wider group of researchers of dynamical systems. The subsequent parts of the proposal contain blueprints for further research in this direction by generalizing the original Boltzmann-Sinai Hypothesis to cylindric billiards, and billiards in physically more relevant containers, i. e. rectangular boxes, convex domains, etc. These sections also address some further open problems in the theory of mathematical billiards that still remain open after my attampts to prove them in my previous research period. Their list can be found in the detailed description of the current proposal.
