Ergodic Theory of Parabolic Flows

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Dynamical systems can roughly be classified according to the speed of divergence of nearby trajectories. Systems with sub-exponential, polynomial divergence of nearby orbits are often called parabolic. Parabolic dynamical systems arise in many mathematical models of scientific phenomena and in applications of dynamical systems to other branches of mathematics, in particular to number theory and geometry. The main long term goal of the research is to explore how far the ideas and methods developed in the study of the above-mentioned systems, based on renormalization and harmonic analysis, can be generalized towards a theory of parabolic dynamics. In the near future, Forni plans to organize his research around three main themes: ergodic theory on finite-area translation surfaces, compact and non-compact, and of billiards in polygons; ergodic theory of smooth time-changes of nilflows and horocycle flows; renormalization and quantitative equidistribution of (higher-step) nilflows and Weyl sums. Among the questions considered are long-standing open problems such as ergodicity of billiards in polygons, the spectral type of smooth time-changes of horocycle flows and optimal bounds on Weyl sums.The proponent's research is in the field of dynamical systems, that is, the study of motion of a deterministic system with time. The most classical example is the motion of the planets, but dynamical systems arise in all area of sciences, for instance in physics, biology, economics. In addition, the methods of dynamical systems can be applied to study problems in other fields of mathematics, in particular in geometry and number theory, as it has been done very successfully in recent decades as dynamical systems has moved closer and closer to the core of pure mathematical research (without abandoning its original strong connection to natural sciences and to applied mathematics). In dynamical systems there is a well-established theory of chaos which applies to systems whose nearby trajectories diverge exponentially fast with time (the weather is perhaps the most famous example.) At the other end of the spectrum, there is regular motion, characterized by trajectories that move all together. The proponent's goal is to advance fundamental research on the intermediate case of weakly chaotic systems, that is, systems that have some measure of chaotic behavior, but whose nearby trajectories diverge at most polynomially fast with time. The motion of an idealized billiard ball on a polygonal table is an example. Systems of this kind, called parabolic, are especially important in applications to geometry, number theory and to mathematical models coming from several branches of physics: solid-state physics, celestial mechanics, statistical mechanics. For instance, in recent years in number theory and, to a lesser extent, in geometry many questions have been reformulated (sometimes solved) as questions on the dynamics of certain parabolic flows. In physics, the motion of an electron on the the so-called Fermi surface of an atom (solid-state physics), the motion of planets near a singularity (celestial mechanics), the motion of an atom in a box (statistical mechanics) are related to parabolic systems. This project also has an important training component with the goal of forming researchers with wide mathematical knowledge.