Ergodic Theory of Parabolic Flows

动力系统大致可以根据附近轨迹的发散速度进行分类。附近轨道具有次指数、多项式发散的系统通常被称为抛物线。抛物型动力系统出现在许多科学现象的数学模型中，以及动力系统在其他数学分支，特别是数论和几何中的应用中。这项研究的主要长期目标是探索基于重整化和调和分析的上述系统研究中发展起来的思想和方法在多大程度上可以推广到抛物动力学理论。在不久的将来，Forni计划围绕三个主题组织他的研究：有限区域平移曲面上的遍历理论，紧的和非紧的，以及多边形中的台球的遍历理论；零流和圆周流的光滑时变的遍历理论；(高阶)尼尔流和Weyl和的重整化和定量等分布。在所考虑的问题中，有诸如多边形中台球的遍历性、圆周流时变光滑的谱型和Weyl和的最优界等长期悬而未决的问题。最经典的例子是行星的运动，但动力系统出现在所有科学领域，例如物理学、生物学、经济学。此外，动力系统的方法可以应用于研究其他数学领域的问题，特别是在几何和数论方面，因为动力系统已经越来越接近纯数学研究的核心(而不放弃它最初与自然科学和应用数学的密切联系)，这一点在最近几十年里已经非常成功地完成了。在动力系统中，有一个久负盛名的混沌理论，它适用于其附近轨迹随时间以指数级速度发散的系统(天气可能是最著名的例子)。在光谱的另一端，有规则的运动，其特征是所有运动的轨迹都在一起。支持者的目标是推进对弱混沌系统中间情况的基础研究，即具有某种混沌行为的系统，但其附近的轨迹随着时间最多以多项式的速度发散。理想的台球在多边形桌上的运动就是一个例子。这种被称为抛物线的系统在几何学、数论和来自几个物理学分支的数学模型的应用中尤其重要：固体物理、天体力学、统计力学。例如，最近几年，在数论中，在较小程度上，在几何中，许多问题被重新表述(有时被解决)为关于某些抛物线流动的动力学的问题。在物理学中，电子在原子的所谓费米面上的运动(固态物理)，行星在奇点附近的运动(天体力学)，盒子里的原子的运动(统计力学)都与抛物系统有关。该项目还有一个重要的培训部分，目的是培养具有广泛数学知识的研究人员。

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.