Homogeneous and Teichmuller Dynamics: A Quantitative Viewpoint

动力系统起源于经典力学，已成为现代数学不可缺少的组成部分。粗略地说，动力系统是预测几何空间中一点运动的规则；动力系统数学模型的例子包括流体动力学、气流动力学、台球轨迹等。作为一个具体的例子，考虑一个多边形台球桌，如l形台球桌；为了简化情况，我们假设桌子没有摩擦，入射角等于反射角。台球轨迹的行为一直吸引着数学家和物理学家——注意，不同的轨迹可以有不同的行为，例如，一些轨迹是周期性的，一些轨迹是密集的（它们在表中的每个点附近访问）。例如，我们可以问台球桌是否有周期轨迹，或者更定量地说，我们可以问一定长度的周期轨迹有多少？令人惊讶的是，这类问题已被证明是出了名的难以回答，并且与现代数学的几个领域有关。事实证明，即使要理解单个台球桌上的台球轨迹，也需要研究具有相同组合的台球桌族，例如，台球桌的边数相同。这些族具有新的对称性，并具有一定的刚性，这有助于解开原来的问题。这个项目研究的问题具有类似的哲学观点。该项目寻求某些刚性结果的扩展和扩展，其中关于对象的相当弱的初始数据产生了对象的几乎完整的分类；目标是在考虑应用程序的情况下，对这些刚性结果提供更定量的说明。以下将是主要目标：(i)动力系统已成为现代数学中几个意想不到的领域的主要参与者。然而，依靠动态论证的证明通常是存在性的，而不是建设性的。对于各种应用，动态参数的定量和有效的解释是非常需要和具有挑战性的。（ii）有限协体积李群的离散子群是数学中的中心对象，已被广泛研究。然而，在一些自然环境中——例如，Kleinian群，几个单群，Apollonian圆填充，和Zaremba的猜想——人们需要考虑具有无穷余体积的半单李群的Zariski密集离散子群（和半群）。这些研究常常导致对无限体积空间上群体行为的研究。紧黎曼曲面的模空间是现代数学的中心之一。尽管模空间是非齐次的，但是在这个空间和齐次空间之间有一些相似之处。模空间动力学是一个非常令人兴奋的研究方向，有许多突出的和具有挑战性的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Dynamical systems, which originated from classical mechanics, have become an indispensable part of modern mathematics. Roughly speaking, a dynamical system is a rule which predicts the motions of a point in a geometric space; examples of mathematical models of dynamical systems include the study of fluid dynamics, airflow dynamics, billiard trajectories, etc. As a concrete example, consider a polygonal billiard table, e.g., an L-shaped billiard table; and to simplify the situation, let us assume that the table has no friction and the angle of incidence equals the angle of reflection. Behavior of billiard trajectories have long been fascinating to mathematicians and physicists--note that different trajectories can have different behaviors, e.g., some trajectories are periodic and some trajectories are dense (they make a visit near every point in the table). For instance, one can ask whether a billiard table has any periodic trajectories, or more quantitatively, one can ask how many periodic trajectories of a certain length exist? Surprisingly, questions of this kind have proven to be notoriously difficult to answer and are connected to several areas of modern mathematics. It turns out that even to understand billiard trajectories on one single table one is led to the study of families of billiard tables which have same combinatorics, e.g., same number of edges for the table. These families turn out to have new symmetries and enjoy certain rigidity properties which help to unravel the original problem. This project studies questions with similar philosophical point of view.The project seeks extensions and extensions of certain rigidity results where a rather weak initial data about an object yields an almost complete classification of the object; the goal is to provide a more quantitative account of these rigidity results with applications in mind. The following will be the main objectives: (i) Dynamical systems have become a major player in several unexpected areas in modern mathematics. Proofs relying on dynamical arguments, however, are usually existential and not constructive. For various applications, quantitative and effective accounts of dynamical arguments are much sought after and challenging. (ii) Discrete subgroups of Lie groups with finite covolume are central objects in mathematics and have been extensively studied. In some natural settings -- e.g., Kleinian groups, several monodromy groups, Apollonian circle packings, and Zaremba's conjecture -- however, one needs to consider Zariski dense, discrete subgroups (and semigroups) of semisimple Lie groups with infinite covolume. These investigations often lead to the study of group actions on infinite volume spaces. (iii) The moduli space of a compact Riemann surface is one of the hubs of modern mathematics. Despite the very inhomogeneous nature of the moduli space, there are several similarities between this space and homogeneous spaces. Dynamics on moduli spaces presents a very exciting avenue for research with many outstanding and challenging problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.