Coarse Differentiation and Teichmuller Dynamics

该提案包括两个主要部分。 第一部分涉及粗糙的几何和几何群论。PI（与D。Fisher和K. Whyte）最近发展了一种新的方法--“粗微分法”，它可以被看作是拟等距的一种可微性替代。当然，传统的导数对这样的地图没有意义，因为它们甚至没有在小尺度上定义;相反，我们必须去更大的尺度。使用这种技术，我们能够解决三个长期存在的公开问题，即证明了准等距刚性的三维可解组溶胶，展示了一个传递图，这是不是准等距的任何凯莱图，并表明，两个点灯组Z圈产品Z mod 2和Z圈产品Z mod 3不是准等距的。我们列出了该方法的一些最新进展和其他潜在应用，其中许多是以前似乎完全遥不可及的问题。第二部分涉及有理多边形中台球的相关分析研究，交换和二次微分的模空间，SL（2，R）作用和测地线流在这些模空间上的动力学。 在最近的工作与M。Mirzakhani，PI能够解决V. Veech在这一领域的一个20年前的猜想。一些技术是基于一个松散的类比流局部对称空间。 尽管微分的模空间有很大的不同，但PI过去和现在都参与了将一些对称空间技术转移到这种设置中。我们建议在这个方向进行更多的研究。提案第一部分中的一些粗略几何与计算机科学有着意想不到的联系，特别是存在有效的算法，可以找到通过切割尽可能少的边来断开图的方法。事实上，我们的一些想法已经被用来解决这个领域的问题。有些自然现象是“混沌的”（即不可预测的）。这些通常是通过统计方法来研究的。另一些是“可整合的”（即可预测的和有规律的）。其他现象介于两者之间。多边形台球系统，这是我们的主要研究课题之一，在第二部分的建议，是一个很好的模型，中间行为。因此，它在物理学中也得到了广泛的研究，特别是与“量子混沌”有关。

This proposal consists of two main sections. The first section deals with coarse geometry and geometric group theory. The PI (together with D. Fisher and K. Whyte) has recently developed a new technique, "coarse differentiation", which can be viewed as a sort of differentiability substitute for quasi-isometries. Of course, conventional derivatives do not make sense for such maps, since they are not even defined on small scales; instead we must go to larger and larger scales. Using this technique, we were able to resolve three longstanding open problems in the field, namely proving the quasi-isometric rigidity of the three-dimensional solvable group Sol, exhibiting a transitive graph which is not quasi-isometric to any Cayley graph, and showing that the two lamplighter groups Z wreath product Z mod 2 and Z wreath product Z mod 3 are not quasi-isometric. We list some recent progress and other potential applications of the method, many of which are to problems which seemed completely out of reach before. The second section concerns the interrelated analytic study of billiards in rational polygons, moduli spaces of abelian and quadratic differentials, and the dynamics of the SL(2,R) action and the geodesic flow on these moduli spaces. In recent work with M. Mirzakhani, the PI was able to resolve a twenty year old conjecture by V. Veech in this area. Some of the techniques are based on a loose analogy with flows on locally symmetric spaces. Even though the moduli spaces of differentials are substantially different, the PI was and is involved in transferring some of the symmetric space techniques to this setting. We propose additional research in this direction. Some of the coarse geometry in the the first part of the proposal has unexpected connections to computer science, in particular the existence of efficient algorithms for finding ways to disconnect a graph by cutting as few edges as possible. In fact, some of our ideas were already used to solve problems in this field. Some natural phenomena are "chaotic" (i.e. unpredictable). These are often studied by statistical methods. Others are "integrable" (i.e. predictable and regular). Other phenomena fit somewhere in between. The polygonal billiard system, which is one of our main subjects of study in the second section of the proposal, is a good model of intermediate behavior. As such it has been studied extensively in physics as well, in particular in connection to "quantum chaos".