Coarse Differentiation and Teichmuller Dynamics

这一建议包括两个主要部分。第一部分讨论粗几何和几何群论。PI（与D. Fisher和K. Whyte一起）最近开发了一种新技术，“粗微分”，它可以被视为准等距的一种可微性替代品。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比例尺上定义；相反，我们必须走向越来越大的尺度。利用该技术，我们能够解决该领域三个长期存在的开放性问题，即证明了三维可解群Sol的准等距刚性，展示了一个与任何Cayley图都不是准等距的传递图，并证明了两个点灯群Z环积Z模2和Z环积Z模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无法解决的问题。第二部分涉及有理多边形、阿贝尔微分和二次微分模空间中台球的相关解析研究，以及这些模空间上SL（2，R）作用和测地流的动力学。在最近与M. Mirzakhani的合作中，PI解决了V. Veech在这一领域20年前的猜想。有些技术是基于局部对称空间上的流动的松散类比。尽管微分的模空间本质上是不同的，但PI过去和现在都涉及到将一些对称空间技术转移到这个设置中。我们建议在这方面进行进一步的研究。提案第一部分中的一些粗糙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，可以通过尽可能少地切割边缘来找到断开图形的方法。事实上，我们的一些想法已经被用于解决这个领域的问题。有些自然现象是“混乱的”（即不可预测的）。这些通常用统计方法来研究。其他则是“可积的”（即可预测和规则的）。其他现象介于两者之间。多边形台球系统是一种很好的中间行为模型，是本文第二部分的主要研究对象之一。因此，它在物理学中也得到了广泛的研究，特别是在“量子混沌”方面。

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".