Growth, Gap, and Geometry

奖项：DMS 1611758，首席研究员：陶静几何是人类研究的一个基础领域，起源于我们对世界的视觉认知；它或隐或显地存在于每一门科学学科中。对称及其用途的发现可以说是人类知识最伟大的成就之一。具体地说，理解和控制几何形状最有效的方法之一就是利用它们的对称性。现代的方法是将这些信息编码在我们称之为基本群的东西中。拓扑学家和几何群论家的一个目标是了解所有可能的基本群，对它们进行分类，并研究它们自身固有的几何性质。传统上，研究最多的几何物体都是低维的，尤其是二维的。这被称为表面理论，是拟议项目的重点领域之一。首席研究员还对从研究基本表面群中获得的直觉感兴趣，并将其推断到更复杂的物体上。这个领域被称为几何群论，在过去的几十年里，这是数学中一个日益活跃的学科。研究项目涉及几何群论和Teichmuller理论一般领域的广泛研究问题，与双曲几何，低维拓扑和动力学相互作用。具体内容包括：(1)度量空间上群作用的生长紧性。这个概念最初是由Grigorchuk和de la Harpe为单词度量引入的，它与群的Hopfian性质和秩刚性猜想有关。(2)直角Artin群及相关群的稳定换向子长度。(3)配备Thurston度规的Teichmuller空间几何。

Award: DMS 1611758, Principal Investigator: Jing TaoGeometry is a basic area of human research, originating with our visual awareness of the world; it is present implicitly or explicitly in every scientific discipline. The discovery of symmetries and their usefulness is arguably one the greatest achievements of human knowledge. Concretely, one of the most effective ways of understanding and controlling geometric shapes is to exploit their symmetries. The modern approach is to encode this information in something we call the fundamental group. One goal of topologists and geometric group theorists is to understand all possible fundamental groups, classify them, and study their own intrinsic geometric properties. Classically, the most studied geometric objects have been in low dimensions, particularly in dimension two. This is called surface theory and is one of the focus areas of the proposed projects. The principal investigator is also interested in taking the intuitions that one gains from studying the fundamental groups of surfaces and extrapolating them to more complex objects. This area is called geometric group theory, an increasingly active subject within mathematics over the last few decades.The research projects address a broad spectrum of research problems in the general areas of geometric group theory and Teichmuller theory, with interactions with hyperbolic geometry, low-dimensional topology, and dynamics. The specific topics include: (1) Growth tightness of group actions on metric spaces. This notion was first introduced by Grigorchuk and de la Harpe for word metrics and it has connections to the Hopfian property for groups and the Rank Rigidity Conjecture. (2) Stable commutator lengths in right-angled Artin groups and related groups. (3) The geometry of Teichmuller space equipped with the Thurston metric.