Research in hyperbolic geometry and mapping class groups

双曲几何与映射类群研究

• 批准号: 1207873
• 负责人: Kenneth Bromberg
• 金额: $ 20万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207873&HistoricalAwards=false
• 关键词: Research; hyperbolic; geometry; mapping; class

Since Thurston's work on the geometrization conjecture, the study of hyperbolic structures on surfaces and 3-manifolds has been a central topic in low-dimensional topology and geometry. Thurston made a number of conjectures that have been a driving force in the field, not only due to their intrinsic interest but because the study of these conjectures has produced new mathematics that has had wide ranging impact in many fields of mathematics. In the previous decade many of Thurston's original conjectures have been solved, introducing both new techniques and new problems into the field. The principal investigator will use these new tools to answer further questions about hyperbolic 3-manifolds and in particular will study the topology of spaces of hyperbolic 3-manifolds attempting to answer questions such as if these spaces are locally connected and if not where locally connectivity fails. One area where the study of hyperbolic 3-manifolds has had great influence is in the study of mapping class groups. This is perhaps the simplest example of a naturally occurring group that is not a lattice in a Lie group and has been much studied by geometric group theorists. The PI will apply many of the tools developed in the study of hyperbolic 3-manifolds, such as the curve complex, to study problems about the mapping class group.Three-manifolds are a central object of mathematical study for several reasons. The most obvious is that the space we live in is a 3-manifold and one basic goal of mathematics is to gain a better understanding of the real world. On a more abstract level, 3-manifolds have a deep and intricate structure that connects to many other areas of mathematics. This project will focus on hyperbolic 3-manifolds. When looking at the simplest examples it first appears that hyperbolic 3-manifolds are quite rare. However, a deeper analysis shows that, in a natural sense, hyperbolic 3-manifolds are the most prevalent type of 3-manifolds. While 3-manifolds themselves are topological objects to understand hyperbolic 3-manifolds them one is quickly led to algebra, geometry and analysis. By studying hyperbolic 3-manifolds we learn more about mathematics as a whole.

自从Thurston的工作的几何化猜想，研究双曲结构的表面和3流形一直是一个中心议题，在低维拓扑和几何。瑟斯顿提出了一些programmures一直是一个驱动力领域，不仅是由于其内在的利益，但因为研究这些programmures产生了新的数学，已广泛的影响，在许多领域的数学。在过去的十年中，瑟斯顿的许多原始问题都得到了解决，为该领域引入了新技术和新问题。首席研究员将使用这些新工具来回答有关双曲3-流形的进一步问题，特别是将研究双曲3-流形空间的拓扑结构，试图回答诸如这些空间是否局部连通以及局部连通性是否失败等问题。其中一个领域的研究双曲3-流形有很大的影响是在研究的映射类群。这也许是自然发生的群不是李群中的格的最简单的例子，并且已经被几何群理论家研究了很多。PI将应用在双曲3流形研究中开发的许多工具，例如曲线复形，来研究关于映射类组的问题。三流形是数学研究的中心对象，有几个原因。最明显的是，我们生活的空间是一个三维流形，数学的一个基本目标是更好地理解真实的世界。在更抽象的层面上，三维流形有一个深刻而复杂的结构，连接到许多其他数学领域。这个项目将集中在双曲三维流形。当看最简单的例子时，首先出现的是双曲3-流形是相当罕见的。然而，更深入的分析表明，在自然意义上，双曲三维流形是最普遍的三维流形类型。虽然3-流形本身是拓扑对象，以了解双曲3-流形，他们很快导致代数，几何和分析。通过研究双曲三维流形，我们可以从整体上了解更多的数学知识。

项目成果

Schwarzian derivatives, projective structures, and the Weil–Petersson gradient flow for renormalized volume

施瓦茨导数、射影结构和重正化体积的 Weil-Petersson 梯度流

• DOI: 10.1215/00127094-2018-0061
• 发表时间: 2019
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Bridgeman, Martin;Brock, Jeffrey;Bromberg, Kenneth
• 通讯作者: Bromberg, Kenneth