Geometry and Topology of Hyperbolic 3-Manifolds

双曲3流形的几何和拓扑

• 批准号: 0603711
• 负责人: Peter Storm
• 金额: $ 10.98万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2007-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603711&HistoricalAwards=false
• 关键词: Geometry; Topology; Hyperbolic; Manifolds

The study of hyperbolic 3-manifolds has made tremendous progress in the past five years. The new tools available have great potential for attacking previously inapproachable questions. These new tools include Perelman's Ricci flow techniques, the presently quite sophisticated theory of discrete subgroups of PSL(2,C), and interactions with number theory via arithmetic lattices. Peter Storm will work to apply these new tools to the study of 3-manifolds, and more general negatively curved spaces such as Gromov hyperbolic spaces, or higher dimensional negatively curved manifolds. A 3-manifold is a mathematical object which models the three dimensional world we occupy. Thus questions about the nature of 3-dimensional space become mathematical questions about 3-manifolds. For this and other reasons, 3-manifolds are carefully studied by many mathematicians. They occupy a special niche between strongly intuitive subjects, such as geometry in the plane, and more abstract subjects, such as the study of mathematical objects with more than 4 dimensions, where meaningful mental images are difficult to produce. The research of Peter Storm tries to benefit from both the intuitive and the more abstract points of view, with the goal of gaining a better understanding the nature of three dimensional objects, and their mathematical cousins.

双曲型3流形的研究在过去的五年中取得了巨大的进展。可用的新工具在解决以前无法解决的问题方面具有很大的潜力。这些新工具包括Perelman的Ricci流技术，目前相当复杂的PSL离散子群理论（2，C），以及通过算术格与数论的相互作用。Peter Storm将致力于将这些新工具应用于3流形，以及更一般的负弯曲空间（如Gromov双曲空间）或高维负弯曲流形的研究。三维流形是一个数学对象，它模拟了我们所处的三维世界。因此，关于三维空间本质的问题就变成了关于三维流形的数学问题。由于这样或那样的原因，许多数学家仔细研究了3流形。它们在强烈直觉的学科（如平面几何）和更抽象的学科（如研究4维以上的数学对象）之间占据了一个特殊的位置，在这些学科中，很难产生有意义的心理图像。Peter Storm的研究试图从直观和更抽象的观点中获益，其目标是更好地理解三维物体的本质，以及它们的数学表兄弟。