Deformations of Geometric Structures and Related Topics

几何结构的变形及相关主题

• 批准号: 0722450
• 负责人: Morwen Thistlethwaite
• 金额: $ 5.97万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-15 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0722450&HistoricalAwards=false
• 关键词: Deformations; Geometric; Structures; Related; Topics

Modern low-dimensional topology is largely concerned with studying the various kinds of geometry that spaces can possess. Two rather different geometries are familiar to us all, specifically Euclidean plane geometry, where the sum of the angles of a triangle is 180 degrees, and the geometry on a sphere (e.g. the Earth), where the sum of angles of a triangle formed from arcs of great circles is greater than 180 degrees. A third, perhaps less familiar geometry, is hyperbolic or non-Euclidean geometry, discovered in the early nineteenth century; here the angle sum of a triangle is in fact less than 180 degrees. Hyperbolic geometry has since assumed a position of pre-eminence in mathematics, owing to its richness and beauty. It should be noted that the mathematics underlying hyperbolic geometry is also important in physics, in particular with regard to special and general relativity. The investigators will acquire a powerful multiprocessor computer with an unusually large amount of memory, and will implement methods they have devised for analyzing deformations of geometries. Knowing the behavior of a geometry under deformations, and the extent to which it can be deformed before it collapses, is an important factor in understanding its nature. The investigators are particularly interested in projective geometry, and a fascinating variant of hyperbolic geometry known as complex hyperbolic geometry (the relation between the "standard" and complex versions is somewhat analogous to the relation between the real numbers and the complex numbers.) The investigators will need to handle complicated objects, including fractal curves, in 3, 4 or 5 dimensions, necessitating much computer power.The goal of this project is to pursue, with the help of a powerful multiprocessor computer, two avenues of research in low-dimensional topology, each concerned with representations of manifold and orbifold fundamental groups. These two topics share an urgent need for an exceptional amount of computer memory, namely a minimum of 64 GB. The first topic concerns deformations of real hyperbolic structures on 2- and 3-dimensional manifolds and orbifolds to structures arising from real projective or complex hyperbolic geometry. The investigators had previously discovered that certain sporadically occurring closed hyperbolic 3-manifolds with small volume admitted such deformations; however, the underlying cause remains a mystery, and the study of degenerations of the resulting structures is virtually uncharted territory. On the other hand, the existence of complex hyperbolic deformations for the 2-sphere with three cone points (of suitable orders) is well established, but again almost nothing is known about degenerations of these structures. The investigators will use the new computer to implement a battery of methods for establishing discreteness of the associated holonomy representations, including a promising new technique based on the natural fibration from (2n+1)-dimensional real projective space to n-dimensional complex projective space. The second topic of research concerns various conjectures related to representations of 3-manifold groups, including knot groups, onto finite groups.The need to store large group conjugacy and multiplication tables again mandates the availability of unusual amounts of computer memory.

现代低维拓扑学主要关注研究空间可能具有的各种几何形状。我们都熟悉两种截然不同的几何，特别是欧几里得平面几何，其中三角形的内角和是180度，而球面几何（例如地球）中，由大圆的弧组成的三角形的内角和大于180度。第三种，也许不太熟悉的几何，是双曲或非欧几里得几何，发现于19世纪早期；这里三角形的角和实际上小于180度。双曲几何由于其丰富性和美丽性，在数学中占据了卓越的地位。应该指出的是，双曲几何的数学基础在物理学中也很重要，特别是在狭义相对论和广义相对论方面。研究人员将获得一台功能强大的多处理器计算机，拥有异常大的内存，并将实现他们设计的分析几何变形的方法。了解几何形状在变形下的行为，以及它在崩溃之前可以变形的程度，是理解其本质的重要因素。研究人员对射影几何和双曲几何的一种迷人的变体——复双曲几何（“标准”和复版本之间的关系有点类似于实数和复数之间的关系）特别感兴趣。研究人员将需要处理复杂的对象，包括3、4或5维的分形曲线，这需要大量的计算机能力。这个项目的目标是在一个强大的多处理器计算机的帮助下，追求低维拓扑的两种研究途径，每一种都涉及流形和轨道基本群的表示。这两个主题都迫切需要大量的计算机内存，即至少64 GB。第一个主题是关于二维和三维流形和轨道上的实际双曲结构对由实际投影或复杂双曲几何产生的结构的变形。研究人员先前已经发现，某些偶发的小体积闭双曲3-流形允许这种变形；然而，潜在的原因仍然是一个谜，而对由此产生的结构退化的研究实际上是未知的领域。另一方面，具有三个圆锥点（适当阶数）的2球的复杂双曲变形的存在是很好的确定的，但是对于这些结构的退化仍然几乎一无所知。研究人员将使用新计算机实现一系列方法来建立相关完整表示的离散性，包括一种基于从（2n+1）维实射影空间到n维复射影空间的自然纤维的新技术。研究的第二个主题涉及与3流形群（包括结群）在有限群上的表示有关的各种猜想。存储大群共轭和乘法表的需要再次要求使用异常多的计算机内存。