Deformations of Geometric Structures and Related Topics

现代低维拓扑学在很大程度上涉及到研究空间可以拥有的各种几何。有两个非常不同的几何我们都很熟悉，特别是欧几里得平面几何，其中三角形的角度之和是180度，以及球面上的几何(例如地球)，其中由大圆弧组成的三角形的角度之和大于180度。第三种可能不太为人所知的几何是19世纪初发现的双曲线或非欧几里德几何；在这里，三角形的角度和实际上小于180度。自那以后，双曲几何因其丰富和美丽而在数学中占据了突出的地位。应该指出的是，作为双曲几何基础的数学在物理学中也很重要，特别是关于狭义相对论和广义相对论。研究人员将获得一台功能强大的多处理器计算机，具有异常大的内存，并将实施他们设计的分析几何变形的方法。了解几何体在变形时的行为，以及它在塌陷之前可以变形的程度，是了解其本质的重要因素。研究人员对射影几何和被称为复双曲几何的一种迷人的双曲几何(“标准”和复数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，包括三维、四维或五维的分形曲线，这需要大量的计算机能力。这个项目的目标是在一台强大的多处理器计算机的帮助下，探索低维拓扑的两种研究途径，每种途径都与流形和奥尔布里基本群的表示有关。这两个主题都迫切需要超大容量的计算机内存，即最低GB。第一个主题涉及二维和三维流形上的实双曲结构到实射影几何或复双曲几何中的结构的变形。研究人员此前发现，某些零星发生的小体积闭合双曲3-流形允许这种变形；然而，潜在原因仍然是一个谜，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，具有三个圆锥点的2-球面的复双曲变形的存在是公认的(合适的阶数)，但关于这些结构的退化几乎一无所知。研究人员将使用新的计算机来实现一系列建立相关完整表示的离散性的方法，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有前途的新技术。第二个研究主题涉及与有限群上的3-流形群(包括纽结群)的表示有关的各种猜想。存储大的群共轭和乘法表的需要再次要求具有非常大量的计算机内存。

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.