Deformations of Geometric Structures and Related Topics

现代低维拓扑学主要研究空间所能拥有的各种几何。 两个相当不同的几何是熟悉的，我们所有人，特别是欧几里德平面几何，其中的角度总和的三角形是180度，和几何上的一个领域（例如地球），其中的角度总和的三角形形成的弧的大圆是大于180度。 第三种几何，也许不太熟悉，是双曲或非欧几里德几何，发现于世纪初;这里的三角形的角和实际上小于180度。 由于其丰富和美丽，双曲几何从此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，具有异常大的内存量，并将实现他们设计的用于分析几何形状变形的方法。 了解几何体在变形下的行为，以及它在塌陷之前可以变形的程度，是理解其性质的一个重要因素。 研究者对射影几何特别感兴趣，以及双曲几何的一个迷人的变体，称为复双曲几何（“标准”和复版本之间的关系有点类似于真实的数和复数之间的关系）。 研究人员将需要处理复杂的对象，包括分形曲线，在3，4或5维，需要大量的计算机能力。这个项目的目标是追求，在一个强大的多处理器计算机的帮助下，在低维拓扑研究的两个途径，每一个都涉及表示的流形和orbifold基本群。 这两个主题都迫切需要大量的计算机内存，即至少64 GB。 第一个主题涉及变形的真实的双曲结构的2-和3-维流形和orbifolds结构所产生的真实的投影或复杂的双曲几何。 研究人员先前发现，某些偶尔发生的小体积封闭双曲三维流形承认这种变形;然而，根本原因仍然是一个谜，对由此产生的结构退化的研究几乎是未知的领域。 另一方面，具有三个锥点（适当阶数）的2-球面的复双曲变形的存在性是确定的，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）维真实的投影空间到n维复投影空间的自然纤维化的有前途的新技术。 研究的第二个主题涉及到的各种illustrures有关的表示3-流形组，包括结组，到有限groups.The需要存储大组共轭和乘法表再次授权的可用性不寻常的数额的计算机内存。

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.