Topics in low-dimensional topology

摘要奖：DMS-0104039首席研究员：Darren Long提出者计划继续他们在各种问题上的工作，这些问题与理解低维流形的几何和拓扑方面有关。在拓扑学方面，研究领域包括Heegaard分裂、隧道解结和相关主题；几何方面处理与双曲流形、表面子群的构造和融合有关的问题，发展对高维双曲流形的理解，以及与其他领域的一些相互作用，例如代数数论。流形在物理、数学和某种程度上在其他科学中发挥着核心作用，因为它们在小尺度上看起来像n维的欧几里德空间。例如，我们生活的空间是一个三维流形，时空是四维流形。由于这个和其他原因，这些维度在数学和物理学中引起了很大的关注。一个基本的悬而未决的问题是，哪个流形才是正确的宇宙模型--有一些猜测认为它属于所谓的双曲流形，在某种意义上似乎是通用的一类三维空间。关于宇宙起源的宇宙学理论限制了时空的形状，以及此后可能出现的流形的限制。这个项目的目标之一是改进现有的方法(这些方法已经排除了许多可能性)，以进一步缩小搜索范围。几何思想的其他应用也是不明显的，但事实上，仍然是直接重要的。毫不夸张地说，几乎任何可以定性表述的问题都可以用几何方法来研究，因此，过去三十年来发展起来的强大的低维拓扑学工具可以应用于它。

AbstractAward: DMS-0104039Principal Investigator: Darren LongThe proposers plan to continue their work on a variety ofproblems bearing on the understanding of the geometric andtopological aspects of low dimensional manifolds. On thetopological side, the areas of study include Heegaard splittings,unknotting tunnels and related topics; the geometric side dealswith issues related to orbifolds, the construction and melding ofsurface subgroups, developing an understanding of higherdimensional hyperbolic manifolds, as well as some interactionswith other areas, for example algebraic number theory.Manifolds play a central role in physics, mathematics and to someextent in other sciences, since they are objects which on smallscales look like Euclidean space of dimension n. For example,the space that we live in is a three manifold and space-time is afour manifold. For this and other reasons, these dimensions haveattracted a good deal of attention in mathematics and physics.One basic unsolved problem is exactly which manifold is thecorrect model for the universe - there has been some speculationthat it falls into the class of so-called hyperbolic manifolds, acertain class of three dimensional spaces which in some senseappear to be generic. Cosmological theories about the origins ofthe universe put constraints on the shape of space-time and henceconstraints on which manifolds could occur. One of the goals ofthis project is refine current methods (which have already ruledout many possibilities) to narrow the search down further. Thereare also other applications of geometric ideas which are lessobvious but still, in fact, directly important. It is noexaggeration to say that almost any problem which can beformulated qualitatively can be studied with geometric methodsand as a result the powerful tools developed over the last thirtyyears in low-dimensional topology can be brought to bear upon it.