Geometry and computability in low-dimensional topology and group theory.

低维拓扑和群论中的几何和可计算性。

• 批准号: 2283616
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2283616
• 关键词: Geometry; computability; low; dimensional; topology

With the proof of Thurston's Geometrisation Conjecture, 3-manifolds are in some sense classified. However, the non-constructive nature of the proof gives no direct method for deciding whether two given 3-manifolds are the same. In this project, Ascari will examine these computational questions. He will approach it using several techniques. On the one hand, topological tools such as hierarchies will prove to be useful. On the other hand, algebraic approaches, such as the use of the profinite completion of the fundamental group, will also be fruitful. Overlying both these approaches is the pervasive role of hyperbolic geometry, which is an area of expertise of Ascari. A field where these questions are particularly of interest is knot theory, which Ascari will also examine. We now explain these approaches in a bit more detail. The profinite completion of a finite generated group is a compact topological group that is the inverse limit of its finite quotients. It is conjectured that the profinite completion of the fundamental group of a hyperbolic 3-manifold should completely determine the manifold. If true, this would provide a new solution to the homeomorphism problem for 3-manifolds. Both of Dario's supervisors have expertise in this area. Lackenby has used profinite completions to study finite-sheeted covers of 3-manifolds, and Bridson has recently discovered the first example of a hyperbolic 3-manifold that is determined by its profinite completion. In fact, it is determined by its profinite completion among all finitely generated residually finite groups. A hierarchy for a 3-manifold is a finite sequence of decompositions along incompressible surfaces that cuts the manifold into 3-balls. Haken showed that many 3-manifolds have a hierarchy; in particular, all knot complements have one. He used these to produce the first solution to the equivalence problem for knots and links. Lackenby has used hierarchies to produce quantitative bounds on the computational complexity of this and related problems. For example, he showed that the problem of recognising the unknot lies in the complexity class co-NP. In his project, Dario will use these methods to analyse more complicated knots. This will tie in with the study of profinite completions, since it seems likely that a proof of profinite rigidity will need to use incompressible surfaces in some way. This project will draw on many different areas of expertise of Dario's supervisors Lackenby and Bridson. It will require sophisticated methods in low-dimensional topology, hyperbolic geometry and geometric group theory. The project lies in the EPSCR Research Areas Geometry & Topology and Algebra.

通过对Thurston几何化猜想的证明，在某种意义上对三维流形进行了分类。然而，证明的非建设性性质并没有直接的方法来决定两个给定的三维流形是否相同。在这个项目中，阿斯卡里将研究这些计算问题。他将使用几种技术来处理它。一方面，拓扑工具，如层次结构将被证明是有用的。另一方面，代数方法，如使用基本群的profinite完成，也将是富有成效的。覆盖在这两种方法之上的是双曲几何的普遍作用，这是阿斯卡里的一个专业领域。一个领域，这些问题是特别感兴趣的是纽结理论，其中阿斯卡里也将审查。我们现在更详细地解释这些方法。有限生成群的profinite完备化是一个紧拓扑群，它是它的有限子群的逆极限。证明了双曲三维流形的基本群的有限完备化应完全决定该流形。如果是真的，这将为三维流形的同胚问题提供一个新的解决方案。达里奥的两个上司都有这方面的专业知识。Lackenby使用profinite完备化来研究3-流形的有限单覆盖，而Bridson最近发现了由其profinite完备化确定的双曲3-流形的第一个例子。事实上，它是由它在所有的有限生成的剩余有限群中的profinite完备化决定的。一个3-流形的层次是一个有限序列的分解沿着不可压缩的表面，切割成3-球的流形。哈肯证明了许多三维流形都有一个层次，特别是所有的纽结补都有一个层次。他用这些产生的第一个解决方案的等价问题的结和链接。Lackenby使用层次结构来产生这个和相关问题的计算复杂性的定量界限。例如，他证明了识别unknot的问题在于复杂性类co-NP。在他的项目中，Dario将使用这些方法来分析更复杂的结。这将与profinite完备化的研究联系起来，因为profinite刚性的证明似乎需要以某种方式使用不可压缩曲面。该项目将借鉴达里奥的主管Lackenby和Bridson的许多不同领域的专业知识。这将需要复杂的方法在低维拓扑，双曲几何和几何群论。该项目位于EPSCR研究领域几何与拓扑和代数。