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

在瑟斯顿几何化猜想的证明下，3-流形在某种意义上是分类的。然而，证明的非构造性没有给出确定两个给定的3流形是否相同的直接方法。在这个项目中，Ascari将研究这些计算问题。他将使用几种技巧来接近它。一方面，诸如层次结构之类的拓扑工具将被证明是有用的。另一方面，代数方法，如利用基本群的无限完备性，也将是富有成效的。覆盖这两种方法的是双曲几何的普遍作用，这是Ascari的专业领域。对这些问题特别感兴趣的一个领域是结理论，Ascari也将对此进行研究。现在我们更详细地解释这些方法。有限生成群的有限补全是一个紧拓扑群，它是其有限商的逆极限。推测双曲3-流形的基群的无限完备性应完全确定该流形。如果成立，这将为3流形的同胚问题提供一个新的解。达里奥的两位主管都是这方面的专家。Lackenby使用无限完井来研究3-流形的有限层盖，而Bridson最近发现了由无限完井决定的双曲3-流形的第一个例子。实际上，它是由它在所有有限生成的剩余有限群之间的无限完备性决定的。3流形的层次结构是沿着不可压缩表面将流形切割成3球的有限分解序列。Haken证明了许多3-流形都有层次结构；特别地，所有结补都有一个。他利用这些给出了结点和连杆等效问题的第一个解。Lackenby使用层次结构来对这个问题和相关问题的计算复杂性产生定量界限。例如，他指出，识别解结的问题在于复杂度类co-NP。在他的项目中，达里奥将使用这些方法来分析更复杂的结。这将与无限完井的研究联系在一起，因为无限刚性的证明似乎需要以某种方式使用不可压缩表面。该项目将借鉴达里奥的主管拉肯比和布里德森的许多不同领域的专业知识。这将需要在低维拓扑、双曲几何和几何群论方面的复杂方法。该项目属于EPSCR研究领域几何与拓扑和代数。

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.