Mathematical Sciences: Low-dimensional Manifolds and Knot Theory

9303229戈登教授将研究关于从纽结和链环构造3-流形的德恩手术的各种问题。在一般情况下，结或连接补语将具有双曲线结构，这在大多数手术中都存在。如果(且仅当，模几何猜想)所得到的3-流形包含本质球面或环面，或者是Seifert纤维的，则这种持久性是失败的。该项目将继续与John Luecke教授合作，通过曲面交点的组合分析来理解和限定这些特殊的手术。到目前为止所得到的结果包括纽结补猜想、可约性猜想以及对本质环面产生的强烈限制。该项目的具体焦点将包括布线猜想和塞弗特纤维空间(环状和环状)何时出现的问题。该项目的总体背景是试图加深我们对3-流形的理解。三维流形是一个局部类似于普通三维欧几里德空间的“空间”，但其全局拓扑结构可能相当复杂。例如，我们的宇宙是一个三维流形，其整体结构目前尚不清楚。如何合理地刻画所有的三维流形是一个重要的问题，虽然已经取得了很大的进展，但仍然没有得到解决。这是一个内容丰富的学科，它利用了广泛的数学技术，包括一些源自量子物理的方法。该项目的主要焦点是Dehn手术，这是一种从一个纽结(即以某种方式嵌入到三维欧几里德空间中的闭合环)或更广泛地说，一个链接(即几个纽结连接在一起)来构造三维流形的过程。粗略地说，就是把结或链环去掉，然后用不同的方法把它们“缝回去”。由于证明了所有的三维流形都可以这样构造，因此对Dehn运算的充分理解将对三维流形的一般理论产生重要的影响。***

9303229 Gordon Professor Gordon will investigate various questions about the Dehn surgery construction of 3-manifolds from knots and links. In the generic case, the knot or link complement will have a hyperbolic structure, which persists under most surgeries. This persistence fails if (and only if, modulo the Geometrization Conjecture) the resulting 3-manifold contains an essential sphere or torus, or is Seifert fibred. This project is to continue a joint program with Professor John Luecke of understanding and circumscribing these exceptional surgeries via the combinatorial analysis of intersections of surfaces. Results obtained so far include the Knot Complement Conjecture, the Reducibility Conjecture, and strong restrictions on the creation of essential tori. Specific foci of the project will include the Cabling Conjecture and the question of when Seifert fibred spaces (both toroidal and atoroidal) arise. The general context of the project is the attempt to further our understanding of 3-manifolds. A 3-manifold is a "space" which is locally like ordinary 3-dimensional Euclidean space, but whose global topological structure may be quite complicated. For example, our universe is a 3-manifold, whose global structure is at present unknown. The problem of describing all 3-manifolds in a reasonable way is an important one that is still unsolved, although much progress has been made. The subject is a rich one, which draws on a wide range of mathematical techniques, including some derived from quantum physics. The main focus of the project is Dehn surgery, which is a procedure for constructing 3-manifolds from a knot (i.e. a closed loop embedded somehow in 3-dimensional Euclidean space) or, more generally, a link (i.e. several knots linked together). Roughly speaking, the knot or link is removed and "sewn back" differently. Since it turns out that all 3-manifolds can be constructed in this way, a sufficiently good understanding of Dehn surgery wou ld have important implications for the general theory of 3-manifolds. ***