Mathematical Sciences: Low-dimensional Manifolds and Knot Theory

小行星9303229 戈登教授将调查各种问题的德恩手术建设3流形从结和链接。 在一般情况下，结或连接补语将具有双曲线结构，这在大多数手术中仍然存在。 当（且仅当）模几何化猜想（英语：Geometriization Conjecture）得到的3-流形包含一个本质球面或环面，或者是塞弗特流形时，这种持续性失败。 该项目将继续与John Luecke教授的联合项目，通过表面交叉的组合分析来理解和限制这些特殊的手术。 到目前为止，得到的结果包括纽结补猜想，约化猜想，和强限制的创建本质环面。 该项目的具体重点将包括电缆猜想和塞弗特的问题时，空间（包括环面和环面）出现。 该项目的总体背景是试图进一步理解三维流形。 三维流形是一个局部类似于普通三维欧氏空间的“空间”，但其整体拓扑结构可能相当复杂。 例如，我们的宇宙是一个三维流形，其整体结构目前还不为人知。 用合理的方法描述所有三维流形是一个重要的问题，尽管已经取得了很大的进展，但仍然没有解决。 这是一个内容丰富的学科，它利用了广泛的数学技术，包括一些来自量子物理学的技术。 该项目的主要焦点是Dehn手术，这是一种从结（即以某种方式嵌入三维欧几里得空间的闭环）或更一般地说，链接（即连接在一起的几个结）构建三维流形的过程。 粗略地说，结或链接被删除和“缝回”不同。 既然所有的三维流形都可以用这种方式构造，那么对Dehn手术的充分理解将对三维流形的一般理论具有重要的意义。 ***

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. ***