Mathematical Sciences: Low-dimensional Manifolds and Knot Theory

9303229 Gordon Gordon 教授将研究有关 Dehn 手术从结和连杆构建 3 歧管的各种问题。 在一般情况下，结或连接补体将具有双曲线结构，该结构在大多数手术中持续存在。 当（且仅当以几何化猜想为模）所得到的 3 流形包含基本球体或环面，或者是 Seifert 纤维时，这种持久性会失败。 该项目将继续与 John Luecke 教授开展联合项目，通过表面相交的组合分析来理解和限制这些特殊的手术。 迄今为止获得的结果包括结补猜想、可约化猜想以及对本质环面创建的严格限制。 该项目的具体重点将包括布线猜想以及 Seifert 纤维空间（环形和环形）何时出现的问题。 该项目的总体背景是试图进一步加深我们对三流形的理解。 3-流形是一个局部类似于普通3维欧几里得空间的“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流形，其整体结构目前未知。 以合理的方式描述所有 3 流形的问题是一个尚未解决的重要问题，尽管已经取得了很大的进展。 这门学科内容丰富，利用了广泛的数学技术，包括一些源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 3 维欧几里得空间中的闭环）或更一般地是链接（即连接在一起的几个结）构建 3 流形的过程。 粗略地说，结或链接被移除并以不同的方式“缝回”。 由于事实证明所有 3 流形都可以通过这种方式构建，因此充分理解 Dehn 手术将对 3 流形的一般理论产生重要影响。 ***

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. ***