Mathematical Sciences: Low-dimensional Manifolds and Knot Theory

9303229 Gordon教授将研究关于结点和连杆构造3流形的Dehn手术的各种问题。在一般情况下，结或连接补体将具有双曲结构，在大多数手术中持续存在。当（且仅当，对几何化猜想取模）得到的3流形包含一个基本球体或环面，或者是Seifert纤维时，这种持久性失效。这个项目将继续与John Luecke教授合作，通过对曲面相交的组合分析来理解和限制这些特殊的手术。迄今为止得到的结果包括结补猜想、可约猜想和对基本环面的创建的强限制。该项目的具体焦点将包括布线猜想和塞弗特光纤空间（包括环形和环形）何时出现的问题。这个项目的总体背景是试图进一步加深我们对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得空间的“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流形，其整体结构目前是未知的。合理地描述所有3-流形的问题是一个重要的问题，但仍然没有解决，尽管已经取得了很大的进展。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，这是一种从一个结（即以某种方式嵌入三维欧几里德空间的闭环）或更一般地说，一个链接（即连接在一起的几个结）构建3-流形的过程。粗略地说，打结或连接被移除并以不同的方式“缝回”。由于事实证明所有的3-流形都可以用这种方式构造，对Dehn手术的充分理解将对3-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. ***