Knot Concordance and Metric Spaces

这个研究项目涉及低维拓扑，它研究被称为流形的空间的性质，流形在局部看起来像我们生活的世界。许多成功用于研究高维流形的数学工具并不适用于四维或更少维的流形，重要的基本问题仍未解决。这个项目研究的是结和链，在三维空间中打结的圆形弦。三维流形在数学上可以用框架链接表示，其中所谓的框架是修改链接附近空间的指令。因此，对结和连杆的研究是理解三维流形的基本工具。在这个项目中，研究者的目标是更好地理解结在第四维空间中随着时间的推移是如何运动的。这条研究路线构成了低维拓扑学的一个重要分支，称为一致性。可以在结点集合上执行具有类似于数字加法性质的代数操作，这将导致构造称为和谐群的代数对象。一致性群是一个极其复杂的实体，尽管许多数学家付出了巨大的努力，但对它的理解却很少。为了获得对其结构的透视，研究者将引入距离的概念，并研究某些几何操作如何改变这个距离。这个项目的主要目标是获得一个更好的理解结和链接的一致性。研究者将通过将一致性群作为具有相关过滤（如n可解过滤和双极过滤）的阿贝尔群，以及具有离散和非离散度量的度量空间来研究这一点。具体项目如下：(1)证明了n可解过滤的“另一半”是非平凡的；(2)确定链协调群的n可解过滤的连续商是否为阿贝尔；(3)了解结的导数Milnor不变量与结的调和类之间的关系；(4)将光滑结和谐群视为度量空间，研究作用于其上的自然几何算子（特别是证明了拓扑切片结子群T上存在一个非离散度量，并证明了双极过滤的连续商在T上是非平凡的）；(5)定义结复合体并研究其代数拓扑。

This research project concerns low-dimensional topology, which studies the properties of spaces called manifolds that locally look like the world we live in. Many of the mathematical tools successfully used to study high-dimensional manifolds do not apply to manifolds in four or fewer dimensions, and important fundamental problems remain unresolved. This project studies knots and links, knotted circular strings in three-dimensional space. Three-dimensional manifolds can be mathematically represented by framed links, where the so-called framings are instructions to modify the space near the link. As a consequence, the study of knots and links is a fundamental tool in understanding three-dimensional manifolds. In this project, the investigator aims to get a better understanding of how knots behave as they move through time as a fourth dimension. This line of investigation constitutes an important subfield of low-dimensional topology known as concordance. An algebraic operation with properties similar to the addition of numbers can be performed on the collection of knots, and this leads to the construction of an algebraic object known as the concordance group. The concordance group is an extremely complicated entity, which is poorly understood despite great efforts by many mathematicians. To gain perspective on its structure, the investigator will introduce a notion of distance and investigate how certain geometric operations change this distance.The primary goal of this project is to gain a better understanding of knot and link concordance. The investigator will do this by studying the concordance group as both an abelian group with associated filtrations, such as the n-solvable and bipolar filtrations, and also as a metric space with discrete and non-discrete metrics. Specific projects are as follows: (1) show that the "other half'" of the n-solvable filtration is nontrivial; (2) determine if successive quotients of the n-solvable filtration of the link concordance group are abelian; (3) understand the relationship between the Milnor invariants of derivatives of a knot and the concordance class of the knot; (4) view the smooth knot concordance group as a metric space and study natural geometric operators acting on it (in particular, show that there is a non-discrete metric on the subgroup of topologically slice knots T and, in addition, show that the successive quotients of the bipolar filtration are nontrivial on T); (5) define the knot complex and study its algebraic topology.