Knot and Link Concordance

拓扑学是研究空间形状的学科，称为流形，局部看起来像欧几里得空间，但整体上可能非常复杂或弯曲。 例如，地球的表面或甜甜圈的表面是流形的例子。 在这种情况下，在表面上有两个方向移动，因此它们被称为二维流形。 该奖项将用于研究三维和四维流形的拓扑结构，以及两个维度的自然子流形如何相互作用。 三维和四维流形的研究在科学中是极其重要的，因为它揭示了我们生活的世界（三维）以及可以被视为时空的四维物体的信息。 这个项目的主要目标是更好地理解哪些节点和链接可以在三维空间中作为球体在时空中移动时的切片，称为切片节点和链接。 这在研究四维流形的拓扑学时尤其重要，因为它涉及到拓扑学中最重要的问题之一：光滑四维庞加莱猜想。 这个猜想说，如果一个人可以计算关于一个四维流形的某些信息，并且得到与标准四维球面相同的答案，那么它实际上是一个球面。 这是唯一一个未知的维度。此外，PI的目标是向著名的切片丝带猜想（Slice-ribbon Conjecture）迈进，该猜想认为，如果一个结是切片的，那么我们可以在时空中变形4-球体，这样它就“漂亮”了。 理解拓扑和同调代数的方法在这个时候特别重要，因为它对拓扑数据分析的研究至关重要，这是数据科学的一个前沿领域。 PI将在会议上介绍这项工作，并将文章发表在公共存储库arxiv.org上，以便所有人都可以获得结果。 PI将举办会议和特别会议，其中大多数发言者是女性或其他代表性不足的数学领域。该奖项提供资金，培养研究生的研究。结和链接是基本的重要性，在研究3-和4维流形。 例如，我们可以将任何4-流形表示为加权链路的Kirby图。 此外，拓扑结和链接协调有很强的关系弗里德曼的手术理论计划分类4维流形。虽然对3-流形的理解很多，但由于几何化猜想和Agol的工作，光滑四维流形的研究仍然知之甚少。 这个项目将进一步我们的知识结和链接一致性，以及一般的3-和4-维拓扑结构，在数学中至关重要的主题。该项目的目标是有一个更好的了解结和链接一致性，以及有关3-流形组的问题。 要做到这一点，PI将研究结的代数结构，并将协调群沿着与它们的过滤（如n-可解过滤和双极过滤）联系起来。 与此相关，PI将研究它们的各种度量空间，以表明它们具有分形空间的结构。 此外，PI还将研究太空中的打结和绳链卫星运营商的行为。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Topology is the study of the shape of spaces, called manifolds, that locally look like Euclidian space but globally can be quite complicated or curved. For example, the surface of the earth or the surface of a donut are examples of manifolds. In this case, on the surface there are two directions to move in and hence they are called 2-dimensional manifolds. This award will be used to study the topology of manifolds in dimensions 3 and 4 as well as how the natural submanifolds in both dimensions interact. The study of manifolds in dimensions 3 and 4 are of utmost important in science as it uncovers information about the world we live in (3-dimensions) as well as 4-dimensional objects which can be seen as space-time. The main goal of this project is to better understand which knots and links can be the slice of sphere in 3-dimensional space as it moves through space-time, called slice knots and links. This is especially important in studying the topology of 4-dimensional manifolds as it relates to one of the most important questions in topology: the Smooth 4-dimensional Poincare Conjecture. This conjecture says that if one can compute certain information about a 4-dimensional manifold and one gets the same answer as for the standard 4-dimensional sphere, then it is actually a sphere. This is the only dimension where it is still unknown. In addition, the PI aims to make progress to the famous Slice-ribbon Conjecture, which says that if a knot is slice then we can deform the 4-sphere in space-time so that is it "nice". Understanding topology and methods for homological algebra is especially important at this time as it is crucial to the study of topological data analysis, a leading area of data science. The PI will present the work at conferences, as well as publish and put the article on the public repository, arxiv.org so that the results are available to all. The PI will run conferences and special sessions where the majority of speakers are women or other underrepresented areas of mathematics. The award provides funds to train graduate students in research.Knots and links are of fundamental importance in the study of 3- and 4-dimensional manifolds. For example, we can present any 4-manifold as a Kirby diagram of weighted links. Furthermore, topological knot and link concordance has a strong relationship to Freedman's surgery-theoretic scheme for classifying 4-dimensional manifolds. While much is understood about 3-manifolds, because of the geometrization conjecture and work of Agol, the study of smooth 4-dimensional manifolds is still poorly understood. This project will further our knowledge of knot and links concordance, as well as general 3- and 4-dimensional topology, subjects of vital importance in mathematics. The goal of the project is to have a better understanding of knot and link concordance as well as questions about 3-manifold groups. To do this, the PI will study the algebraic structure of the knot and links concordance groups along with their filtrations like the n-solvable and bipolar filtrations. Related to this, the PI will study their various metric spaces to show that they have the structure of a fractal space. In addition, the PI will study the behavior of knot and string link satellite operators on the space.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.