Structure of knot and link concordance

低维拓扑学是一个引人入胜的学科，关注于理解3维和4维空间，就像我们所居住的宇宙一样。我研究了3维和4维流形的拓扑，特别是纽结和链接协调。这是对3-空间中圆的嵌入和4-空间中曲面的嵌入的研究。我的建议旨在更好地理解协调空间的结构。在4维流形中嵌入曲面的问题中，纽结和环的一致性是关键的局部问题，这是尝试对4维流形进行分类的核心问题。我主要使用代数和几何工具，但有时也使用分析方法来解决问题。 该提案的宏伟目标是对结点和链接和谐进行分类。我将从几个角度来处理这个问题。一种是构造和计算一个决定两个纽结是否相容的代数阻塞群。另一种观点认为，和谐的失败可以通过嵌入在四维空间中的表面塔来衡量，这种塔被称为摸索和惠特尼塔。通过将节点空间视为度量空间，将度量定义为摸索，我想了解其所谓的分形结构。惠特尼塔的复杂性可用于过滤链接协调空间，了解某些链接如何适合过滤，以及过滤是多么的不平凡，应该促进我们的理解，并为潜在的分类框架提供信息。从另一方面来看，我的目标是获得积极的结果，表明纽结和链环是使用从弗里德曼的工作中导出的专门的拓扑工具来切片的。这与著名的4-流形的外科猜想有关。拨款的另一个目的是增加我们对相关协调问题的理解。高维节点的双协调群就是这样一个问题。我还打算研究一般三维流形中的节点的协调，以及高维非球面节点的协调。目的是找到经典问题中没有表现出来的新现象。

Low dimensional topology is a fascinating subject, concerned with understanding spaces of dimension 3 and 4, much like the universe we inhabit.  I investigate the topology of 3- and 4-dimensional manifolds, in particular knot and link concordance.  This is the study of embeddings of circles in 3-space and embeddings of surfaces in 4-space.  My proposal aims to achieve a greater understanding of the structure of concordance spaces. Knot and link concordance is the key local question in the problem of embedding surfaces in 4-manifolds, which is central to attempts to classify 4-dimensional manifolds. I use principally algebraic and geometric tools, but sometimes analytic methods as well, to attack the questions. The grand goal of the proposal is to classify knot and link concordance. I will approach this from several viewpoints. One is to try to construct and compute an algebraic obstruction group which decides whether two knots are concordant. Another uses the observation that the failure of concordance can be measured by towers of surfaces embedded in 4-space called gropes and Whitney towers. By considering the space of knots as a metric space, with the metric defined in terms of gropes, I would like to understand its purported fractal structure.  The complexity of Whitney towers can be used to filter the link concordance space, and understanding how certain links fit into the filtrations, and indeed how nontrivial the filtrations are, should promote our understanding and inform potential classification frameworks. Approaching from the other side, I aim to achieve positive results, showing that knots and links are slice using specifically topological tools derived from the work of Freedman.  This has relations to the famous surgery conjecture for 4-manifolds. Another goal of the grant is to increase our understanding of related concordance problems.  The double concordance group of high dimensional knots is one such problem.  I also intend to study the concordance of knots in general 3-manifolds, and the concordance of non-spherical knots in high dimensions.  The goal is to find new phenomena which are not exhibited in the classical problem.