CAREER: Extending and unifying modern homological invariants in low dimensional topology

该项目将集中在两个现代同调结不变量：Khovanov同调和结Floer同调。这两个不变量都将链复形与其链同伦类型（因此，同调群）是结不变量的结相关联。该项目有两个主要的研究目标。第一个目标是将这些同调不变量的各个方面扩展到稳定的同伦类型，即，构造新的纽结不变拓扑空间（定义良好的稳定同伦等价），其同调群是现有的不变量。这将产生关于同调不变量的更高结构，这将有助于研究结的某些几何性质，例如它们的四球亏格。本项目的第二个目标是研究同调不变量之间的关系。该项目旨在寻找新的谱序列，以及现有谱序列的组合重构，从Khovanov同源不变量到结Floer同源不变量。这将导致更好地理解为什么这两个不变量来自非常不同的起源有这么多的相似之处。拓扑学是数学的分支，研究空间的形状;低维拓扑学集中在空间到四维。纽结理论是低维拓扑学的一个重要子领域，它研究三维空间中的一维物体，例如，我们生活的（三维欧几里得）空间中的字符串打结。除了在低维拓扑学中是一个非常有价值的工具外，结理论在真实的世界中也被证明是有用的：从分析DNA中的打结到研究液体中的混合，从估计磁场内轨道的能量到创建新的数据加密方案。纽结理论中的一个基本问题是纽结合痕问题：确定一个给定的纽结是否可以在不撕裂或交叉自身的情况下变换为另一个纽结（这样的变换称为纽结合痕）。纽结不变量是数学对象，如数或群，它与纽结相关联，并且在这样的纽结合痕中保持不变。因此，纽结不变量在纽结同伦问题中被广泛使用：如果一个人发现一些纽结不变量在两个给定的纽结上取不同的值，那么他就得出结论，这两个纽结不是同伦的。目前的项目是基于纽结理论，它试图研究某些以前已知的纽结不变量的属性，并将其扩展到构建新的纽结不变量。该项目将通过各种手段传播数学知识，特别是在低维拓扑学领域。这个项目将资助本科生进行暑期研究，为期一周的低维拓扑研讨会，以及一个基于维基的纽结理论网站。

The project will concentrate on two modern homological knot invariants: Khovanov homology and knot Floer homology. Both invariants associate chain complexes to knots whose chain homotopy types (and consequently, homology groups) are knot invariants. This project has two major research goals. The first goal is to extend various aspects of these homological invariants to stable homotopy types, i.e., construct new knot invariant topological spaces (well-defined up to stable homotopy equivalences) whose homology groups are the existing invariants. This will produce higher structures on the homological invariants which will be useful for studying certain geometric properties of knots, such as their four-ball genus. The second goal of this project is to study the relationship between the homological invariants. The project seeks to find new spectral sequences, and combinatorial reformulations of the existing spectral sequences, from the Khovanov homology invariants to the knot Floer homology invariants. This will lead to a better understanding of why these two invariants coming from very different origins share so many similarities.Topology is the branch of mathematics that studies shapes of spaces; and low dimensional topology concentrates on spaces up to dimension four. Knot theory is an important sub-field of low dimensional topology where one studies one-dimensional objects inside three-dimensional spaces, for example, knotted pieces of strings inside the (three-dimensional Euclidean) space that we live in. In addition to being an extremely valuable tool in low dimensional topology, knot theory has also proven to be useful in real world applications: from analysing knotting in DNA to studying mixing in liquids, and from estimating energies of orbits inside a magnetic field to creating new data encryption schemes. A fundamental problem in knot theory is the knot isotopy problem: to determine if a given knot can transform into another one without tearing or crossing itself (such a transformation is called a knot isotopy). Knot invariants are mathematical objects, such as numbers or groups, that one associates to knots, and which remain unchanged during such a knot isotopy. Therefore, knot invariants are used extensively in the knot isotopy problem: if one finds some knot invariant that takes different values on the two given knots, then one concludes that the two knots are not isotopic. The current project is based on knot theory, and it seeks to study properties of certain previously known knot invariants, and to extend them to construct new knot invariants. The project will lead to dispersion of mathematical knowledge, particularly in the area of low-dimensional topology, via a variety of means. This project will fund undergraduate students for summer research, week-long workshops on low-dimensional topology, and a wiki-based website on knot theory.