Extensions of Modern Homological Invariants in Low Dimensional Topology

拓扑学是研究空间形状的数学分支。拓扑学在现实世界中有很多应用，比如研究DNA打结、构建新的数据加密算法、分析大型数据集、机器人运动规划、发展物理学中的量子场论等等。根据Smale的一个著名定理，高维拓扑比低维拓扑要简单一些。因此，只关注四维空间是很有趣的；这个子领域被称为低维拓扑。这些低维也与我们生活的空间相对应——我们生活在一个三维空间中，如果包括时间，在一个四维时空中——这使得低维拓扑更加相关。在低维拓扑中，我们对打结理论特别感兴趣，研究三维空间中的一维物体，比如打结的弦。结理论是低维拓扑学中一个非常重要的课题，也是许多实际拓扑学应用中不可或缺的一部分。结理论研究一个结是否可以在不撕裂或交叉的情况下转化为另一个结（这种转化称为同位素），如果不能，需要进行什么样的修改以确保它们成为同位素。结不变量是与结相关的数学对象（如数字或组），在这种同位素过程中保持不变，因此被广泛用于研究结。目前的项目集中在结理论上，将探索现有的结不变量并构建新的结不变量。该项目将集中研究低维拓扑中的两个现代结不变量族，结花同调和Khovanov同调，自千年之交发现以来，它们已被用于各种应用。该项目的主要目的是构建这些现有不变量的各种版本的新扩展，例如空间细化。具体来说，该项目有以下四个目标：构建Khovanov同调不变量及其扰动的进一步空间细化；利用网格表示构造结花同源性的空间细化；拉格朗日花同源上的群作用研究利用Khovanov同源构造新的组合谱序列。此外，作为该项目的一部分，将组织一些将研究与教育和其他更广泛影响相结合的活动，例如提高洛杉矶数学圈儿童的数学意识和兴趣。该奖项反映了美国国家科学基金会的法定使命，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 branch of mathematics that studies shapes of spaces. Topology has several real-world applications, such as studying DNA knotting, constructing new data encryption algorithms, analyzing large data sets, motion planning for robotics, and developing quantum field theories in physics, to name a few. Due to a famous theorem by Smale, topology in higher dimensions is somewhat simpler than topology in lower dimensions. Therefore, it is interesting to concentrate only on spaces up to dimension four; this sub-field is called low-dimensional topology. These low-dimensions also correspond to the spaces that we live in---we live in a three-dimensional space, and if one includes time, in a four-dimensional spacetime---making low-dimensional topology even more pertinent. In low-dimensional topology, we are specifically interested in knot theory, where one studies one-dimensional objects inside three-dimensional spaces, such as knotted pieces of strings. Knot theory is a fundamentally important topic in low-dimensional topology, and it is also an integral part of many of the real-world topological applications. Knot theory studies whether a knot can be transformed into another without tearing or crossing itself (such a transformation is called an isotopy), and if not, what sort of modifications need to be made to ensure they become isotopic. Knot invariants are mathematical objects (such as numbers or groups) associated to knots which remain unchanged during such an isotopy, and consequently, are extensively used in studying knots. The current project is focused on knot theory and will explore existing knot invariants and construct new ones.This project will concentrate on two modern families of knot invariants in low-dimensional topology, knot Floer homology and Khovanov homology, which have been employed for a variety of applications ever since their discovery at the turn of the millennium. The main aim of the project is to construct new extensions, such as spatial refinements, of various versions of these existing invariants. Specifically, the project has the following four goals: construct further spatial refinements of Khovanov homology invariants and their perturbations; construct a spatial refinement of knot Floer homology using grid presentations; study group actions on Lagrangian Floer homology; and construct new combinatorial spectral sequences from Khovanov homology. Additionally, several activities combining research with educational and other broader impacts will be organized as part of this project, such as increasing mathematical awareness and interest among children at Los Angeles Math Circle.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.