Extensions of Modern Homological Invariants in Low Dimensional Topology

拓扑学是研究空间形状的数学的分支。拓扑学有几个现实世界的应用，例如研究DNA打结、构建新的数据加密算法、分析大型数据集、机器人运动规划以及发展物理学中的量子场论等。由于一个著名的定理Smale，拓扑结构在较高的层面是有点简单的拓扑结构在较低的层面。因此，只专注于四维空间是很有趣的;这个子领域被称为低维拓扑。这些低维也对应于我们生活的空间--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--使低维拓扑更加相关。在低维拓扑学中，我们特别感兴趣的是纽结理论，其中一个研究三维空间中的一维物体，例如打结的弦。纽结理论是低维拓扑学中一个非常重要的研究课题，也是许多实际拓扑应用中不可或缺的一部分。纽结理论研究一个纽结是否可以在不撕裂或交叉的情况下转化为另一个纽结（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。 纽结不变量是与纽结相关的数学对象（如数或群），它们在这种合痕中保持不变，因此广泛用于研究纽结。本项目的重点是纽结理论，并将探索现有的纽结不变量，并构建新的。本项目将集中在两个现代家庭的纽结不变量在低维拓扑结构，结Floer同源性和Khovanov同源性，这已被用于各种应用，因为他们发现以来，在世纪之交。该项目的主要目的是构建新的扩展，如空间细化，这些现有的不变量的各种版本。具体来说，该项目有以下四个目标：构建进一步的空间细化的Khovanov同源不变量和他们的扰动;构建一个空间细化的结Floer同源使用网格表示;拉格朗日Floer同源研究组行动;和构建新的组合谱序列从Khovanov同源。此外，作为该项目的一部分，还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NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.