Homological Invariants in Low Dimensional Topology

这一国家科学基金奖支持低维拓扑学的一个项目，低维拓扑学是研究三维和四维空间形状的数学领域。有趣的是，理解和分类空间形状的基本问题在这些较低的维度中比在较高的维度中更难。事实上，研究这些维度中发生的一些新现象需要使用更现代的不变量，换句话说，与形状相关的量可以区分那些具有不同性质的形状。本课题也与纽结理论密切相关，纽结理论主要研究三维空间中纽结圆的形状。低维拓扑学和纽结理论在物理学(量子理论)、宇宙学(宇宙的形状)、化学(分子结)和生物学(DNA和DNA-蛋白质相互作用的纽结)中有着不同的意义。这个项目旨在利用PI和她的合作者定义的不变量来研究三维和四维空间的形状以及其中打结的圆和曲面的配置。在另一个方向上，PI研究了结圆的不同不变量之间的关系。PI计划组织研讨会和会议，并让本科生参与这个项目的组合和计算方面的工作。Heegaard Floer同调是低维对象(如3维和4维流形、纽结、链环等)的代数不变量的集合，通过计数全纯圆盘来定义。特别地，对于带边界的3-流形，引入了不同类型的这种不变量。例如，Eftekhary和PI将缠绕Floer同调定义为(负)Heegaard Floer同调的推广。在一个方向上，这个项目的目的是(1)研究图的嵌入和利用Tangle Floer同调的同调柱面的同调上边群；(2)引入协和曲面和Seifert曲面的不变量，并利用Tangle Floer同调的Cobordism映射得到解结数的界。在不同的方向上，PI的目的是(3)给出一种利用边界Floer同调(Lipshitz-Ozsvath-瑟斯顿定义的具有边界的三维流形的Heegaard Floer不变量的不同扩展)检测手体的计算有效方法，以期为同伦带状纤维结的检测提供一种新的方法。这个项目还建议研究纽结和链环的不变量的性质，称为辛sl(N)同调。这个不变量在猜想上等于sl(N)同调，PI计划(4)研究辛sl(N)同调中特定对合的谱序列，并建立对这个猜想的更好理解。这个奖项反映了NSF的法定使命，并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，被认为值得支持。

This National Science Foundation award supports a project in low dimensional topology, an area of mathematics that studies shapes of three- and four-dimensional spaces. Interestingly, the fundamental problem of understanding and classifying shapes of spaces is more difficult in these lower dimensions compared to higher dimensions. In fact, investigating some of the new phenomena that happen in these dimensions require the use of more modern invariants, or in other words, quantities associated to shapes that can distinguish between those with different properties. This subject is also closely related to knot theory, that is focused on studying the shapes of knotted circles in three-dimensional spaces. Low dimensional topology and knot theory have various implications in physics (quantum theory), cosmology (the shape of the universe), chemistry (molecular knots) and biology (knotting of DNA and DNA-protein interactions). This project aims to study the shapes of three- and four-dimensional spaces and configurations of knotted circles and surfaces in them using an invariant defined by the PI and her collaborator. In another direction, the PI investigates the relationship between different invariants for knotted circles. The PI plans to organize seminars and conferences and involve undergraduate students in the combinatorial and computational aspects of this project. Heegaard Floer homology is a collection of algebraic invariants for low dimensional objects (e.g. 3- and 4-manifolds, knots, links, etc.), defined by counting holomorphic disks. In particular, different types of such invariants have been introduced for 3-manifolds with boundary. For example, Eftekhary and the PI defined tangle Floer homology as a generalization of (minus) Heegaard Floer homology. In one direction, this project aims to (1) study embeddings of graphs and homology cobordism group of homology cylinders using tangle Floer homology and (2) introduce invariants for concordances and Seifert surfaces, and get bounds for unknotting number using cobordism maps for tangle Floer homology. In a different direction, the PI intends to (3) give a computationally effective way for detecting handlebodies using bordered Floer homology (different extension of Heegaard Floer invariants for 3-manifolds with boundary defined by Lipshitz-Ozsvath-Thurston) with the hope of providing a new way for detecting homotopically ribbon fibered knots. This project also proposes to study the properties of an invariant for knots and links, called symplectic sl(n) homology. This invariant is conjecturally equal to sl(n) homology, and the PI plans to (4) investigate spectral sequences for specific involutions in symplectic sl(n) homology and establish a better understanding of this conjecture.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.