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Homological Invariants in Low Dimensional Topology

低维拓扑中的同调不变量

基本信息

批准号：
1811210
负责人：
Akram Alishahi
金额：
$ 14.23万
依托单位：
Columbia University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2019-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811210&HistoricalAwards=false
关键词：
Homological Invariants Low Dimensional Topology

项目摘要

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.

这个国家科学基金会奖支持低维拓扑学的一个项目，这是一个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维度中比在较高维度中更困难。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，或者换句话说，与形状相关的数量可以区分具有不同属性的形状。本课程也与纽结理论密切相关，纽结理论主要研究三维空间中的打结圆的形状。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的打结和DNA-蛋白质相互作用）中有着各种各样的含义。这个项目的目的是研究三维和四维空间的形状和打结的圆和曲面的配置，其中使用PI和她的合作者定义的不变量。在另一个方向上，PI研究了打结圆的不同不变量之间的关系。 PI计划组织研讨会和会议，并让本科生参与该项目的组合和计算方面。Heegaard Floer同调是低维对象（例如3-和4-流形，结，链接等）的代数不变量的集合，通过计算全纯圆盘来定义。特别是，不同类型的这种不变量已被引入3-流形的边界。例如，Eftekhary和PI将tangle Floer同调定义为（减）Heegaard Floer同调的推广。一方面，本项目的目标是（1）利用tangle Floer同调研究图的嵌入和同调圆柱的同调协边群;（2）引入协调和Seifert曲面的不变量，并利用tangle Floer同调的协边映射得到解结数的界。在另一个方向上，PI打算（3）给出一种计算有效的方法来检测使用边界Floer同调（Heegaard Floer不变量的不同扩展，用于具有Lipshitz-Ozsvath-Thurston定义的边界的3-流形），希望为检测同伦带状纤维结提供一种新的方法。这个项目还建议研究结和链接的不变量的性质，称为辛sl（n）同调。这个不变量在理论上等于sl（n）同调，PI计划（4）研究辛sl（n）同调中特定对合的谱序列，并建立对这一猜想的更好理解。该奖项反映了NSF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。