Homological Invariants in Low Dimensional Topology

低维拓扑中的同调不变量

• 批准号: 2000506
• 负责人: Akram Alishahi
• 金额: $ 8.68万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2000506&HistoricalAwards=false
• 关键词: Homological; Invariants; Low; Dimensional; Topology

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.

该国家科学基金会奖支持低维拓扑学项目，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，与高维度相比，在低维度中理解和分类空间形状的基本问题更加困难。事实上，研究在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。该学科也与纽结理论密切相关，纽结理论专注于研究三维空间中打结圆的形状。 低维拓扑和结理论在物理学（量子理论）、宇宙学（宇宙的形状）、化学（分子结）和生物学（DNA 打结和 DNA-蛋白质相互作用）中具有多种含义。该项目旨在使用 PI 及其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的结圆和表面的配置。在另一个方向上，PI 研究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，通过计算全纯盘来定义。特别是，针对具有边界的 3 流形引入了不同类型的此类不变量。例如，Eftekhary 和 PI 将 tangle Floer 同源性定义为（减去）Heegaard Floer 同源性的概括。在一个方向上，该项目旨在 (1) 使用缠结 Floer 同源性研究图的嵌入和同调圆柱体的同调共边群；(2) 引入索引和 Seifert 曲面的不变量，并使用缠结 Floer 同源性的共边图获得解结数的界限。在不同的方向上，PI 打算 (3) 给出一种计算上有效的方法，用于使用有界 Floer 同调（由 Lipshitz-Ozsvath-Thurston 定义的边界的 3 流形的 Heegaard Floer 不变量的不同扩展）来检测手柄，希望提供一种检测同伦带状纤维结的新方法。 该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。这个不变量推测上等于 sl(n) 同源性，PI 计划 (4) 研究辛 sl(n) 同源性中特定对合的谱序列，并更好地理解这一猜想。该奖项反映了 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。

项目成果

A friendly introduction to the bordered contact invariant

有界接触不变量的友好介绍

• DOI: 10.2140/obs.2022.5.1
• 发表时间: 2022
• 期刊: Open Book Series
• 作者: Alishahi, Akram;Licata, Joan E.;Petkova, Ina;Vértesi, Vera
• 通讯作者: Vértesi, Vera

Bordered Floer homology and contact structures

有界弗洛尔同源性和接触结构

• DOI: 10.1017/fms.2023.19
• 发表时间: 2023
• 期刊: Sigma
• 作者: Alishahi, Akram;Földvári, Viktória;Hendricks, Kristen;Licata, Joan;Petkova, Ina;Vértesi, Vera
• 通讯作者: Vértesi, Vera

Knot Floer homology and the unknotting number

Knot Floer 同源性和解结数

• 发表时间: 2021
• 期刊: Geometry topology
• 作者: Alishahi, A.;Eftekhary, E.
• 通讯作者: Eftekhary, E.