Floer Homology and Low-Dimensional Topology

Florer 同调和低维拓扑

• 批准号: 2104309
• 负责人: Linh Truong
• 金额: $ 9.04万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-09-01 至 2023-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104309&HistoricalAwards=false
• 关键词: Floer; Homology; Low; Dimensional; Topology

Low-dimensional topology is the study of spaces of dimensions three and four and their qualitative geometric properties. The classification of these spaces remains a fundamental problem today. The main theme of this project is to study topological properties of spaces using certain algebraic invariants, called Floer homology and Khovanov homology. These invariants have become central tools in modern topology and have connections to fields ranging from symplectic geometry to quantum physics to biology. In addition to its research component, the project includes plans for mentoring and outreach efforts, with a focus on increasing the accessibility of mathematics to groups underrepresented in the mathematical sciences. The project is devoted to studying three- and four-dimensional manifolds by further developing techniques in Floer homology and Khovanov homology. The first part of the project is to study homology cobordism and knot concordance, including constructing new concordance homomorphisms for knots in homology spheres. The second part of the project concerns properties of the monodromy of open book decompositions and Stein fillability of contact three-manifolds. The third part is to study properties of a link invariant called symplectic sl(n) homology and its connection to Khovanov-Rozansky homology.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.

低维拓扑是对第三和四维空间及其定性几何特性的研究。这些空间的分类仍然是当今的基本问题。该项目的主要主题是使用某些代数不变式（称为Floer同源性和Khovanov同源性）研究空间的拓扑特性。这些不变的人已成为现代拓扑的中心工具，并与从象征性几何到量子物理学再到生物学的领域建立了联系。除了其研究部分外，该项目还包括指导和推广工作的计划，重点是增加数学对数学科学中代表性不足的小组的可访问性。该项目致力于通过进一步开发Floer同源性和Khovanov同源性的技术来研究三维和四维流形。该项目的第一部分是研究同源性恢复和结一致性，包括为同源性领域的结构建新的一致性同态。该项目的第二部分涉及开放式书籍分解的单片的特性和触点三元人的斯坦因填充性。第三部分是研究称为Symplectic SL（N）同源性的链接不变的属性及其与Khovanov-Rozansky同源性的联系。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响审查标准通过评估来进行评估的。

项目成果

Comparing Bennequin-type inequalities

比较 Bennequin 型不等式

• 发表时间: 2021
• 期刊: New York journal of mathematics
• 影响因子: 0.6
• 作者: Aceves, Elaina;Kawamuro, Keiko;Truong, Linh
• 通讯作者: Truong, Linh

More concordance homomorphisms from knot Floer homology

更多来自结弗洛尔同源性的索引同态

• DOI: 10.2140/gt.2021.25.275
• 发表时间: 2021
• 期刊: Geometry & Topology
• 影响因子: 2
• 作者: Dai, Irving;Hom, Jennifer;Stoffregen, Matthew;Truong, Linh
• 通讯作者: Truong, Linh

A combinatorial description of the LOSS Legendrian knot invariant

LOSS 传奇结不变量的组合描述

• DOI: 10.1142/s0218216522500936
• 发表时间: 2022
• 期刊: Journal of Knot Theory and Its Ramifications
• 影响因子: 0.5
• 作者: He, Dongtai;Truong, Linh
• 通讯作者: Truong, Linh

On the Upsilon invariant of fibered knots and right-veering open books

关于纤维结和右转向打开书籍的 Upsilon 不变量

• DOI: 10.4310/mrl.2022.v29.n2.a5
• 发表时间: 2022
• 期刊: Mathematical Research Letters
• 影响因子: 1
• 作者: He, Dongtai;Hubbard, Diana;Truong, Linh
• 通讯作者: Truong, Linh

Annular Link Invariants from the Sarkar–Seed–Szabó Spectral Sequence

Sarkar–Seed–Szabó 谱序列的环形链接不变量

• DOI: 10.1307/mmj/20205862
• 发表时间: 2021
• 期刊: Michigan Mathematical Journal
• 影响因子: 0.9
• 作者: Truong, Linh;Zhang, Melissa
• 通讯作者: Zhang, Melissa