Floer Homology and Low-Dimensional Topology

Florer 同调和低维拓扑

• 批准号: 2005539
• 负责人: Linh Truong
• 金额: $ 10.79万
• 依托单位: Institute For Advanced Study
• 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: 2020-06-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005539&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.

低维拓扑学是对三维和四维空间及其定性几何性质的研究。这些空间的分类今天仍然是一个基本问题。该项目的主题是研究空间的拓扑性质，使用某些代数不变量，称为Floer同调和Khovanov同调。这些不变量已经成为现代拓扑学的核心工具，并与从辛几何到量子物理学再到生物学的各个领域都有联系。除了研究部分外，该项目还包括辅导和外联工作计划，重点是增加数学科学代表性不足的群体对数学的可及性。该项目致力于通过进一步发展Floer同调和Khovanov同调技术来研究三维和四维流形。第一部分是研究同调配边和纽结协调，包括构造同调球面上纽结的协调同态。该项目的第二部分关注的单值性的开卷分解和斯坦因填充接触三流形。第三部分是研究称为辛sl（n）同调的链接不变量的性质及其与Khovanov-Rozansky同调的联系。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。

项目成果

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

Braids, fibered knots, and concordance questions

辫子、纤维结和索引问题

• 发表时间: 2021
• 期刊: Association for Women in Mathematics series
• 作者: Hubbard, Diana;Kawamuro, Keiko;Kose, Feride Ceren;Martin, Gage;Plamenevskaya, Olga;Raoux, Katherine;Truong, Linh;Turner, Hannah
• 通讯作者: Turner, Hannah

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

A refinement of the Ozsváth-Szabó large integer surgery formula and knot concordance

Ozsváth-Szabó 大整数手术公式和结索引的改进

• DOI: 10.1090/proc/15212
• 发表时间: 2021
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Truong, Linh