Low-Dimensional Topology, Floer Homology, and Categorification

低维拓扑、Floer 同调和分类

• 批准号: 1806437
• 负责人: Adam Levine
• 金额: $ 18万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2022-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1806437&HistoricalAwards=false
• 关键词: Low; Dimensional; Topology; Floer; Homology

This project investigates a variety of questions in low-dimensional topology, the study of the global shapes of 3- and 4-dimensional spaces and of knots and surfaces contained within them. This subject lies at the crossroads of many disparate areas of mathematics, and it has a wide variety of applications ranging from cosmology (the shape of the universe) to biochemistry (the knotting of DNA molecules) to mathematical physics. Surprisingly, many problems in low dimensions are usually more difficult than their analogues in higher dimensions and require the use of invariants that go beyond traditional algebraic topology. The PI's particular area of expertise is in Heegaard Floer homology, a collection of invariants for 3- and 4-dimensional manifolds, which has been one of the most fruitful areas of research in low-dimensional topology since the early 2000s. These tools bring together several different fields of mathematics, including representation theory, differential geometry, and analysis, and the PI hopes to elucidate the connections between these different areas and expand the discourse among researchers in these fields.The specific goals of the project are (1) to make progress on a variety of concrete problems in 4-manifold topology, including knot concordance, exotic smooth structures, and embeddings of non-orientable surfaces; (2) to understand the relationship between the Heegaard Floer homology of a 3-manifold and topological properties such as the existence of incompressible surfaces, taut foliations, and left-orderings on the fundamental group; (3) to establish relationships between the knot invariants arising from gauge theory and symplectic geometry and those coming from representation theory and quantum algebra.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.

这个项目研究了低维拓扑学中的各种问题，即研究三维和四维空间的全局形状以及其中包含的节点和曲面。这门学科位于许多不同的数学领域的十字路口，它的应用范围很广，从宇宙学(宇宙的形状)到生物化学(DNA分子的打结)再到数学物理。令人惊讶的是，许多低维问题通常比高维问题的类似问题更难，并且需要使用超越传统代数拓扑的不变量。PI的特殊专业领域是Heegaard Floer同调，这是三维和四维流形的不变量集合，自21世纪初以来一直是低维拓扑研究中最有成果的领域之一。这些工具汇集了几个不同的数学领域，包括表示理论、微分几何和分析，PI希望阐明这些不同领域之间的联系，并扩大这些领域的研究人员之间的讨论。该项目的具体目标是(1)在4-流形拓扑中的各种具体问题上取得进展，包括纽结协调、奇异光滑结构和不可定向曲面的嵌入；(2)了解3-流形的Heegaard Floer同调与基本群上不可压缩曲面、张叶和左序等拓扑性质的关系；(3)建立来自规范理论和辛几何的纽结不变量与来自表示理论和量子代数的不变量之间的关系。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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

SIMPLY CONNECTED, SPINELESS 4-MANIFOLDS

简单连接、无骨架 4 歧管

• DOI: 10.1017/fms.2019.11
• 发表时间: 2019
• 期刊: Sigma
• 作者: LEVINE, ADAM SIMON;LIDMAN, TYE
• 通讯作者: LIDMAN, TYE

Khovanov homology and ribbon concordances

霍瓦诺夫同源性和带状索引

• DOI: 10.1112/blms.12303
• 发表时间: 2019
• 期刊: Bulletin of the London Mathematical Society
• 影响因子: 0.9
• 作者: Levine, Adam Simon;Zemke, Ian
• 通讯作者: Zemke, Ian

Khovanov homology detects the figure‐eight knot

霍瓦诺夫同源性检测数字——八结

• DOI: 10.1112/blms.12467
• 发表时间: 2021
• 期刊: Bulletin of the London Mathematical Society
• 影响因子: 0.9
• 作者: Baldwin, John A.;Dowlin, Nathan;Levine, Adam Simon;Lidman, Tye;Sazdanovic, Radmila
• 通讯作者: Sazdanovic, Radmila