Four-Manifolds and Categorification

四流形及其分类

• 批准号: 2203860
• 负责人: Adam Levine
• 金额: $ 25万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203860&HistoricalAwards=false
• 关键词: Four; Manifolds; Categorification

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 more difficult than their analogues in higher dimensions and require the use of invariants that go beyond traditional algebraic topology. The PI’s primary tools come from Heegaard Floer homology and Khovanov homology, two important packages of invariants developed in 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. In addition, the PI is deeply committed to integrating his research with a passion for education at a variety of levels and mathematical outreach. He has numerous projects in mind for both undergraduate and graduate students in the coming years, and he is deeply invested in expanding the pipeline of women and underrepresented minorities in the field.The specific research goals of the project are organized around two main areas. (1) The PI has made numerous contributions regarding knot concordance, the study of which knots in 3-dimensional space bound smoothly embedded disks in 4 dimensions, along with various other problems concerning smoothly embedded surfaces in 4-dimensional manifolds. These questions are deeply tied to the fundamental strangeness of 4-dimensional topology, as compared with higher dimensions. The PI plans to investigate a number of open questions in this area, including homology slice knots, piecewise-linear concordance of knots in homology 3-spheres, ribbon concordance, new constructions of exotic 4-manifolds, and invariants for knotted 2-spheres in 4-dimensional space. (2) The PI also works on questions that are more internal to the structures of Heegaard Floer homology and Khovanov homology and the relationship between them. While rather technical, these problems may have more topological applications down the road. In particular, the PI plans to investigate some technical issues that arise in the behavior of Heegaard Floer homology under surgery and to continue a long-standing effort to construct a spectral sequence between Khovanov homology and knot Floer homology in characteristic 2.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.

该项目研究低维拓扑中的各种问题，研究 3 维和 4 维空间的全局形状以及其中包含的结和表面。该学科位于许多不同数学领域的十字路口，具有广泛的应用范围，从宇宙学（宇宙的形状）到生物化学（DNA 分子的打结）再到数学物理学。令人惊讶的是，低维中的许多问题比高维中的类似问题更困难，并且需要使用超越传统代数拓扑的不变量。 PI 的主要工具来自 Heegaard Floer 同源性和 Khovanov 同源性，这是 2000 年代初期开发的两个重要的不变量包。这些工具汇集了几个不同的数学领域，包括表示论、微分几何和分析，PI 希望阐明这些不同领域之间的联系，并扩大这些领域研究人员之间的讨论。此外，PI 致力于将他的研究与对各个层次的教育和数学推广的热情结合起来。他在未来几年为本科生和研究生制定了许多项目，并大力投资扩大该领域的女性和代表性不足的少数族裔的人才梯队。该项目的具体研究目标围绕两个主要领域。 (1) PI 在结一致性方面做出了许多贡献，研究了 3 维空间中的结在 4 维空间中绑定平滑嵌入的圆盘，以及有关 4 维流形中平滑嵌入表面的各种其他问题。与更高维度相比，这些问题与 4 维拓扑的基本奇异性密切相关。 PI 计划研究该领域的一些悬而未决的问题，包括同源切片结、同源 3 球体中结的分段线性一致性、带状一致性、奇异 4 流形的新构造以及 4 维空间中的结 2 球体的不变量。 (2) PI 还研究 Heegaard Floer 同调和 Khovanov 同调结构以及它们之间关系的更内部的问题。虽然这些问题相当技术性，但将来可能会有更多的拓扑应用。特别是，PI 计划调查手术中 Heegaard Floer 同源性行为中出现的一些技术问题，并继续长期努力构建特征 2 中 Khovanov 同源性和 Node Floer 同源性之间的谱序列。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。