Four-Manifolds and Categorification

该项目研究了低维拓扑中的各种问题，研究了三维和四维空间的整体形状以及其中包含的结和表面。这门学科处于许多不同数学领域的交叉点，它有各种各样的应用，从宇宙学（宇宙的形状）到生物化学（DNA分子的结）到数学物理学。令人惊讶的是，许多低维问题比高维问题更困难，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 flower同调和Khovanov同调，这是21世纪初发展起来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，并扩大这些领域研究人员之间的讨论。此外，PI致力于将他的研究与对各级教育和数学推广的热情相结合。在未来的几年里，他为本科生和研究生准备了许多项目，他在扩大女性和该领域未被充分代表的少数族裔的渠道方面投入了大量资金。该项目的具体研究目标围绕两个主要领域进行组织。(1) PI在结的一致性方面做出了许多贡献，研究了三维空间中的结在4维空间中约束光滑嵌入的磁盘，以及关于4维流形中光滑嵌入表面的各种其他问题。与高维拓扑相比，这些问题与四维拓扑的基本奇异性密切相关。PI计划研究该领域的一些开放问题，包括同调切片结，同调3球中结的分段线性一致性，带状一致性，奇异4流形的新结构，以及4维空间中结2球的不变量。(2) PI还研究Heegaard flower同调和Khovanov同调的结构以及它们之间的关系的内部问题。虽然技术性很强，但这些问题今后可能会有更多的拓扑应用。特别是，PI计划研究Heegaard花同源性在手术中出现的一些技术问题，并继续长期努力构建特征2的Khovanov同源性和knot花同源性之间的谱序列。该奖项反映了美国国家科学基金会的法定使命，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。

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.