Four-Manifolds and Categorification

该项目研究低维拓扑中的各种问题，研究3维和4维空间的整体形状以及其中包含的结和曲面。这门学科处于许多不同数学领域的交叉点，它有着广泛的应用，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，低维的许多问题比高维的类似问题更困难，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 homology和Khovanov homology，这是21世纪初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话语。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。他在未来几年为本科生和研究生准备了许多项目，并在扩大该领域女性和代表性不足的少数民族的管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已经作出了许多贡献，关于结和谐，研究结在三维空间的约束顺利嵌入磁盘在4维，沿着与其他各种问题有关的顺利嵌入表面在4维流形。这些问题与四维拓扑学的基本奇异性密切相关，与高维拓扑学相比。PI计划研究这一领域的一些悬而未决的问题，包括同调切片结，同调3-球面中结的分段线性协调，带状协调，奇异4-流形的新构造，以及四维空间中打结2-球面的不变量。(2)PI还研究了Heegaard Floer同源性和Khovanov同源性的结构以及它们之间的关系。虽然相当技术性，这些问题可能有更多的拓扑应用的道路。特别是，PI计划调查Heegaard Floer同源性在手术下的行为中出现的一些技术问题，并继续长期努力构建Khovanov同源性和knot Floer同源性之间的光谱序列，特征2。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。

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.