Corks, Concordance, and Complex Curves

软木塞、一致性和复杂曲线

• 批准号: 2243128
• 负责人: Kyle Hayden
• 金额: $ 15.17万
• 依托单位: Rutgers University Newark
• 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: 2022-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2243128&HistoricalAwards=false
• 关键词: Corks; Concordance; Complex; Curves

Differential topology is concerned with manifolds — objects such as circles, spheres, and tori (and their higher-dimensional analogs). Manifolds arise naturally in physics (our universe as a 4-dimensional manifold), computer science (graphics), and biology (investigating how the shape and "knottedness" of molecules like DNA affect their function). In many ways, this subject is especially interesting for 4-dimensional spaces. Fortunately, a manifold can often be studied using the lower-dimensional manifolds that sit inside of it. Remarkably, several of the most important questions about 4-dimensional manifolds can be reduced to asking if a given knotted circle in 3-space arises as the boundary of a 2-dimensional disk in 4-space. This project aims to develop new techniques to tackle this latter problem and related questions, with a range of applications in 4-dimensional topology. Of particular interest are the disks and surfaces in 4-space that arise as solution sets to equations with two complex variables, known as "complex plane curves". Such surfaces exhibit surprising connections to foundational questions about 4-manifolds, as well as connections to other areas of mathematics, such as the mathematical theory of braids. In addition, aspects of the project aim to illuminate the connections between complex plane curves and important topological tools that have deep connections to mathematical physics. These projects include accessible entry points for undergraduate research.The project is guided by three interrelated questions: (1) When does a knot in 3-space bound a disk (or another low-genus surface) in 4-space? (2) How unique is an embedded surface in a 4-manifold, up to isotopy? (3) Which smooth surfaces in complex manifolds are isotopic to complex curves? For all three questions, the PI aims to blend constructive techniques (such as handle calculus and branched coverings) to enhance the power of obstructive tools like Heegaard Floer homology and Khovanov homology. Further investigation of the cobordism maps in the aforementioned homology theories will help shed light on whether these tools can be used to detect pairs of orientable surfaces in 4-dimensional space that are "exotically knotted", i.e., isotopic through ambient homeomorphisms but not diffeomorphisms. In particular, the PI will use these homology theories to investigate uniqueness problems for complex curves. In addition, the PI will continue to develop topological techniques for studying complex curves in 4-space and other 4-manifolds, including novel techniques using singular foliations inspired by the theory of characteristic foliations in contact geometry.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.

微分拓扑与流形有关——诸如圆、球体和环面（以及它们的高维类似物）等对象，流形自然地出现在物理学（我们的宇宙是一个 4 维流形）、计算机科学（图形）和生物学（研究 DNA 等分子的形状和“打结”如何影响其功能）在许多方面，这个主题对于 4 维空间特别有趣，幸运的是，通常可以使用流形来研究。值得注意的是，关于 4 维流形的几个最重要的问题可以简化为询问 3 空间中的给定结圆是否作为 4 维中的 2 维圆盘的边界出现。该项目旨在开发新技术来解决这个问题和相关问题，在 4 维拓扑中的一系列应用特别令人感兴趣的是 4 空间中的圆盘和表面，它们作为方程的解集而出现。两个复变量，被称为“复平面曲线”，这样的曲面与 4 流形的基本问题以及与其他数学领域（例如辫子的数学理论）的联系表现出令人惊讶的联系。这些项目旨在阐明复杂平面曲线和与数学物理有深刻联系的重要拓扑工具之间的联系，包括本科生研究的可访问切入点。该项目以三个相互关联的问题为指导：(1) 3- 中的结何时出现。空间限制了磁盘（或(2) 4 流形中的嵌入表面在同位素方面有多独特？ (3) 复杂流形中的哪些光滑表面是复杂曲线的同位素？对于这三个问题，PI 旨在混合构造性技术（例如处理）微积分和分支覆盖），以增强 Heegaard Floer 同调和 Khovanov 同调等阻塞性工具的能力。上述同源理论中的图将有助于阐明这些工具是否可用于检测 4 维空间中“奇异结”的可定向表面对，即通过环境同胚而非微分同胚检测同位素。将使用这些同调理论来研究复杂曲线的唯一性问题。此外，PI 将继续开发用于研究 4 空间和其他空间中的复杂曲线的拓扑技术。 4-流形，包括受接触几何特征叶理理论启发而使用单一叶理的新技术。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。