Corks, Concordance, and Complex Curves

软木塞、一致性和复杂曲线

• 批准号: 2114837
• 负责人: Kyle Hayden
• 金额: $ 15.17万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2022-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2114837&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.

差异拓扑与流形有关 - 电路，球体和托里（及其较高维度类似物）等物体。在物理学（我们的宇宙为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）中，自然而然地具有歧管。在许多方面，对于四维空间，这个主题尤其有趣。幸运的是，通常可以使用位于其内部的低维歧管来研究歧管。值得注意的是，关于4维流形的一些最重要的问题可以简化为询问在3空间中的给定打结圆是否出现为4个空间中的2维磁盘的边界。该项目旨在开发新技术，以解决此后的问题和相关问题，并在4维拓扑中使用一系列应用。特别令人感兴趣的是四空间中的磁盘和表面，这些磁盘和表面是将解决方案设置为两个复杂变量的方程，称为“复杂平面曲线”。这样的表面暴露了与有关4个manifolds的基本问题以及与其他数学领域的联系，例如桥梁的数学理论。此外，该项目的各个方面旨在阐明复杂平面曲线和与数学物理具有深厚连接的重要拓扑工具之间的联系。这些项目包括本科生研究的可访问入口点。该项目以三个相互关联的问题为指导：（1）何时在3空间中结合4个空间的磁盘（或另一个低生物表面）？ （2）在4个manifold中嵌入的表面如何独特？ （3）复杂歧管中哪些平滑表面是同位素到复杂曲线的？对于所有三个问题，PI旨在融合建设性技术（例如处理微积分和分支覆盖范围），以增强Heegaard Floer同源性和Khovanov同源性等阻塞性工具的功能。在优先的同源性理论中进一步调查了合并图，将有助于阐明这些工具是否可以用于检测4维空间中“外部打结”的可定向表面，即通过环境正同形的同位素，而不是差异化。特别是，PI将使用这些同源性理论来研究复杂曲线的独特性问题。 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 precious of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.