Corks, Concordance, and Complex Curves

微分拓扑学关注的是流形--诸如圆、球和环面（以及它们的高维类似物）之类的对象。流形在物理学（我们的宇宙是一个四维流形）、计算机科学（图形学）和生物学（研究DNA等分子的形状和“打结”如何影响它们的功能）中自然出现。在许多方面，这个主题对于四维空间特别有趣。幸运的是，流形通常可以用它内部的低维流形来研究。值得注意的是，关于四维流形的几个最重要的问题可以简化为：三维空间中给定的打结圆是否是四维空间中二维圆盘的边界。该项目旨在开发新技术来解决后一个问题和相关问题，并在四维拓扑中有一系列应用。特别令人感兴趣的是4-空间中的圆盘和曲面，它们作为具有两个复变量的方程的解集而出现，称为“复平面曲线”。这样的曲面与四维流形的基础问题有着惊人的联系，也与其他数学领域有着联系，比如辫子的数学理论。此外，该项目的各个方面旨在阐明复杂平面曲线与重要拓扑工具之间的联系，这些工具与数学物理有着深刻的联系。这些项目包括本科生研究的切入点。该项目由三个相互关联的问题指导：（1）3-空间中的纽结何时约束4-空间中的圆盘（或另一个低亏格曲面）？(2)4-流形中的嵌入曲面直到合痕有多唯一？(3)复流形中哪些光滑曲面与复曲线是同素的？对于所有这三个问题，PI的目标是混合建设性的技术（如处理演算和分支覆盖），以提高像Heegaard Floer同源性和Khovanov同源性的障碍性工具的力量。对上述同源性理论中的配边映射的进一步研究将有助于阐明这些工具是否可以用于检测四维空间中“奇异打结”的可定向表面对，即，通过环境同胚而不是环同胚的同位素。特别是，PI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还将继续开发用于研究四维空间和其他四维流形中复杂曲线的拓扑技术，包括利用接触几何中特征叶理理论启发的奇异叶理的新技术。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。

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.