Dehn Surgery, Four-Manifolds, and Symplectic Topology

拓扑学是一个数学领域，研究对象的内在形状，如呼啦圈、海量数据集、我们的宇宙或DNA链。例如，拓扑学可以用来测量生物过程如何改变我们DNA的形状。令人惊讶的是，这些形状的变化可以通过三维和四维流形来分析，这些流形是整个数学和物理中出现的拓扑学中的基本形状。这导致了试图完全理解和分类这些三维和四维流形的根本问题。虽然我们了解三维物体的许多方面，但在四维空间中，我们仍然大多处于黑暗之中。研究这些天体的一个有效工具被称为Floer同调，它来自于求解物理学中的强大方程。在这个项目中，PI将使用Floer同调来通过分析这些形状内部的结环和曲面的配置来研究这些形状的复杂性。这个项目的潜在结果将是加强纽结和三维/四维流形之间的联系，同时发现四维流形拓扑中的新结构。该项目随后将通过与北卡罗来纳州一家非营利性舞蹈公司黑匣子舞蹈剧院的合作向公众传播。通过舞蹈表演和公共工作坊，这将向公众展示拟议项目中使用的基本拓扑学概念。这个联合项目还将致力于改善公众对数学的看法，并加强STEM与艺术之间的联系。通过一种名为Dehn手术的手术，可以使用纽结来产生新的三维和四维流形，即一个人移除管状邻域，并通过同胚进行调节。低维拓扑中的许多不变量和工具，特别是Floer同调，在Dehn手术下表现得特别好，PI将使用这些工具来提高我们对三维和四维流形的理解，并进一步加深它们与纽结理论的联系。本课题的三个主要目标是：1)在二元链环上构造Dehn运算不能得到的同调三球；2)找到具有辛结构的四流形的代数拓扑上的新约束；3)进一步探索同调球面上的纽结的调和群的代数结构。两个潜在的结果将是更好地理解四维流形的复杂性，其边界是根据它们的手柄结构来衡量的，以及纽结的性质与其Dehn手术的拓扑之间更紧密的联系。

Topology is an area of mathematics that studies the intrinsic shapes of objects, such as a hula hoop, a massive data set, our universe, or a strand of DNA. For example, topology can be used to measure how biological processes change the shape of our DNA. Surprisingly, these shape changes can be analyzed through three- and four-dimensional manifolds, fundamental shapes in topology that appear throughout mathematics and physics. This leads to the fundamental problem of trying to completely understand and classify these three- and four-dimensional manifolds. While we understand many aspects of three-dimensional objects, in dimension four we are still mostly in the dark. One effective tool for studying these objects is called Floer homology, which comes from solving powerful equations from physics. In this project, the PI will use Floer homology to study the complexity of these shapes by analyzing the configurations of knotted loops and surfaces inside of them. The potential outcome of this project will be to strengthen connections between knots and three-/four-dimensional manifolds while discovering new structure in the topology of four-dimensional manifolds. This project will then be disseminated to the public through a collaboration with Black Box Dance Theater, a North Carolina non-profit dance company. By way of dance performances and public workshops, this will present to the public the fundamental notions of topology used in the proposed project. This joint project will also work to improve public perception of mathematics and enhance connections between STEM and the arts.Knots can be used to produce new three- and four-manifolds by an operation called Dehn surgery, where one removes a tubular neighborhood and reglues via a homeomorphism. Many invariants and tools in low-dimensional topology, especially Floer homology, are particularly well-behaved under Dehn surgery, and the PI will use these tools to improve our understanding of three- and four-manifolds and further their connections with knot theory. Three major goals of this project are to: 1) construct homology three-spheres which cannot be obtained by Dehn surgery on a two-component link, 2) find new constraints on the algebraic topology of four-manifolds admitting symplectic structures, and 3) further explore the algebraic structure of the concordance group of knots in homology spheres modulo concordance in homology cobordisms. Two potential outcomes would be a better understanding of the complexity of four-manifolds with boundary measured in terms of their handlebody structures and a stronger connection between properties of a knot and the topology of its Dehn surgeries.