Dehn Surgery, Four-Manifolds, and Symplectic Topology

Dehn 手术、四流形和辛拓扑

基本信息

批准号：
1709702
负责人：
Tye Lidman
金额：
$ 18万
依托单位：
North Carolina State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1709702&HistoricalAwards=false
关键词：
Dehn Surgery Four Manifolds Symplectic

项目摘要

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.

拓扑是数学领域，研究对象的内在形状，例如呼啦圈，大量数据集，我们的宇宙或一条DNA。例如，拓扑可用于测量生物过程如何改变DNA的形状。令人惊讶的是，这些形状变化可以通过三维和四维流形，拓扑中的基本形状来分析，这些形状出现在整个数学和物理学中。这导致了试图完全理解和分类这些三维和四维流形的基本问题。尽管我们了解三维对象的许多方面，但在维度四中，我们仍然大部分位于黑暗中。研究这些对象的一种有效工具称为浮点同源性，它来自从物理学中求解强大的方程式。在该项目中，PI将通过分析它们内部打结的环和表面的配置来研究这些形状的复杂性。该项目的潜在结果将是加强结与三维流形之间的连接，同时在四维流形的拓扑结构中发现新结构。然后，该项目将通过与北卡罗来纳州非营利性舞蹈公司Black Box Dance Theatre合作将其传播给公众。通过舞蹈表演和公共研讨会，这将向公众展示拟议项目中使用的拓扑基本概念。该联合项目还将致力于提高公众对数学的看法，并通过称为Dehn Surgery的手术来使用STEM与艺术之间的联系。开口可用于生产新的三杆和四个manifolds，在该手术中，人们可以通过同音形态去除管状邻居并进行恢复。在Dehn手术下，许多低维拓扑的不变性和工具尤其是表现出色，并且PI将使用这些工具来提高我们对三个和四个manifolds的理解，并进一步与结理论联系。 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. 两个潜在的结果将更好地理解四个manifolds与其手柄结构测量的边界的复杂性，并在结的特性与其Dehn手术的拓扑之间建立更强的联系。