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Knots and surfaces in three- and four-manifolds: Applications of symplectic topology and quantum algebra to low dimensional topology

三流形和四流形中的结和表面：辛拓扑和量子代数在低维拓扑中的应用

基本信息

批准号：
0906258
负责人：
Matthew Hedden
金额：
$ 14.57万
依托单位：
Michigan State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2009
资助国家：
美国
起止时间：
2009-08-15 至 2012-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906258&HistoricalAwards=false
关键词：
Knots surfaces three four manifolds

项目摘要

The theme of this project is to study topological and geometric objects in low-dimensions. One of the primary ways the project will accomplish this is by using tools from symplectic geometry, gauge theory, and quantum algebra. Many of these tools come packaged as algebraic invariants e.g. Floer homology theories or combinatorial knot invariants. A byproduct of the project will be a deepened understanding of the invariants themselves, clarifying the relationships between them and the internal structure of their underlying theories. Specific goals include understanding the knotted curves in the three-dimensional sphere on which a surgery procedure produces simple manifolds, a general exploration of the knots which arise from algebraic curves in projective surfaces, and understanding the extent of, and method by which, combinatorial invariants detect geometric objects e.g. whether Khovanov homology detects the unknot.The study of three- and four-dimensional spaces, and knotted curves and surfaces within them, is a central task to our understanding of both large and small scale aspects of the universe. Determining the shape of the universe depends upon a mathematical understanding of the possible shapes that could occur and the properties these shapes have. These properties are known as invariants, and the project furthers our understanding of space by the discovery of new invariants and the study of existing invariants. In this pursuit it has been quite fruitful to examine the way in which curves and surfaces can be tied in knots. This kind of knotting is not only relevant to understanding the shape of space, but has recently become significant in the study of DNA. Confined to a small space, long strands of DNA naturally become knotted, and certain processes depend upon an understanding of the complexity of these knots. Applications of this project include effective ways of measuring different types of complexity of knots.

这个项目的主题是研究低维的拓扑和几何对象。该项目将实现这一目标的主要方法之一是使用辛几何，规范理论和量子代数的工具。这些工具中的许多都被打包为代数不变量，例如Floer同源理论或组合结不变量。该项目的副产品将是对不变量本身的深入理解，澄清它们之间的关系及其基础理论的内部结构。具体的目标包括理解三维球面上的打结曲线，手术过程产生简单的流形，从投影曲面中的代数曲线产生的结的一般探索，以及理解组合不变量检测几何对象的程度和方法，例如Khovanov同调是否检测unknot。以及其中的曲线和曲面，是我们理解宇宙大小尺度方面的中心任务。确定宇宙的形状取决于对可能出现的可能形状以及这些形状所具有的属性的数学理解。这些属性被称为不变量，该项目通过发现新的不变量和研究现有的不变量来进一步加深我们对空间的理解。在这种追求中，研究曲线和曲面可以打结的方式是相当有成效的。这种打结不仅与理解空间的形状有关，而且最近在DNA研究中变得重要。DNA长链被限制在一个很小的空间里，自然会打结，而某些过程取决于对这些结的复杂性的理解。该项目的应用包括测量不同类型的复杂的结的有效方法。