Knots and Surfaces in Three-Manifolds

三流形中的结和曲面

• 批准号: 1237324
• 负责人: Joan Licata
• 金额: $ 10.61万
• 依托单位: Institute For Advanced Study
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-11-01 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1237324&HistoricalAwards=false
• 关键词: Knots; Surfaces; Three; Manifolds

The proposed research focuses on applying Floer-theoretic invariants to study objects in low-dimensional topology and contact geometry. The first project develops combinatorial approaches to symplectic field theory. The foundation for this study is a combinatorial formulation of Legendrian contact homology for Seifert fibered spaces which will be used to explore Legendrian knot theory in rational homology three-spheres. The second project studies generalizations of the notion of knot genus. We will develop techniques for studying non-orientable surfaces in link complements, adapting existing Heegaard Floer invariants to detect minimal-complexity non-orientable surfaces. Without the sensory apparatus to detect quantum phenomena, our experience of the universe is as a three-dimensional physical space or as a four-dimensional spacetime; this project studies the properties and interactions of three-, and four-dimensional mathematical objects called manifolds. The last decade has seen success in studying these objects using techniques from Floer theory, a powerful yet technically difficult approach. One of the primary goals of the proposed research is developing versions of Floer-theoretic tools which are easier to use. This will facilitate the study of low-dimensional manifolds and shed new light on the original constructions.

拟议的研究重点是应用弗洛尔理论不变量来研究低维拓扑和接触几何中的对象。 第一个项目开发辛场论的组合方法。 这项研究的基础是 Seifert 纤维空间的传奇接触同调的组合公式，它将用于探索有理同调三球体中的传奇结理论。第二个项目研究结属概念的概括。我们将开发研究链接补体中不可定向表面的技术，调整现有的 Heegaard Floer 不变量来检测最小复杂度的不可定向表面。 如果没有感知装置来检测量子现象，我们对宇宙的体验就是三维物理空间或四维时空；该项目研究称为流形的三维和四维数学对象的属性和相互作用。在过去的十年里，使用弗洛尔理论的技术研究这些物体取得了成功，弗洛尔理论是一种强大但技术上困难的方法。拟议研究的主要目标之一是开发更易于使用的弗洛尔理论工具版本。 这将有助于低维流形的研究，并为原始结构提供新的线索。