Knots and Surfaces in Three-Manifolds

提出的研究重点是将Floer理论不变量应用于研究低维拓扑和接触几何中的对象。第一个项目发展了辛场理论的组合方法。本研究的基础是Seifert纤维空间的Legendrian接触同调的一个组合形式，它将被用来探索有理同调三球面上的Legendrian纽结理论。第二个项目研究纽结亏格概念的泛化。我们将开发技术来研究链接补中的不可定向曲面，采用现有的Heegaard Floer不变量来检测最小复杂性的不可定向曲面。没有探测量子现象的感官设备，我们对宇宙的体验是作为一个三维物理空间或作为一个四维时空；这个项目研究被称为流形的三维和四维数学对象的性质和相互作用。在过去的十年里，使用弗洛尔理论的技术研究这些天体取得了成功，这是一种强大但技术困难的方法。这项拟议研究的主要目标之一是开发更容易使用的Floer理论工具的版本。这将有助于低维流形的研究，并为研究原始结构提供新的线索。

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.