Knots and Surfaces in Three-Manifolds

拟议的研究重点是应用Floer理论不变量来研究低维拓扑和接触几何中的对象。 第一个项目发展辛场论的组合方法。 本研究的基础是一个组合的塞弗特纤维空间的Legendrian接触同调，这将被用来探索Legendrian结理论在合理的同调三球。第二个项目研究结属概念的推广。我们将开发技术，研究非定向表面的链接补充，适应现有的Heegaard Floer不变量，以检测最小的复杂性非定向表面。 如果没有感官设备来检测量子现象，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;这个项目研究三维和四维数学对象的属性和相互作用称为流形。在过去的十年里，人们已经成功地使用Floer理论的技术来研究这些物体，Floer理论是一种强大但技术上困难的方法。提出的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。 这将促进低维流形的研究，并为原始构造提供新的思路。

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.