CAREER: Symplectic Field Theory and Low-Dimensional Topology

这个项目使用全纯曲线来研究辛几何和低维拓扑的交叉问题。一个主要目标是为辛场论的相对版本开发一个代数框架，并将其应用于接触几何和结理论；首席研究员和其他研究人员最近取得了重大的部分进展。一个相关的目标是通过共切束的辛场论研究低维物体（例如，结和三流形和四流形）的光滑拓扑。首席研究员之前使用这种策略引入了一种称为结接触同调的结不变量，该策略部分地理解了弦拓扑的连接。本课题将发展结接触同调理论，研究其在辛场论中的推广，并将其推广到低维流形的不变量上。可能的应用包括与相似同调不变量的关系，如Heegaard flower同调和Khovanov同调，以及解决低维拓扑问题的新方法，如结调和和区分流形上的光滑结构。低维拓扑学，或对三维和四维形状的研究，是数学中最重要的经典学科，也是其他科学，特别是物理学的自然兴趣。近年来，低维拓扑中许多基本的和长期存在的开放问题已被证明非常适合于来自不同数学领域辛几何的新技术。这一进展大部分涉及辛流形中的全纯曲线的研究，这与物理学中的弦理论有着密切的联系。首席研究员将利用他过去在辛场论框架下对全纯曲线的研究成果来开发拓扑结构（如结和三维空间）的新不变量。这可能会对低维拓扑有有趣的应用，包括使用辛技术解决仍然未解决的光滑四维庞加莱猜想的长期目标。首席研究员还将从事一些教育工作，包括为本科生开发一门新的通用课程，通过分析自然、艺术和日常生活中的模式来介绍数学推理的方法。

This project uses holomorphic curves to study questions in the intersection of symplectic geometry and low-dimensional topology. One main goal is developing an algebraic framework for the relative version of Symplectic Field Theory, with applications to contact geometry and knot theory; significant partial progress has recently been made by the Principal Investigator and other researchers. A related goal is studying the smooth topology of low-dimensional objects (e.g., knots and three- and four-manifolds) via the Symplectic Field Theory of cotangent bundles. The Principal Investigator has previously used this strategy to introduce a knot invariant called knot contact homology, which has partially understood connections to string topology. This project will develop the theory of knot contact homology by studying its generalization in Symplectic Field Theory and extending it to invariants of low-dimensional manifolds. Possible applications include relations to similar homological invariants such as Heegaard Floer homology and Khovanov homology, and new approaches to problems in low-dimensional topology such as knot concordance and distinguishing smooth structures on manifolds.Low-dimensional topology, or the study of shapes in three and four dimensions, is a classical subject of paramount importance in mathematics and of natural interest to other sciences, especially physics. In recent years, many fundamental and longstanding open problems in low-dimensional topology have proven remarkably amenable to new techniques from a different mathematical field, symplectic geometry. Much of this progress has involved the study of holomorphic curves in symplectic manifolds, which has close ties to string theory in physics. The Principal Investigator will use his past work on holomorphic curves within the framework of Symplectic Field Theory to develop new invariants of topological structures such as knots and three-dimensional spaces. This will likely have interesting applications to low-dimensional topology, including a long-term goal to resolve the still-unsolved smooth four-dimensional Poincare conjecture using symplectic techniques. The Principal Investigator will also pursue several educational endeavors, including the development of a new general-interest course for undergraduates, introducing methods of mathematical reasoning through analysis of patterns in nature, the arts, and everyday life.