CAREER: Symplectic Field Theory and Low-Dimensional Topology

本计画利用全纯曲线来研究辛几何与低维拓扑的交集问题。一个主要目标是开发一个代数框架的相对版本的辛场论，与应用接触几何和纽结理论;重大的部分进展最近已取得的主要研究人员和其他研究人员。一个相关的目标是研究低维对象的光滑拓扑（例如，结和三个和四个流形）通过余切丛的辛场论。首席研究员之前曾使用这种策略引入了一种称为结接触同调的结不变量，它部分理解了与弦拓扑的连接。本计画将研究纽结接触同调理论在辛场论中的推广，并将其延伸至低维流形的不变量，以发展纽结接触同调理论。可能的应用包括关系到类似的同调不变量，如Heegaard Floer同调和Khovanov同调，和新的方法来解决低维拓扑问题，如结协调和区分流形上的光滑结构。低维拓扑，或三维和四维形状的研究，是数学中至关重要的经典课题，也是其他科学，特别是物理学的自然兴趣。近年来，低维拓扑学中许多基本的和长期存在的开放问题已经被证明非常适合于来自不同数学领域的新技术，辛几何。这些进展大多涉及辛流形中全纯曲线的研究，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内对全纯曲线的研究来开发新的拓扑结构不变量，如结和三维空间。这可能会在低维拓扑中有有趣的应用，包括使用辛技术解决尚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几项教育工作，包括为本科生开发新的通用兴趣课程，通过分析自然，艺术和日常生活中的模式引入数学推理方法。

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.