Holomorphic Curves and Low-Dimensional Topology

该项目侧重于低维拓扑和结理论之间的相互作用，以及辛几何和接触几何之间的相互作用。最近已经证明计数辛流形中的全纯曲线是研究三流形和四流形拓扑的有力工具。特别是，与共切束相关的辛场论在获得关于低维流形上光滑结构的信息方面显示出很大的希望。该项目提出在这种情况下寻找辛场论的组合和计算可访问的公式，并将其应用于诸如结理论和（尚未解决的）光滑四维庞加莱猜想等领域。该项目还将研究这些辛味光滑不变量与其他最近发展的几何不变量（如Heegaard flower理论和弦拓扑）之间的关系。一个密切相关的项目将为Legendrian子流形组合制定辛场论，解决接触几何中的一个重要开放问题。本文研究了拓扑学中的一些问题，如四维空间上“奇异”光滑结构的存在性猜想（“光滑四维庞加莱猜想”）。虽然这些问题是经典的，但采用辛场论提出的方法是相当新的。它涉及最近从辛几何、代数几何和弦理论等相关领域发展起来的技术。这些技术在过去十年中取得了惊人的成功，并在拓扑结构中显示出巨大的希望，以理解三维和四维空间的分类，以及结的分类。这可能会在物理学的相关方面有有趣的应用，比如弦理论和镜像对称。

This project focuses on the interplay between low-dimensional topology and knot theory on one side, and symplectic and contact geometry on the other. It has recently been demonstrated that counting holomorphic curves in symplectic manifolds is a powerful tool in the topology of three- and four-manifolds. In particular, the Symplectic Field Theory associated to cotangent bundles shows great promise in yielding information about smooth structures on low-dimensional manifolds. This project proposes to find combinatorial and computationally accessible formulations for Symplectic Field Theory in this circumstance, with applications to areas such as knot theory and the (still unsolved) smooth four-dimensional Poincare conjecture. The project will also investigate the relationship between these symplectic-flavored smooth invariants and other recently developed geometric invariants such as Heegaard Floer theory and string topology. A closely related project will formulate Symplectic Field Theory combinatorially for Legendrian submanifolds, addressing a significant open problem in contact geometry.This proposal studies problems in topology such as the conjectured existence of an "exotic" smooth structure on four-dimensional space (the "smooth four-dimensional Poincare conjecture"). Although the problems are classical, the proposed approach using Symplectic Field Theory is quite new. It involves recently developed techniques from the interrelated fields of symplectic geometry, algebraic geometry, and string theory. Such techniques have been spectacularly successful in the last decade and show great promise in topology for understanding the classification of spaces in three and four dimensions, as well as the classification of knots. This will likely have interesting applications to related aspects of physics such as string theory and mirror symmetry.