Mathematical Sciences: Symplectic and Contact Geometry and Topology, and Their Applications

数学科学：辛几何和接触几何与拓扑及其应用

• 批准号: 9626430
• 负责人: Yakov Eliashberg
• 金额: $ 20.36万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626430&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Symplectic; Contact; Geometry

9626430 Eliashberg This project will concentrate on several areas of symplectic and contact geometry and topology in interaction with other areas of mathematics. In particular, there will be continuing study of 3-dimensional contact structures and the relationship between contact geometry and the theory of codimension-1 foliations. The theory of confoliations initiated in joint work with W. P. Thurston should help to clarify this relationship. Another important part of the project will be devoted to symplectic geometry of Stein manifolds, where one of the main tools should be an analog of Morse-Smale theory for plurisubharmonic functions on Stein manifolds. Yet another focus (joint with H. Hofer) will be contact homology theory, which should deliver new invariants of contact manifolds in dimension 3 and higher. Finally, a significant part of the project (joint with M. Gromov) will involve the symplectization of Morse theory. The results here should help in understanding which part of stable Morse theory can be generalized to the problems of Lagrangian intersections. Symplectic geometry and topology have played an important role in such different areas of mathematics as low-dimensional topology, Hamiltonian dynamics, foliation theory, sub-Riemannian geometry, complex geometry and the theory of functions of several complex variables. Despite much recent progress, many basic questions remain unanswered. Moreover, the discovery of new interactions with these other fields creates new demands for the interior development of symplectic geometry and topology. The research under this project should answer some of the old problems of the field and discover further applications. ***

这个项目将集中在与其他数学领域相互作用的辛几何和接触几何以及拓扑学的几个领域。特别是，将继续研究三维接触结构以及接触几何与余维-1叶理理论之间的关系。在与w·p·瑟斯顿共同工作中提出的组合理论应该有助于澄清这种关系。该项目的另一个重要部分将致力于斯坦因流形的辛几何，其中一个主要工具应该是斯坦因流形上多次谐波函数的莫尔斯-小理论的模拟。然而，另一个焦点（与H. Hofer联合）将是接触同调理论，它应该提供3维及更高维度的接触流形的新不变量。最后，项目的一个重要部分（与M. Gromov联合）将涉及摩尔斯理论的简化。这里的结果应该有助于理解稳定莫尔斯理论的哪一部分可以推广到拉格朗日交点问题。辛几何和拓扑学在低维拓扑学、哈密顿动力学、叶理理论、次黎曼几何、复几何和复变函数理论等不同的数学领域中发挥着重要的作用。尽管最近取得了许多进展，但许多基本问题仍未得到解答。此外，与这些其他领域的新相互作用的发现为辛几何和拓扑的内部发展创造了新的需求。本项目的研究应回答该领域的一些老问题，并发现进一步的应用。＊＊＊