Mathematical Sciences: Symplectic and Contact Geometry in the Interaction with Topology and Complex Analysis

该项目设想继续发展 辛几何与拓扑相互作用 数学领域。 在先前支助下取得的主要进展是 实现了三维接触几何，新的 高维闭和开接触流形的不变量， 在描述边界的不变性质时， 辛流形，在辛流形的应用中， 本文对真实的四维空间中的2-纽结理论进行了研究。 本 继续，提出了几个新的研究领域。 在 特别是，研究新发现的联系， 辛几何、伪合痕理论和代数 K-理论，拉格朗日交理论在莫尔斯中的应用 复流形上的多重次调和函数理论， 接触同调理论的发展和应用。 其他预测的研究方向是：三维 接触拓扑和辛方法的进一步发展 4-维拓扑学 在现代数学中， 研究具有非常特殊和重要的作用。 许多最近 发现的区域不适合任何 传统分类。 例如， 4-唐纳森的维拓扑学来自规范理论， 量子物理学中的主题，以及许多其他最近的拓扑 物理学激发了许多发现。 辛拓扑. 现在是一个美丽的融合不同的数学科学： 拓扑，Hamilton动力学，复分析，微分 它吸引了越来越多的想法，从不同的 领域和回报他们意想不到的新应用程序， 使得这一方向的研究特别有前途。

This project envisions the continuing development of symplectic geometry and topology in its interaction with other fields of mathematics. The main progress under prior support was achieved in 3-dimensional contact geometry, construction of new invariants of high-dimensional closed and open contact manifolds, in the description of invariant properties of boundaries of symplectic manifolds, and in the application of symplectic methods to the theory of 2-knots in real 4-space. In the present continuation, several new areas of research are suggested. In particular, there is the study of newly discovered connections between symplectic geometry, pseudoisotopy theory and algebraic K-theory, applications of Lagrangian intersection theory to Morse theory for plurisubharmonic functions on complex manifolds, and the development and application of contact homology theory. Other projected directions of research are: 3-dimensional contact topology and further development of symplectic methods in 4-dimensional topology. It seems that in modern mathematics interdisciplinary research plays a very special and important role. Many recent discoveries were made in areas which do not fit into any traditional classification. For instance, a breakthrough in 4-dimensional topology by Donaldson came from gauge theory, a topic in quantum physics, and many other recent topological discoveries were motivated by physics. Symplectic topology at present is a beautiful blend of different mathematical sciences: topology, Hamiltonian dynamics, complex analysis, differential geometry, etc. It attracts more and more ideas from different fields and repays them with unexpected new applications, which makes research in this direction especially promising.