Towards the Border of Symplectic Rigidity and Flexibility

辛拓扑学是为了回答有关经典力学的定性问题而发展起来的，例如保守动力系统中周期轨道的存在和数目。自从这门学科开始以来，关于刚性的结果，断言了对动力学的各种限制，一直与关于灵活性的结果共存，产生了乍一看是反直觉的结构。虽然过去三十年来最重要的发展是关于刚性的，但最近又发现了几个新的辛柔性实例。当前研究项目的目标是进一步发展灵活的和刚性的方法，以寻求辛拓扑两部分之间边界的精确描述。该项目的主要目标是：-发展构造辛结构的有效方法；-发展辛和接触伪等距理论，作为理解辛群和接触变换组拓扑的必要步骤；-系统地发展辛拓扑的分段线性版本，作为发展拓扑哈密顿动力学的必要步骤；以及-进一步发展辛场理论。该项目的工作有望为在闭流形上构造辛结构提供新的方法，并为辛场理论的辛不变量和接触不变量的计算提供新的有效技术。

The subject of symplectic topology was developed in order to answer qualitative questions concerning classical mechanics, such as the existence and number of periodic orbits in conservative dynamical systems. Since the inception of the subject, results on rigidity, asserting various constraints on the dynamics, have coexisted with results on flexibility, yielding constructions that at first glance were counter-intuitive. While the most important developments of the last three decades concern rigidity, several new instances of symplectic flexibility have been discovered more recently. The goal of the current research project is to further develop both flexible and rigid methods in a search of precise description of the boundary between the two parts of symplectic topology. The main objectives of the project are: -- development of effective methods of construction of symplectic structures; -- development of symplectic and contact pseudoisotopy theories, as necessary steps in understanding topology of groups of symplectic and contact transformations; -- systematic development of a piecewise linear version of symplectic topology as a necessary step in the development of topological Hamiltonian dynamics; and -- further development of symplectic field theory. The work on the project is expected to provide new methods for construction of symplectic structures on closed manifolds and to develop new effective techniques for computing symplectic and contact invariants of symplectic field theory.