Towards the Border of Symplectic Rigidity and Flexibility

辛拓扑学科的发展是为了回答经典力学中的定性问题，如保守动力系统中周期轨道的存在和数量。自该学科成立以来，刚性的结果，对动力学的各种约束，与灵活性的结果共存，产生了乍一看是反直觉的结构。虽然过去三十年中最重要的发展与刚性有关，但最近发现了几个新的辛柔性实例。目前的研究目标是进一步发展柔性和刚性方法，以寻找辛拓扑两部分之间边界的精确描述。该项目的主要目标是：开发有效的辛结构施工方法；——发展辛和接触伪同位素理论，作为理解辛和接触变换群拓扑结构的必要步骤；——系统地发展了辛拓扑的分段线性版本，作为拓扑哈密顿动力学发展的必要步骤；辛场论的进一步发展。本课题的工作将为在闭流形上构造辛结构提供新的方法，并为辛场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.