Towards the Border of Symplectic Rigidity and Flexibility

辛拓扑学的发展是为了回答有关经典力学的定性问题，例如保守动力系统中周期轨道的存在性和数量。自成立以来的主题，刚性的结果，断言各种约束的动态，已经共存的结果的灵活性，产生的建设，乍一看是违反直觉的。虽然在过去的三十年中最重要的发展涉及刚性，但最近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柔性和刚性方法，以精确描述辛拓扑两部分之间的边界。该项目的主要目标是：--发展构造辛结构的有效方法; --发展辛和接触伪合痕理论，作为理解辛和接触变换群拓扑的必要步骤; --系统地发展辛拓扑的分段线性形式，作为发展拓扑哈密顿动力学的必要步骤;辛场论的进一步发展。该项目的工作有望为闭流形上辛结构的构造提供新的方法，并为计算辛场论的辛和接触不变量开发新的有效技术。

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.