Symplectic embeddings and packing maps, their contact analogues, and other classic symplectic problems

PI建议使用J-全纯曲线理论的最新发展对辛流形和接触流形的拓扑方面进行研究。一方面，继续与R.其次，PI将研究域的辛嵌入和相关的辛填充问题，并着眼于证明一般的填充稳定性。相关问题的研究是几个可能的估计辛能力。第二个项目，与D。Gay，将描述四维椭球嵌入的三维接触类似物。一系列的3维接触技术将用于可能的应用程序的4维辛嵌入问题。在第三个目标PI继续以前的工作使用J-全纯曲线技术追求拓扑方面的辛同构群的流形，如直4维surface.In对比与经典几何，辛几何主要是建立在面积的概念（二维辛子流形）。嵌入和包装问题是基本的刚性问题，最初是由哈密顿动力学，并寻求递归性质的哈密顿自同构。对这种填充问题的回答将指导我们对辛空间的理解。由于其非常计算的性质，这些问题有可能涉及学生在指导计算机实验。

The PI proposes to conduct research on topological aspects of symplectic and contact manifolds using recent developments in the theory of J-holomorphic curves. In one direction, continuing previous work in collaboration with R. Hind, the PI will investigate symplectic embeddings of domains and related symplectic packing questions, with an eye for proving a general packing stability property. Related problems to study are several possible estimates of symplectic capacities. A second project, with D. Gay, will describe 3-dimensional contact analogues of 4-dimensional ellipsoid embeddings. An array of 3-dimensional contact techniques will be used with possible applications to the 4-dimensional symplectic embedding questions. In a third objective the PI continues previous work using J-holomorphic curves techniques to pursue topological aspects of symplectomorphism groups of manifolds such as ruled 4-dimensional surfaces.In contrast with classical geometries, symplectic geometry is predominantly built around the notion of area (of two dimensional symplectic submanifolds). Embedding and packing questions are fundamental rigidity questions originally motivated by Hamiltonian dynamics, and the quest for recurrence properties of Hamiltonian automorphisms. Answers to such packing questions will guide our understanding of symplectic spaces. Due to their very computational nature, these problems have the possibility of involving students in guided computer experimentation.