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

PI建议利用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。在一个方向上，继续之前与R. Hind合作的工作，PI将研究域的辛嵌入和相关的辛填充问题，并着眼于证明一般填充稳定性。要研究的相关问题是辛容量的几种可能的估计。与D. Gay合作的第二个项目将描述四维椭球嵌入的三维接触模拟。一系列三维接触技术将被用于可能的应用于四维辛嵌入问题。在第三个目标中，PI继续先前的工作，使用j全纯曲线技术来研究流形的辛形态群的拓扑方面，如有规四维曲面。与经典几何相比，辛几何主要是围绕面积的概念（二维辛子流形）建立的。嵌入和填充问题是最初由哈密顿动力学和哈密顿自同构的递归性质引起的基本刚性问题。这些包装问题的答案将指导我们对辛空间的理解。由于它们的计算性质，这些问题有可能让学生参与引导计算机实验。

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.