Cut-and-paste contact topology

Bennequin和Eliashberg在20世纪80年代和90年代的开创性工作表明，在所有接触结构中，紧密的结构最能反映环境空间的拓扑结构。只有在过去的十年里，才开发出足够的工具来解决这门学科中的许多基本分类问题。PI建议与本田和Mati 'c合作，使用剪切粘贴技术分析哈肯歧管的接触拓扑结构。本研究的重要成果包括Haken同调球中紧密接触结构的构建，将叶理理论的关系扩展到层合和双曲流形，以及对产品表面下复杂结构和接触结构之间相互作用的理解。三维流形以我们生活的三维空间为模型。接触拓扑学和辛拓扑学最初是由从事经典力学和热力学工作的物理学家提出的。从流体流动到投影屏幕电视的制造，它们的应用范围之广令人惊讶。拟议的研究有可能产生适用于这些领域的新结果。

The pioneering works of Bennequin and Eliashberg from the 1980s and the 1990s have shown that, among all contact structures, the tight ones best reflect the topology of the ambient space. Only in the past decade have sufficient tools been developed to allow many of the basic classification questions in the subject to be attacked. The PI proposes joint work with Honda and Mati\'c to analyze the contact topology of Haken manifolds using cut-and-paste techniques. Important outgrowths of this investigation include constructions of tight contact structures in Haken homology spheres, extensions of the relationship with foliation theory to laminations and hyperbolic manifolds, and an understanding of the interplay between the subsurface complex and contact structures on a product.Three-dimensional manifolds are modeled on the three-dimensional space we live in. Contact and symplectic topology were originally motivated by questions of physicists working in classical mechanics and thermodynamics. They arise in a surprisingly broad array of applied situations ranging from fluid flows to the manufacture of projection screen televisions. The proposed research has the potential to produce new results applicable to these fields.