CAREER: Contact Structures and Low-Dimensional Topology

提案DMS-0237386 PI：Ko本田（南加州大学）标题：接触结构和低维拓扑结构摘要该项目的重点是通过接触结构的三维流形的拓扑结构。 在过去的几年里，研究人员和他的合作者（科林，Etnyre，Giroux，Kazez，Matic）已经开发了一个大规模的剪切和粘贴理论的紧密接触结构。 进一步开展该项目，他计划完成紧接触结构的有限性问题项目（与Colin和Giroux合作），研究Legendrian和横向结（与Etnyre合作），并探索与叶理/层压理论的联系（与Kazez和Matic合作）。 更重要的是，他试图理解紧密性的本质，以及与双曲几何和三维流形有限型不变量的关系。 我们研究的三维空间在局部上类似于标准的（欧几里德）三维空间。 这些物体在整体上可能非常复杂，但一个局部的观察者无法分辨出其中的区别，就像蚂蚁无法分辨出它是坐在一个平面上还是一个非常大的球体上。 在我们的工作中，我们试图更好地理解三维空间，通过采用一种新型的探针，所谓的接触结构。 尽管数学家们认识接触结构已有几十年的历史，但直到最近二十年才逐渐有可能对这种结构有一个很好的理解。 接触结构也与四维几何（时空几何）、量子物理和动力学（如流体动力学）密切相关。

Proposal DMS-0237386PI: Ko Honda (USC)TItle: CONTACT SRUCTURES AND LOW-DIMENSIONAL TOPOLOGYABSTRACTThis project focuses on the topology of 3-dimensional manifolds via contactstructures. Over the last several years, the investigator and hiscollaborators (Colin, Etnyre, Giroux, Kazez, Matic) have developed a largelycut-and-paste theory of tight contact structures. Taking this program further,he plans to conclude the project on finiteness questions for tight contactstructures (with Colin and Giroux), investigate Legendrian and transverse knots(with Etnyre), and explore connections with foliation/lamination theory (withKazez and Matic). More fundamentally, he seeks to understand the nature oftightness, and relations with hyperbolic geometry and finite type invariants of3-manifolds.This project is a continued study of 3-dimensional spaces. The 3-dimensionalspaces we study will locally be similar to the standard (Euclidean)3-dimensional space. These objects may be very complicated globally, but alocal observer cannot tell the difference, just as an ant cannot tell whetherit is sitting on a flat plane or a very large sphere. In our work, we seek tobetter understand 3-dimensional spaces by employing a new type of probe, calleda contact structure. Although mathematicians have been cognizant of contactstructures for decades, a good understanding of such structures has becomegradually possible only over the last twenty years. Contact structures arealso intimately connected with 4-dimensional geometry (the geometry ofspace-time), quantum physics, and dynamics (such as fluid dynamics).