CAREER: Contact Structures and Low-Dimensional Topology

题目：接触结构和低维拓扑（CONTACT structures AND LOW-DIMENSIONAL topology）摘要：本课题主要研究三维流形的接触结构拓扑。在过去的几年里，研究者和他的合作者（Colin, Etnyre, Giroux, Kazez, Matic）已经开发了一个紧密接触结构的剪切粘贴理论。进一步，他计划完成紧接触结构的有限问题项目（与Colin和Giroux），研究Legendrian和横向结（与Etnyre），并探索与叶理/层压理论的联系（与kazez和Matic）。更根本的是，他试图理解紧性的本质，以及与双曲几何和3流形有限型不变量的关系。这个项目是对三维空间的继续研究。我们研究的三维空间将在局部与标准（欧几里得）三维空间相似。这些物体在全局上可能非常复杂，但局部观察者无法分辨其中的区别，就像蚂蚁无法分辨自己是坐在一个平面上还是一个非常大的球体上一样。在我们的工作中，我们试图通过使用一种称为接触结构的新型探针来更好地理解三维空间。尽管数学家们几十年前就已经认识到接触结构，但对这种结构的良好理解只是在最近二十年才逐渐成为可能。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).