CAREER: Contact Structures and Low-Dimensional Topology

提案DMS-0237386PI：Ko Honda(南加州大学)标题：联系SRUCTURES和低维拓扑这个项目专注于通过接触结构来研究三维流形的拓扑。在过去的几年里，这位研究人员和他的合作者(科林、埃特尼尔、吉鲁、卡兹、马季奇)发展了紧密接触结构的大量剪切粘贴理论。在这个项目的进一步发展中，他计划(与Colin和Giroux)完成关于紧密接触结构的有限问题的项目，(与Etnyre)研究Legendrian和横向纽结，并探索与分层/分层理论的联系(与Kazez和马季奇)。更根本的是，他试图理解紧性的本质，以及与三维流形的双曲几何和有限类型不变量的关系。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局部地类似于标准的(欧几里得)三维空间。这些物体在全球范围内可能非常复杂，但局部观察者无法区分，就像蚂蚁无法分辨它是坐在平面上还是坐在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新型的探头，称为接触结构，来更好地理解三维空间。虽然数学家对接触结构的认识已经有几十年了，但对这种结构的良好理解直到最近20年才逐渐成为可能。接触结构还与四维几何(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).