CAREER: Contact Structures and Low-Dimensional Topology

University Of Southern California, Los Angeles CA

Investigators

Abstract

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