CAREER: Contact Structures and Low-Dimensional Topology

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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).