The focus of this proposal is to address several fundamental questions in low and high dimensional contact geometry. In low dimensions, the most basic question asking which three manifolds admit tight contact structures is still open as are questions related to the result of various surgery operations on contact manifolds. As part of this proposal the Principal Investigator will build on past work studying these questions to, among other things, illuminate the nature of tight contact structures on hyperbolic manifolds and Legendrian surgery on tight contact manifolds. In addition, he will extend recent advances in Legendrian knot theory to not only better understand the structure of such knots but also to classify contact structures on some families of three manifolds including some of the much studied and notoriously difficult small Seifert fibered spaces. In higher dimensions even the existence of contact structures is not completely understood. Recent progress in higher dimensional contact geometry makes the time ripe for an intense investigation of these structures. Namely, a few years ago Niederkrueger introduced the notion of a plastikstufe in hopes of finding an analog of the famed three dimensional tight vs. overtwisted dichotomy in all dimensions (other proposed notions, such as bLobs, have even more recently surfaced) and most recently the Principal Investigator has completely answered the existence question for contact structures on five manifolds (as has another team of researchers). Part of the project will involve addressing the existence of contact structures on all odd dimensional manifolds as well as investigating notions of overtwistedness in higher dimensions. The Principal Investigator will also further develop contact homology computations in higher dimensions and study the elegant conormal construction in order to apply contact geometric techniques to the study of knot theory in dimension three and embedding theory more generally.
Contact geometry is a venerable subject that arose as a natural language for geometric optics, thermodynamics and classical mechanics. One encounters contact structures everyday when parallel parking a car, skating, or watching the play of light in a glass of water. Contact geometry has long been studied by mathematicians in physicists but in the last decade or so it has blossomed into a remarkably rich and beautiful theory with close ties to the topology of manifolds (that is the structure of space and space-time), string theory in modern physics, Riemannian geometry, and fluid dynamics. The Principal Investigator will illuminate the connection between contact geometry and Riemannian geometry and the topology of manifolds. He will also explore basic and fundamental questions concerning the existence and uniqueness of contact structures and their submanifolds in high dimensions. In addition the Principal Investigator will continue working with a large group of graduate students and organize conferences and seminars to help educate the next generation of researchers and create fertile environments in which new ideas and collaborations can grow.
