Submanifolds and Metrics in Contact Geometry

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Contact structures on manifolds are natural objects, born over two centuries ago, in the work of Huygens, Hamilton, and Jacobi, on geometric optics. They have been studied by many mathematicians and seem to touch on diverse areas of mathematics and physics, but only in the last few decades have they moved into the foreground of mathematics. This is due to the remarkable breakthroughs in contact topology, resulting in a rich and beautiful theory with many applications. Studying subsets and their interactions with such structures was instrumental in the understanding of three-dimensional spaces, and it led to profound progress. The Principal Investigator will now extend this to higher dimensions, where this exploration is likely to prove equally illuminating. The Principal Investigator will also continue to study properties of contact structures on low-dimensional spaces and their interaction with topology and Riemannian geometry, and he will train the next generation of researchers by working with a large group of graduate students and organizing conferences and seminars.The research supported by this award will focus on problems centered around three broad topics: the interactions of contact geometry and topology in low dimensions, properties and constructions of contact manifolds in higher dimensions, and connections between contact geometry and the more familiar Riemannian geometry. In low dimensions the main motivating question is to determine which three-manifolds admit a tight contact structure. Currently quite a bit is known about this question, but very little is known about it for hyperbolic homology spheres. This problem will be studied using a variety of techniques, from convex surfaces, to holomorphic curves and Riemannian geometry. In addition, understanding interactions between various properties a contact structure can have will be studied. While much is known about contact geometry in low dimensions, there is very little known in higher dimensions. The principal investigator will study constructions and properties of high-dimensional contact manifolds. The starting point for this will be the study of isotropic and contact submanifolds of contact manifolds. Such considerations have led to a wealth of information in low-dimensions and it is expected to be similarly fruitful in higher dimensions as well. In the past few years there have been some interesting and subtle connections between contact geometry and Riemannian geometry. The Principal Investigator will explore this further hoping to find contact geometric analogs of classical results relating topology to Riemannian geometry.