The focus of this proposal is to better understand the nature of contact structures in all (odd) dimensions, with special attention given to dimension three, and to apply contact topological techniques to questions in hydrodynamics. The first main theme of the proposed research is Legendrian knots. Legendrian knots are knots that are tangent to a contact structure and seem to encode a great deal of information about the contact structure. For example, the famed tight vs. overtwisted dichotomy in dimension three can be understood in terms of Legendrian knots. These knots also give an important invariant of a contact structure. As part of this proposal the general structure of Legendrian knots will be studied. The expected outcome will be various classification result for certain Legendrian knots and contact structures; and, moreover, a better understanding of Legendrian surgery (an important surgery construction of contact structures). Legendrian knots in higher dimensions will also be studied using contact homology. There is very little known about contact structures, or Legendrian knots, in dimensions above three. By investigating Legendrian knots in these dimensions the nature of contact structures should be illuminated, just as the corresponding study revealed much about three dimensional contact structures. The final part of the proposed research centers on the connection between contact structures and hydrodynamics discovered a few years ago by the Principal Investigator and R. Ghrist. Here work with Ghrist will continue with the aim of understanding when, and what type of, closed flow lines occur in fluid flows. We shall also study hydrodynamic instability from the contact topological perspective. This naturally leads into the study of energy minimization for fluid flows and relations between contact and Riemannian geometry.
Contact structures are very natural objects, born over two centuries ago, in the work of Huygens, Hamilton and Jacobi on geometric optics. Through the centuries contact structures have touched on many diverse areas of mathematics and physics, including classical mechanics and thermodynamics. In everyday life one encounters contact geometry when ice skating, parallel parking a car, navigating a submarine, using a refrigerator, or simply watching the beautiful play of light in a glass of water. Many great mathematicians have devoted a lot of their work to this subject but only in the last decade or two has it moved into the foreground of mathematics. This renaissance is due to the recent remarkable breakthroughs in contact topology, resulting in a rich and beautiful theory with many applications. The most remarkable feature of all this recent work is the intimate connections between contact structures and topology in dimension three. Moreover, there were important newfound interactions with Hamiltonian mechanics, symplectic and sub-Riemannian geometry, foliation theory, complex geometry and analysis, topological hydrodynamics, and knot theory. The Principal Investigator will explore new connections between contact structures and topology in all (odd) dimensions and continue his study of idealized fluid flows (hydrodynamics) via contact geometry.
