The focus of this proposal is to study the relation between topology and contact geometry in all (odd) dimensions and to apply contact geometric techniques to questions in hydrodynamics. Many of the connections between topology and contact geometry are mediated by Legendrian knots (these are knots that are tangent to a contact structure), thus the first main theme of the proposed research is Legendrian knots. As part of this proposal the general structure of Legendrian knots will be studied. The expected outcome will be various classification results for certainLegendrian knots and contact structures; and, moreover, a betterunderstanding 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 contactstructures, or Legendrian knots, in dimensions above three. By investigating Legendrian knots in these dimensions the nature of contactstructures 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 study of geometric optics and partial differential equations. Through the centuries contactstructures have touched on many diverse areas of mathematics and physics,including classical mechanics and thermodynamics. In everyday life oneencounters contact geometry when ice skating, parallel parking a car,using a refrigerator, or simply watching the beautiful play of light ina 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. Thus by studying this abstract notion of a contact structure one can learn many subtle things about the universe in which we live. For example, the study of contact geometry has recently lead to some unexpected advances in our understanding of the flow of idealized fluids. The Principal Investigator will explore connections between contact structures and topology in all (odd) dimensions, continue his study of idealized fluid flows (hydrodynamics) via contact geometry and analyze intriguing new conjectures concerning string theory and contact geometry. The Principal Investigator will also engage in several educational endeavors, including the support and encouragement of graduates students and the creation of introductory and survey materials to bring the rapidly developing field of contact geometry to a wider audience.
