Shapes of Symplectic and Legendrian Submanifolds

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AbstractAward: DMS-9971374Principal Investigator: Lisa TraynorA central theme throughout all branches of symplectic geometry isthe importance of the special symplectic or lagrangiansubmanifolds of a symplectic manifold and of the speciallegendrian submanifolds of a contact manifold. This proposaloutlines a number of projects that explore the interesting fineline between the rigidity and flexibility of the shapes of thesespecial submanifolds. Projects include studying the possibleefficiency of symplectically packing subsets of euclidean spaceor cotangent bundles of tori and studying the possible evolutionof symplectic balls or legendrian links and tangles undersymplectic or contact transformations. Techniques includecombinatorics and homology theories defined via generatingfunctions and pseudo-holomorphic curves. In addition,connections between symplectic geometry and biology are proposedbased on observed coincidental behaviors of legendrian curves andDNA.Symplectic and contact geometry have their origins in physics:they are the setting for understanding spinning tops, mechanicsof underwater vehicles, and planetary trajectories. A basicproblem is to understand how a system can evolve under thenatural motions imposed by a contact or symplectic structure.One approach to understanding the dynamics is to start with astandard object such as a ball or a looped piece of string and tostudy the possible configurations that this basic shape canattain. The results can be quite surprising. For example,although many deformations of a ball are possible, a classicalresult is that it can never be squeezed in particular directions.This is a geometric version of the Uncertainty Principle fromphysics. This project aims is to give further insight into theflexibility and rigidity of these canonical transformations.