Principal Investigator: Lisa Traynor
A central theme throughout all branches of symplectic geometry is the importance of the special symplectic or lagrangian submanifolds of a symplectic manifold and of the special legendrian submanifolds of a contact manifold. This proposal outlines a number of projects that explore the interesting fine line between the rigidity and flexibility of the shapes of these special submanifolds. Projects include studying the possible efficiency of symplectically packing subsets of euclidean space or cotangent bundles of tori and studying the possible evolution of symplectic balls or legendrian links and tangles under symplectic or contact transformations. Techniques include combinatorics and homology theories defined via generating functions and pseudo-holomorphic curves. In addition, connections between symplectic geometry and biology are proposed based on observed coincidental behaviors of legendrian curves and DNA.
Symplectic and contact geometry have their origins in physics: they are the setting for understanding spinning tops, mechanics of underwater vehicles, and planetary trajectories. A basic problem is to understand how a system can evolve under the natural motions imposed by a contact or symplectic structure. One approach to understanding the dynamics is to start with a standard object such as a ball or a looped piece of string and to study the possible configurations that this basic shape can attain. The results can be quite surprising. For example, although many deformations of a ball are possible, a classical result is that it can never be squeezed in particular directions. This is a geometric version of the Uncertainty Principle from physics. This project aims is to give further insight into the flexibility and rigidity of these canonical transformations.
