Chain level local Floer theory

Symplectic topology is the study of properties of symplectic manifolds which are invariant under Symplectic transformations. A symplectic manifold is a space in which to any two dimensional surface embedded in it there is associated a quantity called symplectic area. A symplectic transformation is a transformation of space which is allowed to stretch and compress space in such a way that symplectic area of any two dimensional surface embedded in space is unchanged. Such transformations play a fundamental role in many of the meeting points of mathematics and physics, from classical mechanics through quantum mechanics to string theory. In pure mathematics, they also he at the crossroads of the central fields of geometry.

Symplectic topology exhibits an intriguing mix of flexibility and rigidity. This makes it very difficult to get a handle on it. Using a simile by Allen Hatcher, Floer Theory is like a lantern forming algebraic images of symplectic manifolds. While topology is elusive, algebra can be manipulated mechanis tically. However, Floer theory forms these algebraic images in a non-explicit way, making it hard to compute.

The aim of the present research is to create tools for studying Floer Theory of a symplectic manifold by chopping it up into smaller building blocks, forming the algebraic images of these, and then piecing together the algebraic image of the global manifold from those of the local pieces.