Award: DMS 1105832, Principal Investigator: Dmitry Tamarkin
These research projects concentrate on issues related to the microlocal approach to categories associated with symplectic manifolds, such as the Fukaya category. It is anticipated that by applying algebraic tools such as differential graded categories and higher categories, will yield geometric simplifications by avoiding transversality and other geometric conditions that can be hard to provide.
Symplectic geometry is the study of the background structure for the Hamiltonian version of classical mechanics. The study of this kind of geometry was revolutionized in the 1980s by the use of analytic techniques involving maps of surfaces into symplectic manifolds, a point of view that led more recently to the Fukaya categories invoked below. These Fukaya categories carry a lot of significant information about the underlying symplectic manifold, and it would be useful to have a less subtle description of them than the original definition.