Symplectic geometry, a branch of mathematics having its origins in classical dynamics, has become an area of growing importance in contemporary mathematics. In this project we propose several plans for research that have as their common theme the application of ideas from symplectic geometry to other areas of mathematics, and in return the ways in which ideas arising from these applications motivate new methods in symplectic geometry.
One of our projects involves extending ideas from the symplectic geometry of compact manifolds to the setting of Banach manifolds appearing in gauge theory and the theory of loop groups. Another is a set of applications to combinatorics and real analysis, and possibly number theory, centered on the generalizations of the classical Euler-Maclaurin formula. Still another project is an attempt to understand the structure of hyperkahler manifolds by reworking the ideas of Atiyah and Bott on gauge theory over Riemann surfaces in this context.
