Applications of Symplectic Geometry

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Symplectic geometry, a branch of mathematics having itsorigins in classical dynamics, has become an area of growing importancein contemporary mathematics. In this project we propose severalplans for research that have as their common theme the applicationof ideas from symplectic geometry to other areas of mathematics, andin return the ways in which ideas arising from these applications motivatenew methods in symplectic geometry.One of our projects involves extending ideas from the symplecticgeometry of compact manifolds to the setting of Banach manifoldsappearing in gauge theory and the theory of loop groups. Anotheris a set of applications to combinatorics and real analysis, andpossibly number theory, centered on the generalizations of theclassical Euler-Maclaurin formula. Still another project is anattempt to understand the structure of hyperkahler manifolds byreworking the ideas of Atiyah and Bott on gauge theory over Riemannsurfaces in this context.