This research program concentrates on three problems in symplectic geometry. First, we hope to show that Hofer's bi-invariant Finsler metric on the group of Hamiltonian diffeomorphisms of a symplectic manifold is unique. The second project seeks to adapt techniques of asymptotic geometric analysis to the study of symplectic capacities. Previous arguments of this kind have made progress toward a symplectic isoperimetric inequality, indicating the power of the method. Third, the recently discovered Calabi quasi-morphisms of a symplectic manifold will be studied and applied to Lagrangian intersection theory and to a study of the dependence of quantum homology on symplectic structure.
These research projects investigate symplectic geometry, the geometric structure that serves as background to the Hamiltonian approach to mechanics and quantum theory. Coordinate systems in this geometry encode both position and momentum of a moving particle, and additional structure automates the derivation of laws of motion from a choice of Hamiltonian function. Symplectic geometry is an old subject that was renewed in the mid-1980s by exceptionally useful work of M. Gromov on surfaces in symplectic manifolds, and is today one of the most active areas of mathematics.
