Symplectic Techniques in Differential Geometry (Symplectic Techniques)

Symplectic geometry is the mathematical framework which was created to most efficiently describe classical mechanics. It dates back to Hamilton's formulation of Newton's laws of motion, in the 1800s. The modern subject, however, was truly born much more recently, in the 1960s and 1970s, following the discovery of surprising "rigid" and "flexible" phenomena. The subject has become a study in the tension between these two conflicting characteristics. In the intervening years, the development of symplectic geometry has accelerated to the point where symplectic methods are applied in a huge number of contexts, far beyond the origins of the subject in classical mechanics. The research proposed here will continue this, attacking problems in a wide range of fields which at first sight are not directly connected to symplectic geometry (Einstein 4-manifolds, minimal surfaces, 3-dimensional conformal geometry, foliations, 3-manifold invariants,...) whilst also addressing questions in the core discipline of symplectic geometry itself (integrable systems, Floer theory, symplectic connections,...).