Complex and Symplectic Geometry of Complexifications

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

DMS-0204634 Richard Hind One can associate to a smooth manifold a symplectic structure on its cotangent bundle, and to a real-analytic Riemannian manifold a canonical complex structure on a domain inside its cotangent bundle. This project will continue work of the investigator studying these symplectic and complex structures. A broad aim is to discover how closely the symplectic geometry relates to the smooth structure on the original manifold, and how the complex geometry reflects the Riemannian properties. Also, since much symplectic infomation is contained in properties of compatible complex structures, it is also interesting to relate the symplectic and complex geometry. Cotangent bundles are the simplest examples of symplectic manifolds, yet many fundamental problems in symplectic geometry are still elusive in these cases. The investigator will conduct research into the uniqueness of symplectic structures and of isotopy classes of Lagrangian submanifolds. The complex manifolds we study arise naturally in many branches of mathematics, notably algebraic geometry and representation theory. Important aspects of their geometry remain to be understood, together with analytical problems which should have applications to number theory. Symplectic geometry originated as the modern mathematical language of classical and quantum mechanics. It is now an exciting area of pure mathematical research, but continues to benefit from current physical ideas. Since the eighties it has been clear that much insight into symplectic geometry can also be gained from studying compatible complex structures. Fortunately the study of complex geometry has long been a main focus of mathematics. This project will conduct research into both symplectic and complex geometry, focussing on interactions between the two. The specific objects of study arise as the first examples of symplectic manifolds, and as examples of complex manifolds occuring frequently in many branches of mathematics and theoretical physics.