Moment maps and J-holomorphic curves

Stanford University, Stanford CA

Investigators

Abstract

DMS-0072267 Kai Cieliebak The focus of this project is the investigation of a new system of nonlinear partial differential equations. These equations arise naturally in the study of J-holomorphic curves in symplectic manifolds with a Hamiltonian group action. Instances of these equations include the vortex equations, the anti-self-dual Yang-Mills equations, and the Seiberg-Witten equations. Solutions of the equations give rise to invariants of Hamiltonian group actions. These invariants are related via an adiabatic limit to the Gromov-Witten invariants of the symplectic quotient by the group action. This project pursues the following goals: (1) achieving a deeper understanding of known connections between invariants; (2) establishing new correspondences between invariants; (3) using the correspondences between invariants to compute invariants. Invariants that can be addressed in this way include: Donaldson invariants and Seiberg-Witten invariants of certain smooth 4-manifolds; instanton Floer homology and Seiberg-Witten Floer homology of certain smooth 3-manifolds; Gromov-Witten invariants, quantum cohomology and Floer homology of symplectic manifolds. A symplectic manifold is a generalisation of the phase space in classical mechanics. Geometric properties of this space influence the behavior of mechanical systems. For example, compactness of the space forces mechanical systems to admit periodic motions. This project focuses on geometric properties of symplectic manifolds with symmetries. Symmetries of a mechanical system lead to conserved quantities represented by a moment map. Classical examples of moment maps are linear and angular momentum. This project lies in the intersection between two large branches of symplectic geometry: the study of J-holomorphic curves, and the study of Hamiltonian group actions. These branches have traditionally had little overlap. Moreover, the project touches several areas adjacent to symplectic geometry: Hamiltonian dynamics, quantum field theory, low-dimensional topology, and algebraic geometry.