ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 53C, 58B, 58D, 83C Fischer, Arthur E., University of California, Santa Cruz, California, USA The Adjoint Formulation and Symplectic Reduction We discuss the geometry underlying the symplectic reduction of an infinitedimensional symplectic manifold (P, ω) with an infinite-dimensional symmetry group G and a momentum map Ψ : P → G^{∗}. Our main tool is a generalized adjoint formulation which asserts that

Z_{A}(p) = (ω_{p}^{♯} ∘ dΨ(p)^{⋆}) · A ,

where Z_{A} : P → T P is the infinitesimal generator corresponding to A ∈ G and dΨ(p)^{⋆} : G → T_{p}^{∗}P is the natural adjoint of dΨ(p) : T_{p}P → G^{∗}. We show how the generalized adjoint formulation plays a fundamental role in symplectic reduction and discuss various applications of this formulation to (1) the construction of a universal symplectic splitting for almost Hermitian G-manifolds, (2) the geometry of symplectic reduction and dynamical reconstruction, (3) linearization stability and bifurcations of momentum maps,

(4) the structure of the constraint space, and (5) equilibria stability and singular reduction.