Lie group actions on symplectic manifolds

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Reyer Sjamaar proposes to investigate topological and geometricaspects of symplectic manifolds, in particular quasi-Hamiltonian Liegroup actions, symplectic Hodge theory, real forms of symplecticmanifolds, and their applications to certain eigenvalue problems.This project is expected to advance knowledge in the fields ofsymplectic geometry and matrix analysis. The proposed methods areborrowed from differential and algebraic topology, index theory andgeometric invariant theory, and they build on results concerningcohomological and convexity properties of momentum mappings obtainedin previous NSF-funded projects.According to an important principle of mathematical physics, known asNoether's principle, symmetries of mechanical systems give rise toconservation laws, such as the well-known laws of conservation ofenergy and momentum. The modern differential-geometric formulation ofNoether's principle is based on the notion of a momentum map. Thefunds for this project will be used to investigate several problemsconcerning momentum maps, with applications to matrix analysis, whichis the study of large systems of linear equations, and the topology ofmoduli spaces, which are parameter spaces for solutions of nonlinearequations such as the Yang-Mills equation.