Reyer Sjamaar proposes to investigate topological and geometric aspects of symplectic manifolds, in particular quasi-Hamiltonian Lie group actions, symplectic Hodge theory, real forms of symplectic manifolds, and their applications to certain eigenvalue problems. This project is expected to advance knowledge in the fields of symplectic geometry and matrix analysis. The proposed methods are borrowed from differential and algebraic topology, index theory and geometric invariant theory, and they build on results concerning cohomological and convexity properties of momentum mappings obtained in previous NSF-funded projects.
According to an important principle of mathematical physics, known as Noether's principle, symmetries of mechanical systems give rise to conservation laws, such as the well-known laws of conservation of energy and momentum. The modern differential-geometric formulation of Noether's principle is based on the notion of a momentum map. The funds for this project will be used to investigate several problems concerning momentum maps, with applications to matrix analysis, which is the study of large systems of linear equations, and the topology of moduli spaces, which are parameter spaces for solutions of nonlinear equations such as the Yang-Mills equation.
