Bryant R.L. An Introduction to Lie Groups and Symplectic Geometry

Durham: Duke University, 1993. — 170 p.A series of nine lectures on Lie groups and symplectic geometry delivered at the Regional Geometry Institute in Park City, Utah, 24 June-20 July 1991.Introduction: Symmetry and Differential Equations
First notions of differential equations with symmetry, classical "integration methods." Examples: Motion in a central force field, linear equations, the Riccati equation, and equations for space curves.
Lie Groups
Lie groups. Examples: Matrix Lie groups. Left-invariant vector fields. The exponential mapping. The Lie bracket. Lie algebras. Subgroups and subalgebras. Classification of the two and three dimensional Lie groups and algebras.
Group Actions on Manifolds
Actions of Lie groups on manifolds. Orbit and stabilizers. Examples. Lie algebras of vector fields. Equations of Lie type. Solution by quadrature. Appendix: Lie's Transformation Groups, I. Appendix: Connections and Curvature.
Symmetries and Conservation Laws
Particle Lagrangians and Euler-Lagrange equations. Symmetries and conservation laws: Noether's Theorem. Hamiltonian formalism. Examples: Geodesics on Riemannian Manifolds, Left-invariant metrics on Lie groups, Rigid Bodies. Poincarg Recurrence.
Symplectic Manifolds
Symplectic Algebra. The structure theorem of Darboux. Examples: Complex Manifolds, Cotangent Bundles, Coadjoint orbits. Symplectic and Hamiltonian vector fields. Involutivity and complete integrability. Obstructions to the existence of a symplectic structure. Rigidity of symplectic structures. Symplectic and Lagrangian submanifolds. Fixed Points of Symplectomorphisms. Appendix: Lie's Transformation Groups, II
Classical Reduction
Symplectic manifolds with symmetries. Hamiltonian and Poisson actions. ment map. Reduction.
Recent Applications of Reduction
Riemannian holonomy. K hler Structures. K hler Reduction. Examples: Projective Space, Moduli of Flat Connections on Riemann Surfaces. HyperK hler structures and reduction. Examples: Calabi's Examples.
The Gromov School of Symplectic Geometry
The Soft Theory: The h-Principle. Gromov's Immersion and Embedding Theorems. Almost-complex structures on symplectic manifolds. The Hard Theory: Area estimates, pseudoholomorphic curves, and Gromov's compactness theorem. A sample of the new results.