Lie theory

1. Introduction to Lie groups
2. Lie Bracket, closure under the Lie Bracket
3. Derivations and the Jacobi identity
4. Lie group homomorphisms
5. Exponential map: injectivity and surjectivity, diffeo, examples of differential correspondence (exp, Taylor, derivations, adjoint), continuous maps are smooth [1], Baker-Campbell-Hausdorff
6. Lie group topology
7. Lie correspondence (exercise: determinant unique)
8. The Killing form; factorising non-Abelian Lie groups
9. Some applications (parallel parking, indefinite Orthogonal group, some isomorphisms, some connected and compact proofs, odd-dimensional spheres, covers and quaternions)
10. Abstract Lie algebra, representation theory [3]
11. Lie derivatives, relations to differential geometry [4]
12. Lie Geometry -- [5]; exponential surjective when compact