Geometry

Lie theory

UPDATED Lie theory course available as a ★★★★★ PDF text. Archived Lie theory course:

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

Diffeomorphisms and smooth manifolds

1. ★★★★☆ Curvature is just the Hessian

Miscellaneous geometries

1. ★★☆☆☆ Introduction to Projective Geometry