Differential geometry

the stupid df/dx vs partial f/partial x stuff -- we need a co-ordinate system to define the notion of "partial f/partial x", so we can talk about the other co-ordinates being context. Why is this invariant? Chain rule. Gallier, Differential geometry and Lie groups

Lie theory

Update: Much of the Lie theory course (1-9) is now also available as a PDF text. Arguably it is better written than the articles below. 
  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

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