**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.

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

**Diffeomorphisms and smooth manifolds**

**Miscellaneous geometries**

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