# Math notes

In 2019 I had some free time and decided to learn a bit of higher math. If I had to go back in time and share with my former self which bits were the most fun and useful to learn from the various areas, this is what I'd say.

## Category theory

### Results

* [The Yoneda Lemma](https://www.youtube.com/watch?v=h64yZs8ThtQ)

The linked lecture is truly a gem.

* [Natural transformations as categorical homotopies](https://math.stackexchange.com/questions/3483323/natural-transformations-as-categorical-homotopies)

### Resources

* [Bartosz Milewski's lectures](https://www.youtube.com/watch?v=I8LbkfSSR58&list=PLbgaMIhjbmEnaH_LTkxLI7FMa2HsnawM_)

## Topology

* Symplectic manifolds and their relationship to Hamiltonian mechanics.
* Lie theory.
* Left-invariant vector fields.
* Simplicial homology.
* Differential forms and [the exterior derivative as an integral](https://math.stackexchange.com/questions/593920/integral-definition-of-exterior-derivative/614473).

## Linear algebra

* Why a finite-dimensional vector space $V$ is canonically isomorphic to its double-dual $V^{**}$, but non-canonically isomorphic to its dual $V^*$
* Why $V\otimes W$ is isomorphic to $Hom(V^*, W)$.