lie algebra

this is just sort of an idea im toying with but i think maybe one of the reasons a lot of people struggle with math is they aren’t being taught math grammar properly?? like, there’s sort of 2 parts to math: the abstract concepts, & the language we use to write them down on paper. the language part really is just a language, but because it’s such a precise one, tiny mistakes like misplaced parentheses completely change the meaning of things. so if you don’t have a good grasp on the grammar it really hinders your ability to work with & understand the abstract concepts

Revision...

Revised the first half of topology today and remembered how much I like it. Its one of the few courses where I can still sit down and scribble pictures and stare in to space and use plenty of intuition and eventually arrive at the correct reasoning; pretty much what got me into maths in the first place. It really is a beautiful and rich subject!

Going to start the algebraic topology part tomorrow, much harder but also really interesting. The thrust of algebraic topology as I understand it is to translate topological problems into problems in group theory. These are generally much easier to understand and a great deal of group theory is very well understood.

And then the ideas of Lie Algebras help us translate harder group theory problems into linear algebra which is easier again! I love maths, why don’t I listen in lectures?

Generalisation is not the point of mathematics. To be honest, it’s usually rather dry. The challenge is to generalise in a rich and revealing direction.
— 

Terry Gannon, Moonshine Beyond the Monster §3.3

(generalisations of the affine algebras: Kac-Moody algebras, Borcherds’ algebras, toroidal algebras)