I spent around two months being completely enamoured by geometric algebra until I realised it flies in the face of some pretty strong trends of mathematics in the last couple of decades.
How come? Well, we’ve been moving towards more and more powerful generalisations and unifying concepts over the years. First, it was the various flavours of algebraic this-and-that, then category theory, then today, people are trying to do hot things with HoTT.
(of course math is much, much bigger than this, but this is the story I particularly have been watching unfold before me)
The trouble with geometric algebra is that it’s merely a special case of a much more general concept, namely tensor algebras. Tensor algebras are the most general algebra you can possibly get from things that behave like vectors; with GA you are stuck with vectors where the distances are measured by something squared as opposed to any other power (or some other crazy function like the taxicab metric).
Using geometric algebra feels like typeless programming because you’re combining everything into a single quantity called the multivector whose components aren’t straightforward to keep track of. Furthermore, you have to decide the dimension in advance because IIRC these types change when you add more dimensions (the formulas however do not, surprisingly).
That said, it’s really marvellous how compact things can get if you stay in 3+1 dimensions. Maxwell’s equations become literally a one-liner, and the horrible, horrible mess that is rotational dynamics becomes so much more tractable for mortals like me. One can literally do the entire physics undergrad curriculum using GA without breaking a sweat. Trouble is, you’d have to learn some basic algebraic theory and that’s a hard sell for people who are already afraid of math because “it’s too abstract”.