I am interested in picking up enough mathematical background in order to easily understand a paper like this one: Growth, geometry and mechanics of a blooming lily, and be easily able to create my own models when necessary. Therefore, a first-principles knowledge of the mathematics behind models for physics is not only aesthetically pleasing to me, but also necessary.
Here is how I have bumbled along thus far: I asked a question similar to this one on Mathematics StackExchange, which was soon closed.
Before it was closed however, I gathered some interesting pointers on topics I could explore (see answer to that question):
I also got pointers to check out general topics like differential geometry and topology.
I picked up some pretty good books, and on the way, got hooked onto geometric algebra. Oh, I loved it! A chance to be free from crappy tensor notation?! I'll take it! I even found a book that approaches differential geometry from a geometric algebra perspective.
As I began to learn though, I became fascinated with the process of setting up an algebra rooted (initially) in the motivation to model physical phenomena. I spent a bit of time thinking about whether for instance, if it would be possible for me to develop an algebra that captures direction and magnitude separately. I wondered what it is about algebraic structures in particular that allows them to "model". I wanted to learn more about Grassmann's "General Theory of Forms".
So I kept on going down the rabbit hole of abstraction. I read some preliminaries on lattice theory, and today I was about to start reading Ravi Vakil's notes on algebraic geometry. I thought about why it is I started down this road though, after reading the preface to the notes, which asked me to always remember this, and I wondered if I should ask for some feedback on my path so far. The fact that I feel a little lost right adds only further motivation to ask:
What are your comments on my path so far? Which direction would you point me in next?
I should note that being a structural engineering undergraduate student in the past, I have a pretty firm grip on the lightly-mathematical introduction undergraduate students get to concepts like "elasticity" and "plasticity".