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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".

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closed as primarily opinion-based by tpg2114, Brandon Enright, David Z Feb 10 at 23:29

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

Sounds interesting, but unfortunately vague resource-recommendation questions tend not to get answered here. You sound like you're on a decent track to learning the necessary stuff though. –  DumpsterDoofus Feb 11 at 2:55