I'm a physics graduate coming out of a physics program that was so poorly structured and disorganized. In the last year and a half of undergraduate school, I developed a very keen interest in the applications of abstract algebra and geometry in constructing physical theories. I don't have a good background in mathematics, I've only taken calculus differential equations, and complex variables, but I'm studying more algebra and geometry on my own at the moment. My problem is that I don't know where to start and what roadmap I should follow to organize my efforts to make up for the lost time in undergrad school. What fundamental mathematics should I start with as a foundation? I studied linear algebra and some group theory and topology, but it all feels so disorganized. Should I have a strong background in real or functional analysis? What would I use that for? What other fields should I be familiar with?
I asked the question in Academia SE but it was closed and I was told that here would be a more appropriate place to ask.
