Sorry this reply got long pretty quickly but I think I might be able to help point you in a few different, albeit interrelated directions.
Differential equations are absolutely essential in physics. So if you haven’t had a true differential equations course, I’d recommend starting there. It does have a considerable overlap with linear algebra, so having that background should work in your favor.
For the physical applications of differential equations aside from looking at them through a purely mathematical lens, I would focus on the “simple harmonic oscillator” and it’s various forms. It shows up repeatedly in nature at all scales—from planetary to atomic motion. It’s also foundational to chaotic systems which are a whole lot of fun.
Quantum mechanics is quite literally written in the language of linear algebra, so perhaps you would find it interesting. It may seem daunting at first, but look up “Dirac Bra-ket notation” or “first quantization” to learn about the conventions that physicists use. You might even find that you already understand much of it once you know what the symbols actually represent.
I would say that Lie algebra is most abundantly found in particle physics. But particle physics uses “second quantization formalism” which is like a further generalization of the Dirac notation I mentioned. Basically, in linear algebra terms: first quantization involves using linear operators for physical *quantities* like position, momentum, and energy in *Hilbert Space*, while second quantization uses linear operators associated with physical *fields* like the gravitational and electromagnetic in *Fock Space*.
Having an electrical background, have you worked with a lot of complex numbers? Do you have any experience with signal processing or Fourier analysis? If yes, those things are definitely helpful in physics. If not, I would maybe even start there to get some background on those topics in the electrical language you are already familiar with.
I do theoretical research in quantum condensed matter physics, so I work with a lot more “pure maths” than an experimentalist would, at least generally speaking. I’m currently researching quantum topology and differential geometry, so maybe check those out too if you’re interested. I absolutely love math and physics, so if you have more specific questions feel free to DM me and I’d be happy to answer your questions. Hope this helps!
