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I'm quite sure that similar questions like this have been asked for more than thousands of times on here but since each person's background and interests are unique I believe questions like this should not be considered duplicates.
I'm a first year pure math undergradudate student. I have a decent knowledge of multi-variable calculus, mathematical analysis, ODE's, linear algebra and probability theory. Recently I've been teaching myself vector analysis because apparently physicists tend to use vector analysis way more often than mathematicians do (As Daniel Fisher on math.SE said: theoretically mathematicians don't need to know vector analysis because they study differential forms on manifolds in details, that's why vector analysis is never taught in math curriculum). I feel comfortable with index notations, Einstein's summation convention and tensor algebra, however, I have no intuition about how physicists think about tensors.
My knowledge of physics is limited to the Physics I and Physics II courses that I've taken. Both of these courses were taught from the book "Fundamentals of Physics, 8th edition" written by "Halliday, Resnick and Walker". Before I go to university, I used to self-study Resnick's introduction to Special Relativity on my own. I feel quite comfortable with the mathematical ideas behind SR to the level explained in that book. I guess I have to add that our physics I course covered all chapters of Halliday's book except the last chapters on thermodynamics and our physics II course covered up to the RLC circuits (It was a summer course and we had shortage of time).
So, having said all this, my main question is this: I think if I ever decide to get into a graduate program in my current university, I prefer to focus on mathematical physics. I like to study physics from a mathematical point of view. Fortunately, my university allows students to take courses from the Physics department and jointly work with Physics professors during a Master degree. I want to strengthen my physics background during my undergraduate studies. I know that physics undergrad students study a lot of physics and I don't have time to cover all those topics, so I prefer to jump directly to the topics I like to study during my master degree. I'm fascinated by General Relativity and Quantum Mechanics and I hope one day I'd be able to understand string theory.
What is the fastest "mathematically oriented" approach to study GR and QM without learning Lagrangian and Hamiltonian mechanics or Advanced Electromagnetism or such complicated stuff?
Thanks for spending your time reading my question. I hope that my question is not closed as off-topic or duplicate because I really need guidance for my own specific situation.