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This is most likely a question for those who have experience/knowledge of theoretical/mathematical physics at the graduate level and can provide recommendations following my criteria.

Here is some background about me:

I am a senior majoring in math and physics at a small university and am required to choose a topic to perform research on for my senior year. My university does not offer any graduate programs in physics, so I am essentially required to come up with a topic by myself as my professors which do perform research are mainly experimentalists and i'm not to interested in their research.

As mentioned in the topic, I am particularly interested in a theory based project as opposed to experiment since 1) beneficial experimental projects here are simply not attainable for financial reasons and 2) I am interested in pursuing a career in theoretical/mathematical physics post undergrad.

My specific goals for this project are simple. I do not expect to do cutting edge, original theoretical research. I would simply like to pick a topic which is well established and would allow me to learn a substantial amount of new physics and mathematics in preparation for graduate school. In particular, I would like to pick a mathematical/theoretical topic which, once an adequate amount of the theory has been learned on my part, has a substantially difficult problem/calculation accessible to an undergraduate I could then apply what I've learned too. Once having done this I would then compare my results with what has been established or a computer simulation.

At the moment, my interests in potential topics include particle physics (Griffiths Book) where I could calculate the lifetime of a particle- although i'm not sold on this problem- or general relativity ( Shutz, hartle, Moore, Thorne etc.) of which i haven't a clue of a potential target problem/calculation.

Finally, I should mention i am quite competent in the following areas at the standard undergraduate level i would assume: calculus(multi variable, vector etc.), linear algebra, differential equations, applied multi variable calculus in statistics, beginner formal maths (number theory, real analysis,geometry), thermodynamics (baierlein level), E&M (Griffiths), classical mechanics (Taylor), quantum mechanics(david mcintyre),modern/atomic/nuclear physics (Townsend/Riesnick).

I have a whole year to complete this project, so it can be fairly large,rigorous,challenging and demanding.

Thank you for any assistance in this matter!