I guess besides your GRE, another reason why you cannot go to graduate school is your recommendation letter. If you are going to do geometry and mathematical physics, you should not find an applied math prof unless he/she is extremely famous or he/she was graduated from the school you applied. This is my experience. And also you should not find a young prof with the same reason.
Well, it's good that you can find some big time profs for the recommendation letters for your next application. But I must tell you something, which you may regard as counterexample. I know there is a student, whose recommenders are huge prof in geometry. (You can see this name on some advanced textbooks in Riemannian geometry or research papers. Well, that prof was doing research and got a lot of famous results of Laplaican of Riemannian manifolds). He applied to the schools which are similar to yours. But he was rejected by MIT, Harvard, Berkeley, etc. He is now at Stanford.
I am not discouraging you. As the several posts before, I said selecting people into graduate schools is not an exact science, perhaps not even a science. Most of us have the experience that got rejected by worse schools. On the other hand, if you can do what you said, I think there is no doubt that you can go to best graduate school in US. But, if I were you, I would not let myself to have a very high expectation. Because I know there is nothing for sure in this world, especially something not entirely within your control.
Study mathematics is not only for going to graduate schools. Nothing is to be sacrificed.
Anyway, good luck for your mathematical career.


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