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From my research, it seems that the answer differs by institution. For example, it would be no problem to apply to the Applied Mathematics program at Chicago and do research in theoretical physics, but Princeton, for instance, does not consider theoretical physics as applied mathematics.

Let me clarify. I'm currently at math major. Last year, as a first year, I took Honors Analysis and Complex Analysis, receiving As in both class. By the end of my second year, I will have taken Topology, Functional Analysis, Algebraic Topology, Differentiable Manifolds, and three quarters of Honors Algebra. (However, I did research in manifold theory last summer, so that I already know quite a bit of differential geometry and algebraic topology.) I have not taken any physics courses here (although I've done quite a bit of independent study), but I'll probably take graduate quantum physics and advanced mechanics by the end of my 4th year. I will have completed graduate analysis and graduate geometry/topology by the end of my 3rd year. Clearly, I'm more of a mathematician than I am a physicist.

The main areas of math and physics that interest me are differentiable geometry and its connections with relativity/cosmology and functional analysis and its connections with quantum physics. So, at least according to Wikipedia's definition, I'm interested in mathematical physics. Unfortunately, this interest doesn't seem pure enough to qualify as pure math, and most institutions apparently don't consider mathematical physics as applied mathematics. This leaves physics, but to be honest, I'm really not interested in any other branches of physics than what I already stated, and I would be required to take graduate classes in many different areas of physics, most of which are completely unrelated to my research area, and in which I have no experience. In addition, it would be difficult to get into the top physics programs in the country considering the fact that by the time I apply to grad school (early in my 4th year), I would only have 3 physics courses max on my transcript. Can anyone give me any advice?