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My question is would you recommand an (average level intelligent) future theoretical physcist to do a Phd in pure mathematics (related to physics such as topology) and go back to physics once the mathematics has been mastered. Or do a Phd in physics and pick up the higher level mathematics such as Topology, abstract algebra etc

Considering that this person is not a genius than I presume that the former (Phd in math) is a better choice since it is much harder to learn higher level mathematics by one self compared with picking up a physics text book without any fear of the mathematics in it and ready to absorb all the physics inside. But I could be wrong. Any suggestions.

Thanks