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Furthermore, could you also apply this question to rigorous math in general? How much will courses such as Lie Algebras and Lie Groups, and Algebraic Topology and Geometry apply to pursuits of Theoretical Physics?

I guess the most basic question I'm trying to ask is: is it necessary to have a deep understanding of these topics to make important/original contributions in physics, or can these insights be achieved with just a basic understanding of the mathematics that I mentioned?