How much pure math should a physics/microelectronics person know I do condensed matter physics modeling in my phd and I was struck up learning quite an amount of physics. But while having done lot of physics courses, I see that if I learn pure math I would understand lot of things better and also probably my modeling skills also might get better. A math prof asked me to do all courses from basic analysis to differential geometry. But that would take time as I see I am taking time doing rudin(principles of mathematical analysis) . So I am little confused whether am going the right way. Any suggestions?
 A: Warning: Biased answer ahead.
I think this greatly depends on the precise area of research you plan to get into, but I can at least list things that I believe are necessary for you:


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*Linear algebra: The theory of vector spaces and linear maps, especially in the infinite-dimensional setting. Some people would call this functional analysis. Regardless, this is important for understanding the formalism behind quantum mechanics.

*Classical analysis: The study of differentiable and integrable functions, as well as a good intuition for how to approximate functions and estimate errors, is extremely important. Since you mentioned modeling, knowing how to control the norms of various errors is very important for numerical simulations. I would lump advanced calculus and differential equations into this category. Study of complex variables is also very useful, as many phenomena in physics are very nicely described using complex functions, such as circuits and electromagnetic waves.

*Point-set topology: You don't need to delve into this area that deeply, but ultimately, all of modern physics has developed a very topological flavor. Study of this area might also help you gain intuition about the general setting most mathematical physicists use.

*Discrete mathematics and probability: Again, this is a broad field, but a physicist should have a good sense for how to translate physical models between "continuous" and "discrete" versions, and what sorts of different phenomena arise from them, such as in lattice models for phase transition, etc. Related to this area is probability, which arises in quantum mechanics, statistical mechanics, and other areas. A solid grasp of principles in probability would probably prove very useful.
Areas commonly studied but which I am not emphasizing from a pure math standpoint:


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*Abstract algebra: I think this really depends on the specific area of physics research, and anyhow, mathematicians study algebra in a much broader way than physics really needs.

*Geometry: Certainly used often in physics, but again not something you necessarily need to know from a mathematician's point of view.

*Computation: The study of algorithms, their complexity and numerical stability. This is actually something you should study, but I'm going to guess that physicists already offer courses in this area with the right amount of physics focus.

