I list in the following some main branches of mathematics that I have already heard spoken. I would like to know that, for a student who would like to become a good researcher in pure mathematics, In which level and in what order is recommended to learn these branches during his four years of study at the university for general culture ? If there are branches that you find important but not in my list, you can add them.
- Real analysis
- Complex analysis
- Theory of Measures and Integration
- Functional analysis
- General Topology
- Algebraic topology
- Abstract algebra
- Number Theory
- Ordinary differential equation
- Partial differential equation
- Riemannian geometry
- Differential geometry
- Logic and set theory
- Algebraic geometry
- Analytical geometry
- Probability and statistics
Assuming you've had your standard calculus sequence, linear algebra, discrete math, and perhaps differential equations, I would unhesitatingly say that the most important, by far, are real analysis, abstract algebra, and general topology. Next up: complex analysis and numerical analysis (which you should add to your list; yes, I think even pure mathematicians should know numerical analysis). The rest are all great choices, but I would not call them the backbone of the curriculum. As for order, I think it would be valuable to take topology before any of the rest of your analysis courses. It's valuable to take abstract algebra before number theory. Certainly real analysis before measure and integration. The differential equation sequence just depends on calculus, and perhaps linear algebra. I think everyone should take probability and statistics, but it is not part of the backbone of the standard math curriculum.