# Math learning after Differential Equations?

I made it up to DE(Pre-Calc, Calc I and Calc II with basic stats before that same school) in community college through a special program(I'm poor). I'll have to self-teach from now on but am still very in to math and engineering. What do people usually go to with math after DE?

I'm interested in topology but it looks like there is a lot of other things I need to do first like advanced algebra.

• you don't need almost anything to start learn topology. – user42912 Oct 9 '14 at 2:36

After differential equations, I think you should try learning real analysis at the level of Rosenlicht or so and abstract algebra at the level of Gilbert and Gilbert. These two topics will greatly improve your mathematical maturity and will prepare you for more advanced mathematics.

From here, I would learn complex analysis and venture into topology. You don't need to learn topology after real analysis, but I feel that topology is best understood after learning real analysis well. However I think you must know real analysis well before complex analysis. Complex analysis is almost impossible to appreciate without seeing how.. yucky real analysis can be. From there, the sky is the limit.

If you haven't had the genuine treat of a basic course in complex variables, do that first. Topology is high in my list, but you really must get complex variables under your belt first. It will completely change your take on lots of stuff you have already done.

• Why don't real analysis first? – user42912 Oct 9 '14 at 2:43
• I think doing complex prior to real is a huge travesty, especially considering how real analysis results are needed a lot in complex analysis. – Cameron Williams Oct 9 '14 at 2:45
• @CameronWilliams: No, you misunderstand me, I'm not talking about Complex Analysis. I'm talking about a basic course in introductory complex variables. This barely requires calculus. – MPW Oct 9 '14 at 3:04
• Even still, I feel you can't appreciate complex variables without knowing calculus on $\Bbb R^2$ and how awful it can be. – Cameron Williams Oct 9 '14 at 3:25
• @CameronWilliams: Truth be told, he should probably finish the calculus sequence (Cal III and Cal IV) before taking any of the more advanced stuff. Then he'll be in a far better position to appreciate higher mathematics. – MPW Oct 9 '14 at 3:40

Real analysis if you have not done it so far.

• Real analysis gives you a foundation for the things that you learned in the (presumably) intuitionistic calc/de courses you took. It is by far the most important topic you can master if you plan to continue doing analysis. – parsiad Oct 9 '14 at 2:40
• Just to point out that "intuitionistic" does not mean "intuitive" or "intuition-based," but rather "denying the law of the excluded middle," so that the OP presumably did not take intuitionistic calculus courses. – Kevin Carlson Oct 9 '14 at 2:47
• I should have used a less overloaded word; I did not mean it in the intuitionistic logic sense. – parsiad Oct 9 '14 at 2:49
• ...it's not impossible that OP knows some en.wikipedia.org/wiki/Constructive_analysis :-) – parsiad Oct 9 '14 at 2:54
• I know a little. DE(MAT 285) was the most rigorous class at the local community college, and I spent about a year doing all the MAT courses this school had and finishing with that. After research I find it's pretty much the same case in all community colleges. After that you learn on your own or pay tuition at a university. I'm mostly using it for economic modeling and game theory, but I'm also interested in pure mathematics. – 4334of Oct 10 '14 at 21:16