What mathematics to learn

A few points with pure math in mind (notes to my past self, perhaps):

• Mathematics is a single discipline, knowing each of its basic topics helps in understanding the other topics. Don't omit anything on undergraduate level, include some topology, set theory, logic, geometry, number theory, category theory, complex analysis, differential equations, differential geometry where some courses skip them (in addition to the more reliably standard linear algebra, analysis, abstract algebra, etc.).
• The goal is fluency, as in learning a language, not mere ability to parse the arguments and definitions. It's possible to follow a text that's much too advanced for your level, but you won't learn nearly as much as if you were ready to read it.
• Reading unfamiliar mathematics is difficult, while familiar material can be rapidly scanned. As a result, reading partially redundant supplementary texts comes at a relatively modest cost, but improves understanding of the material. In particular, some books can be included primarily to connect topics that are already known. Other books can be included as preliminary texts that precede other ostensibly introductory books that you could parse, but would learn less from without the preliminary text.
• Learn every topic multiple times, at increasing levels of sophistication, taking advantage of the improving knowledge of other topics learned in the meantime. The rule of thumb is to read 1-2 books on undergraduate level, and 1-2 books on graduate level.
• Don't be overly obsessive, it's not necessary to repeat all proofs in writing or solve all exercises.
• Don't shy away from revisiting elementary material. It's not there just as a stepping stone to more advanced material, to be forgotten once you're through, it should remain comfortably familiar in itself.

EDIT: As Vladimir Nesov points out below, don't just study what I recommend; this is a list of things that you might look up outside of classes to help stay motivated while doing a degree in mathematics, more than a list of things you should study to learn mathematics outside of school.

Also refined point (7).

I am a mathematics graduate student; I currently focus on number theory and arithmetic geometry. So here are a few areas I'd recommend, coming from different goal structures I have:

1) if you are interested in learning things that are really cool and beautiful I would recommend elementary number theory, for example from Hatcher's visual approach. This doesn't require much heavy grounding and is absolutely awesome and has neat pictures. If you want to continue on this path, p-adic numbers are the place to go.

2) if you are interested in studying advanced mathematics, I'd recommend studying representation theory and category theory; these seem to have lots of applications in almost every area of mathematics, including algebra, number theory, mechanics, geometry, and topology. Maybe less in analysis or logic, although more logicians' perspectives on category theory seems valuable to me. Also complex analysis.

3) If you are interested in abstract concepts that feel like they have universal applicability (I don't know how much the metaphors I draw from these actually help me but I draw them almost constantly): linear algebra, group theory, and basic real analysis. Symmetry and distance are every day concepts; seeing them mathematically derived was very cool to me.

4) if you want to make lots of money, I think calculus and dynamical systems lead most directly into financial modeling; I'm not really sure.

5) if you want to do interesting research with real life applications, you might be better sticking to statistics and probability theory; although dynamical systems have their applications in game theory the impression that I get is that the difficulty mostly comes from differential equations and computation complexity, not from mathematical insights.

6) You probably shouldn't study algebraic geometry. I do a little, but it is filled with technical definitions and complex terminology and it has a reputation for taking people a very long time to be able to understand at all. If you want an intellectual challenge for yourself maybe it's appropriate, but if you want to learn a field quickly and use the insights it is probably more trouble than it is worth, at least until some amazing new text book comes out on it which I doubt may ever happen. It is too late for me, save yourself!

7) if you are in school at a university, I'd suggest looking up math professors on ratemyprofessors, the ratings aren't perfect but it does look like they correlate well with my experiences with my professors. Requiring slightly more effort but giving much better information would be asking other math majors or TAs about different professors' teaching styles. And then, just take courses from good professors. This is probably worthwhile in any subject; better professors are going to mean more than good classes. Taking a class with a good professor means you will probably enjoy the class, taking it with a bad professor means you probably won't. I don't think this is the context of your question but it is probably relevant to others asking similar questions.

the utility of different mathematical concentrations

The answer is highly dependent on one's goals, interests, and personality. Hence there is unlikely to be any particular document which successfully explains the answer in a way that applies to everyone.

What would be better would be for people interested in learning mathematics to post comments explaining the nature of their own particular interest, and to receive individually-tailored replies. If general patterns emerge from those replies, these general patterns can then be extracted and abstracted afterward, preferably with appropriate cautions and disclaimers.

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I am interested in Computer Science, compiler optimization, and machine learning but I know relatively little about these subjects.

I am currently reading Concrete Mathematics.

Don't forget to learn some functional analysis, especially Hilbert spaces and how they're used in, for example, quantum theory and stochastic processes.

The main question which is important here: why do you want to learn mathematics?

Here's one such resource: The simple math of everything

What resources exist detailing which mathematics to learn in what order?

One could acquire a university syllabus for a degree in mathematics and use textbooks and or MIT Opencoursware to learn with.

What resources exist that explain the utility of different mathematical concentrations for the purpose of directing studies?

I am having trouble understanding this question. By the words "mathematical concentrations", could one substitute the term "mathematical subfields"? If so, then I would say the best resource could be more experienced people with similar instrumental goals to one's self.

Really, you should study the math that you think is the most deep and beautiful and fascinating.

Here's one such resource: The simple math of everything