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Someone has made this template for a self taught CS degree. I am a CS grad and found it quite good. As I don't have much idea about a proper pathway to learn degree level mathematics in my own. Please guide me for a similar template for mathematics. The name of the papers would be enough, as I will try to search for MOOCs, however if someone provides that too, it will be great help.

Initially, I posted this question in mathoverflow. Someone suggested to post it here.

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  • $\begingroup$ If you can afford it, hire tutors (i.e. grad students) to help. $\endgroup$
    – Elle Najt
    Commented Aug 4, 2018 at 23:12
  • $\begingroup$ [Not sure I can recommend it but] Here is what I took as an undergraduate by semester: 1, Calc I; 2, Calc II, Discrete Math; 3, Calc III, Linear Algebra; 4, Real Analysis, Abstract Algebra I, Intro to Comp Sci; 5, Complex Analysis, Mathematical Logic; 6, Topology, Philosophy of Math, Theoretical Foundations of Comp Sci; Summer, REU on p-adic Arithmetic Dynamics; 7, Thesis I (p-adic analysis), Abstract Algebra II (Galois Theory); 8, Thesis II, Real Analysis II, Metaphysics. $$ $$ Notably missing are: Differential Equations, Number Theory, Set Theory, Probability, Statistics, Econ[ometrics] etc. $\endgroup$
    – Benjamin Dickman
    Commented Aug 6, 2018 at 20:18

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That's an interesting question and it's a little difficult to answer without knowing what you're starting point is and what your interests are. If you don't have a solid, high school level background then you should start with algebra (often called College Algebra) and some kind of pre-calculus class that includes trigonometry. Once you've got those down, you're ready to get serious with the college level material.

Semester One

  • Calculus I - calculus of two variables, primiarly focused on differentiation with a basic introduction to integration near the end

Semester Two

  • Calculus II - more calculus of two variables primarily focused on integration techniques and convergence of series
  • Transition to Higher Math - High school gives students the very mistaken impression that math is about solving equations and answering word problems. It would be more accurate to say that it's about proving if and when a solution exists. Actually finding the things is a question for engineers. In other words, math is about developing formal proofs about mathematical structures. This is a huge transition for a lot of students and some degree programs offer a specific class intended to help students make it.

Semester Three

  • Calculus III - multivariate calculus

  • Differential Equations

Semester Four

  • Abstract Algebra 1 or Real Analysis 1 - Abstract algebra is the grown up, way more mature cousin of the algebra you learned in high school. Real Analysis is the 20th century version of calculus.

  • Set Theory and/or Logic

Semester Five

  • Abstract Algebra 2 or Real Analysis 2

  • Linear Algebra

That covers what an aspiring mathematician really needs to know although you could arguably replace the two algebra classes with classes in real analysis.

From here, it really becomes a question of choosing electives. If you already have an interest in computer science, you might consider classes in combinatorics and graph theory. If your interests are more abstract, you could go with number theory a semester or two of topology and add real analysis or abstract algebra if you haven't done both already. If engineering interests you, you could do vector analysis, more differential equations and complex analysis.

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    $\begingroup$ Excellent (and kind) answer. Only thing I would add for the questioner is the idea of doing Dover paperbacks or just full cost textbooks (but often older is better, not just for cost but simplicity). Also to emphasize books and drill books that have the answers as you need the feedback. Not sure MOOCS is really the right path as opposed to self study with books. $\endgroup$
    – guest
    Commented Aug 4, 2018 at 20:42
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    $\begingroup$ @guest I'll second your Dover suggestion. I have quite a few of their books on my bookshelf. They're usually both of good quality and reasonably priced. $\endgroup$
    – G. Allen
    Commented Aug 4, 2018 at 23:13
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    $\begingroup$ +1 for restraint, although this is slanted a bit more towards algebra than I would have used. (I'd probably put a very elementary 1-semester post-calculus real analysis class in place of Abstract Algebra 2, such as this.) All too often I see replies to similar questions (especially in Mathematics Stack Exchange) that include everything but the kitchen sink and thus provide little help to the person. $\endgroup$
    – Dave L Renfro
    Commented Aug 5, 2018 at 14:00
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    $\begingroup$ @SueVanHattum A lot is going to depend on how much time the person/student has to invest. I tried to keep each time point relatively small on the assumption that the person is doing this in their free time which would be presumably be limited. I do agree that some topics, e.g. linear algebra, and a number of the electives could be brought forward if the person has the time available. $\endgroup$
    – G. Allen
    Commented Aug 5, 2018 at 18:53
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    $\begingroup$ Just a remark, but what you call "calculus of two variables" I would call "calculus of one variable" (or "calculus of a single variable" or "single variable calculus"). $\endgroup$
    – J W
    Commented Aug 6, 2018 at 7:49

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