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After some initial research on math topics, it seems there are about 4 main streams as follows:

1) calculus -> analysis -> complex variables

2) linear algebra -> abstract algebra -> topology

3) discrete mathematics -> number theory

4) statistics

By "->", I mean "seems to be a good foundation for". So is studying the above 4 "streams" in parallel a good way to self-school in undergrad math?

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In (4), you forgot Multivariate calculus -> Probability -> Statistics. More seriously, the nice thing about math is that you don't get a tree, but a graph that does contain many cycles. So many orderings of topics will yield a feasible plan of study (though finding textbooks that match this order will be trickier). – Thierry Zell Feb 21 '11 at 21:09
My personal view is the the four streams you have identified are, to some extent, artifacts of the way you are learning things, or being taught them, and the fact you have not seen very much of the vast body of mathematical knowledge that has been developed over many years. For a start, professional number theorists know a lot of abstract algebra and ideas from algebraic topology. So I would counsel against having a rigid plan, because you will find during your mathematical development that your views and diagnoses may change – Yemon Choi Feb 21 '11 at 21:11
By the way, I am not sure that this really warrants the tag "mathematics-education", which I always thought was more to do with pedagogy rather than requests about learning – Yemon Choi Feb 21 '11 at 21:12
"analysis -> complex variables" makes exactly as much sense as "analysis -> calculus", for the same reasons. There are approaches to teaching complex variables that are informal in the same way in which first-year calculus is, and they can make sense to the same extent that a first-year calculus course can. – Michael Hardy Feb 21 '11 at 21:42
If you want to teach yourself besides a day job, then following all strands in parallel might a bit much. If it motivates you more you can choose a single of the strands to start with, or follow an approach which is motivated more by content, as those proposed here (especially the number theory strand will lead you to all the things you mentioned above):… – Peter Arndt Feb 22 '11 at 12:59
up vote 5 down vote accepted

If you intend to study on your own the best approach is to follow a structured sequence just like in an ordinary Math degree.... but nevertheless not forgetting that everything is interconnected and prerequisites and applications are highly nonlinear among different subjects (like remarked in some comments above). A more detailed list could be this one (each column to be learned simultaneously within the rest of topics):

(Analysis Undergrad.)

Calculus (one variable) -> Vector Calculus -> Functions of One Complex variable -> Measure Theory

--------------------------------> Ordinary Diff. Eq. -> Partial Diff. Eq. -> Variational Calculus -> Integral Eq.

(Algebra & Discrete Undergrad.)

Linear & Multilinear Algebra -> Group Theory -> Rings & Modules -> Intro to Representation Theory

Combinatorics & Graph Theory

Elementary Number Theory

(Geometry & Topology Undergrad.)

Affine & Euclidean Geometry -> Projective Geometry -> Differential Curves & Surfaces

----------------------------------------> Point Set Topology -> Introduction to Elementary Algebraic Topology

(Probability & Statistics Undergrad.)

Elementary Statistics --> Elementary Probability -> Advanced Statistical Methods

(Analysis Grad.)

Real Analysis -> Functional Analysis -> Complex Analysis (several variables)

Dynamical Systems (and Chaos)

Partial Differential Equations (general theory)

(Algebra Grad.)

Commutative Algebra -> Homological Algebra -> Category Theory

Lie Algebras -> Representation Theory

(Geometry & Topology Grad.)

Smooth Manifolds -> Algebraic Topology

--------------------------> Differential Topology

--------------------------> Algebraic Geometry

--------------------------> Riemannian Geometry -> Complex Geometry -> Symplectic Geometry

I do not know about advanced statistics and probability, and graduate number theory should should deal with analytic number theory and algebraic number theory with class field theory up to diophantine and arithmetic geometry.

May be you could make your own list according to your tastes looking up some course sequences and syllabus offered by good universities.

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wow... that's awesome. I know it's an irritating question for mathematicians to answer because everything is of course interconnected. I appreciate the attempt to come up with some artificially linear paths that a beginner could follow. Thx! – mathmoggy Feb 22 '11 at 0:54
my pleasure...... I am myself a theoretical physicist that has learned pure mathematics by himself, so I have struggled a lot alone. Now I am focusing on algebraic geometry with a mathematician professor in order to apply to graduate school next year. So you would never imagine the amount of good math you can learn by yourself. – Javier Álvarez Feb 22 '11 at 1:03
I would not put combinatorics and graph theory as a prerequisite for elementary number theory. And some kind of analysis course should precede topology, since the whole motivation for most concepts in topology comes from ideas in analysis on the real line (or R^n). – KConrad Feb 22 '11 at 2:41
@KConrad: you are right, I have edited the list to separate discrete math and number theory. About topology: that depends on your previous background. E.g. the Math B.Sc. at the universities of my country has an elementary topology course (point set topology, mostly of metric spaces) given at the same time that analysis in one variable, during the first semester. Even in my Physics degree we studied a chapter in elementary point set topology before diving in the rest of analysis; you just give the motivation and constructions focusing on R^n to later study differential and integral analysis. – Javier Álvarez Feb 22 '11 at 17:00

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