I want to master the field of Pure Mathematics by myself?

Question

I know that I am smart enough to get a Ph.D in pure mathematics, but since I am 16 and I have a different career path in mind, I can't do Pure Maths at university. I want to, however, in my own time, 'master' pure mathematics. To get an idea of where I'm at, I'm still quite the mathematics amateur - I've learned up to Complex numbers (locus, vector analysis, arithmetic, de moivre's theorem, etc.), Conic sections, Differential and Integral calculus (including fourier transform, which I'm learning about now) and volumes. I want to go even further, but I can't find a source of information that tells me what to learn sequentially and where to learn it. I don't know which textbooks to get, where to learn the info required to do the questions in the textbook, etc. I just don't know haha. I know this sounds nerdy, but I feel like my school and limited resources are holding me back, and I have a burning passion for mathematics and a thirst to 'master' it. Can anyone give me tips or websites where I can systematically master maths? Thank you so much in advance!

Answer 1

The blog referenced below has worked this all out for you.



If I were you, I'd find the nearest university and check out the PS section of their library. You don't really understand what you are getting into, but, if you are as good as you think you are, it shouldn't matter.

Answer 2

A standard undergraduate math curriculum might be up your alley. A rough sequence and a few texts I'm familiar follows.



Differential/Integral/Multivariable calc

--I'm teaching from Stewart's calc text at the moment, which is fine. There are tons of books at this level. I have no idea what I used as an undergrad.



Linear algebra/Differential equations

--See first link for discussion.



Intro real analysis

--This is often an "intro to proofs" class. I used Rudin's Principles of Mathematical Analysis. There's only a few standard options I'm aware of. See second link for discussion.



Intro abstract algebra

--I used Dummit and Foote. It's big and has topics through early graduate level.



From there math majors seem to break up into semi-random required and elective courses. I remember taking courses in Differential Geometry (do Carmo's book), Algebraic Geometry (Reid's book), a self-study of Galois theory (Garling's book), some general point-set topology (in-college lecture notes)... oh, some more analysis and measure theory (second half of Rudin's Principles)... and I forget what else. Other popular choices would be complex analysis (tons of books; see third link), Fourier series and the like (Rudin's Real and Complex Analysis, I suppose), combinatorics/graph theory/elementary number theory (no recommendations), PDE's, dynamical systems... many options.



The calculus/intro linear algebra courses tend to be easy and taken by many non-math majors. The real analysis/abstract algebra tend to be your first serious math course. They also lay a foundations for other electives; the order of electives doesn't often matter much.





All that said, this is just the tip of the iceberg. As I've gotten further in math, the amount I've already learned seems paradoxically smaller and smaller compared to what's out there. You will never "master the field of Pure Mathematics"; there is literally not enough time in a human lifetime, since it's already ginormous and it's only growing. Your goal should not be to "reach the finish line", it should be to enjoy the trip.



P.S. The people at Math Stack Exchange or MathOverflow almost always know what they're talking about if you want information of this type. (All my links are from one or the other.)

Answer 3

Mathematics is very big. No one (certainly no one since Gauss) has mastered all of mathematics. This is not meant to discourage you. But you need to be realistic about it. Browse this site http://www.math-atlas.org/ to get some idea of the size of mathematics. I suggest clicking on the last item, Layman´s Guide to the Math Subject Areas. Read that.



The other weird thing about mathematics is that it is all tied together, so that learning it systematically in a linear way is not possible. What you have to do is something like this: learn a lot about subjects A, B, and C and then go back and learn more about A. There is a thing called "mathematical maturity" which you only get bit by bit and so you have to redo things after you are ready, after struggling with them before you were ready. Bit by bit, iteratively, you learn more and more.



There is lots of stuff available on the web. I also recommend you explore the math area of the nearest university library.



If you insist on a list of math books, I recommend this list: http://www.ocf.berkeley.edu/~abhishek/chicmath.htm



Good luck!

Answer 4

You are 16 and you claim you are smart enough to get a PhD in mathematics? If only you knew. Do you even have an undergraduate degree? Heed my advice. Five years from now, you are going to be laughing at yourself for writing this. Get a couple of degrees in math before you consider a doctorate.

Answer 5

Agree with the other answer. MIT's opencourseware is a fantastic tool.



These two might be good next courses for you.



Linear Algebra

http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/index.htm



Multivariable calculus

http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/index.htm

Answer 6

MIT website