Learning higher-mathematics on your own I was hoping someone had an opinion on how to learn higher-mathematics (specific fields that could be of use to me) outside of a classroom setting.
I graduated with an M.S. in Computer science about a decade ago, standard curriculum that I believe is still somewhat taught (Calc, Multivariate Calc, Dif Eq, Linear Algebra, Discrete Math, etc.). I work as a software engineer (they give us a title of Computer Scientist for some reason) for an Contract R&D (gov stuff).
I have found my math skills withering over the years, probably for lack of use of particular fields. For the past couple of years, I am constantly reading research papers (computer science related) for background when developing a new algorithm. What I notice is that I will often get stuck on some mathematical notation or methodology that I am unfamiliar with, when trying to understand the paper. I have been attributing this to my withering math skills, and having to do with fields I never studied in school (or deeply enough).
I try to go back and review what I need to understand the paper, but this leads to a seeming unending link of I need to know this before I can understand that, etc.. With sometimes unsatisfying results.
I was wondering what people have experienced as the best way to learn higher math (advanced calculus, advanced prob and stats, tensor calculus, advanced linear algebra, etc.) as well as refreshing what they were taught in school MANY years ago.
I have tried looking course work on MITs website, to see what graduate math students are being taught. I procure those books and notes, and try to go through the class syllabus myself. But I guess its the lack of rigor, that is failing me the most (school imposed a strict rigor), so I end up just glossing over things when I should be trying to deeply understand thee material (trying to get at the meat of what I am trying to understand, for the task at hand). But over-all this seems ultimately flawed and I only come out with partial understanding.
I want to try to follow a method that would eventually get my math skills on par with a computer science PhD graduate level of understanding of the involved math (say with a focus on computer vision, AI, ML, and computer graphics). What I have been doing over the years is not working for me.
Any suggestions?
 A: I’m an econ major, and I have been self-studying pure mathematics along the way for 3 years, from analysis and algebra years ago, to functional analysis, differential geometry, algebraic topology, and algebraic geometry now. I have never take any math courses beyond calculus, linear algebra and probability that are required for econ major. To assure you, serf-learning is not hard at all. It is the most pleasant thing I enjoy in college. But you do need someone to guide you and provide you the relevant information, point to you what books you should read. After your have more and more mathematical maturity, you become more and more independent and are able to find the resources on your own. My aim here is to share my experience and provide guidance for all who want to self study mathematics. After one have had  a firm mathematical background, he or she can go on to study more relevant fields.
General Advice

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*Math learning is a long term process. There is no shortcut. You need to take serious efforts, learn as much as possible, and build a firm basis of knowledge. Do not expect just picking relevant fields (like optimization), studying it, and ignore others. Otherwise you will likely encounter lots of gaps, then forget what you’ve learned, and then turn to those old materials over and over agin without any new understanding. Time is a must-have investment for success in math learning.

*The deeper you go in pure math, the more you will understand those
elementary concepts in calculus, linear algebra, and other math
tools used by scientists. For example, if you do not study topology,
then it is likely that you will be confused with many convergence
theorems you meet everywhere, and you are likely to forget them. And
without topology, you can’t have a true understanding of calculus.
Once you have had the theoretical depth, it is often a trivial
matter to remember what is really going on.

General Guidance
There are two crucial resources for self learning:

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*You need good books, and you should spend serious time studying them
by yourself;

*Use Internet to ask questions and find lecture notes.

And after absorbing the knowledge on the books for a while, after you have some mathematical maturity, you should gradually become an active learner：
Formulate and ask your own questions, prove theorems listed on the textbooks on your own, using your own notations, or write math blogs to explore your own ideas.
So the path is: Read good books to acquire knowledge and maturity $\Longrightarrow$ At the same time, use Internet to find lecture notes, graphics and videos that can give you less formal and intuitive explanations $\Longrightarrow$ Have your own ideas and your own understandings on what you learned, and at the same time are more skilled at tackling math problems.

Specific Guidance
After Calculus and some basic notion in linear algebra, a self-learner may begin his or her first journey on rigorous mathematics. The first book I recommend is
S. Axler, Linear Algebra Done Right
This is the book that lead me to the fantastic world of pure mathematics. I remember how surprised I was when I saw the beautiful proofs and the powerful abstractions for the first time, and how I enjoy reading it day and night in my fresh year (I even read it the night before an econ course final exam :)  It focus on the theory of linear algebra instead of the determinant approach and calculations of matrices. The ideas and the proofs in the book can quickly increase one’s mathematical maturity. And by the way, linear algebra is used extensively in many areas of mathematics, like differential geometry and functional analysis, so a good understanding of it is very important.
At the same time of reading Axler’s Linear Algebra Done Right, you may also read
T. Apostol,  Mathematical Analysis
Personally I self-studied Rudin’s Principles of Mathematical Analysis in my fresh year, and read the Apostol afterward. But looking back, I would recommend a self-learner to swallow something easier first. The most important lesson I have learned for self-studying is one should not go too fast.
After that, you have abstract algebra, real analysis, ODE, complex analysis and topology waiting for you to learn. You may start learning topology using
J. Munkres, Topology
right after you have finished mathematical analysis. This book is very suitable for self-learning, and it can also greatly increase one’s mathematical maturity.
For abstract algebra I recommend:
M. Artin, Algebra
which is pretty well-known. It also contains materials on linear algebra that are missing in Axler’s book. I have also read Dummit and Foote, Abstract Algebra but I do not recommend it as a first encounter, since although its materials is detailed, it contains far less motivations, which can cause pain on a self-learner.
For complex analysis I recommend
R. Ash, Complex Variables
Self-learners, please do not read the famous Complex Analysis by Ahlfors for a first-time learning in complex analysis. It’s completely useless to you. I also recommend postponing the reading of Rudin’s other two books (real and complex analysis, functional analysis) till much later where you have had enough background and motivations.
A nice book on ODE is
W. Adkins and M. Davidson. Ordinary Differential Equations
For real analysis, I strongly recommend a graduate level book:
J Yeh, Real Analysis: Theory of Measure and Integration
which is super-detailed, zero-gap. A real gem. But warning: while the book is extremely helpful, you should not indulge yourself in the comfortable proofs and go through the material all way long without thinking. Always ask: what theory I have learned, what proof methods I have mastered, and can I remember and reproduce the whole machinery on my own?
After all of these, you may begin exploring more advanced subjects. Here I list several books on each subject
Functional Analysis
E. Kreyszig, Introductory Functional Analysis With Applications
C. Aliprantis and K. Border, Infinite Dimensional Analysis: A Hitchhiker’s Guide
P. Lax, Functional Analysis
J. Conway, A Course in Functional Analysis
R. Megginson, An Introduction to Banach Space Theory
Probability Theory
R. Ash, Probability and Measure Theory
J. Rosenthal, A First Look at Rigorous Probability Theory
S. Resnick, A probability path
D. Williams, Probability with martingales
R. Dudley, Real analysis and probability
K. Chung, A Course in Probability Theory
P. Billingsley, Probability and measure
E. Çinlar, Probability and Stochastics
P. Billingsley, Convergence of Probability Measures
Differential Geometry
J. Lee, Introduction to Smooth Manifolds
M. Spivak, A Comprehensive Introduction to Differential Geometry
Algebraic Topology
K. Jänich, Topology
R. Bott and L. Tu, Differential Forms in Algebraic Topology
Algebraic Geometry
M. Ried, Undergraduate Algebraic Geometry
K. Smith etc, An Invitation to Algebraic Geometry
D. Mumford, The Red Book of Varieties and Schemes
The Stacks Project
Other Subjects, and More Books for References:
G. Hardy and E. Wright, An Introduction to the Theory of Numbers
R. Diestel, Graph Theory (free!)
C. Adams, The Knot Book
L. Evans, Partial Differential Equations
J. Munkres, Analysis on Manifolds
W. Rudin, Real and Complex Analysis
W. Rudin, Functional Analysis
D. Cohn, Measure Theory
E. Weiss and G. Stein, Introduction to Fourier Analysis on Euclidean Space
Princeton Lectures in Analysis by E. Stein and R. Shakarchi
Online resources are also very important
See for example the excellent videos made by Jos Leys:
http://www.josleys.com/galleries.php?catid=13
Other websites:
Virtual Math Museum
Paul Nylander’s Personal Website
Terence Tao’s Blog
The Scientific Graphics Project
Savoir Sans Frontieres
And some excellent notes:
Brad Osgood – The Fourier Transform and its Applications
Paul Garrett – Abstract Algebra
Andreas Gathmann – Algebraic Geometry
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Good luck to all self learners!!!
