I am currently a college student, computer programming, who has developed an intense passion for mathematics. Following my graduation I wish to pursue a University degree in mathematics, the September of 2014, with plans to continue towards a Master's degree and Ph.D. Before I state my question, I would like to state that I have read many posts here on M.SE, I cannot provide a link to them all due to my "limit", respective of my current reputation level, that being 1.
Some of the titles of the posts that I have read include, but are not limited to:
- How to study math to really understand it and have a healthy lifestyle with free time?
- How to effectively and efficiently learn mathematics
- Is there a good “bridge” between high school math and the more advanced topics?
- How Do You Go About Learning Mathematics?
- How can I learn to “read maths” at a University level?
- Books that every student “needs” to go through
- Just how much proof knowledge is necessary to begin Spivak's Calculus?
- Looking for a beginner to advanced maths series
During the next two years I wish to prepare for my future pursuit by learning, not rote, the required mathematics for the "undergraduate" portion, as well as beyond if possible.
I know of some of the various areas of mathematics such as; Algebra (elementary, linear, multi-linear and abstract), Geometry (discrete, algebraic and differential), Calculus (single/multi-variable), Set Theory, Number Theory, Combinatorics, Graph Theory, Topology, Differential Equations, Logic, Proofs & Proof Writing, Etc...
In essence, I wish to make a "road map" of sorts, I understand that various areas may cross paths, for learning these various areas/topics. I am currently uncertain as to how I would plot out this map. I would like to state that I have not studied linear algebra or any form of calculus/pre-calculus yet; though, I am currently brushing up my "elementary" algebra skills.
I have come across the following, How to become a Pure Mathematician, but ran into a peculiar situation. For instance, I had followed the very first link following the "Stage 1" heading which lead to the following book:
Barnard S. and Child J.M., Higher Algebra
After scanning through some of the first pages within the book I realized the notation was unknown to me. After some further browsing, I realized some of the notation was found in topics on Set Theory. This lead me to infer that Set Theory may be a pre-requisite to understanding the information contained within said book.
I myself, am uncertain of all pre-requisites for the various areas of mathematics; hence, my question. With this "road map" of sorts, I wish to include a collection of books and various other forms of reference for the various topics, on, what I feel is a journey of sorts. Regarding reference materials, I have seen others mention materials such as;
- Apostal - Calculus Vol.1
- Spivak M. - Calculus
- G. Chartand, A.D. Polimeni and P. Zhang - Mathematical Proofs: A Transition To Advanced Mathematics
Using mathematical proofs as an example, I know of the various titles of reference materials and can read reviews and opinions on sites such as Amazon, but I wish to here the opinions of the M.SE community of which books would be recommended as a stepping stone into a topic, onto, a more advanced and thorough explanation, and so on and so forth.
Any information that I have missed or you wish to add would be greatly appreciated. My current studying regimen consists of reading the material, taking copious notes in my own words as well as the authors, asking "why", working through all posed questions, creating questions of my own, then, reviewing the material and organizing said material for input into La/Tex. During my college semesters I devote approximately four hours during the night, weekdays, and 4-6 hours on weekends. During summer periods I adjust the timing of my study periods around my work hours, though if possible, I aim for two 4 hour sessions per day when the time is available.
Again, thank you in advance to all replies, comments and recommendations. I do wish the nature of this question is applicable for M.SE.
A posting that may be of great benefit to the more novice of members here on M.SE would be a mathematical equivalent of the following from StackOverFlow:
We could possibly have a vote of sorts amongst the more senior members of the community as to which materials should be listed and where.