A path to more advanced math topics.

Question

I am from Hong Kong and have just finished all the public exams. In my high school life, I've learned some topics on mathematics which are fundamental, yet I think are not in-depth enough.

The topics I've studied:

• Mathematical induction
• Binomial Theorem
• Trigonometry-- Basically CPD Angle Formula, Sum-to-angle/Angle-to-Sum Conversions
• Calculus-- First Principle/ Differentiation (up to 2 variables)/ Integration (1 variables)(by part)(substitution)
• Matrices and Determinants (up to 3x3)/ Systems of Linear Equations (up to 3 variables)
• 2-D/3-D Vectors/ Manipulation of Scalar products & Vector Products

That's all.

Question

1) Which of these topics should be studied in broader field? Is there any advanced subtopics that I am strongly recommended to study?

2) I know there're many fields in math that should also be explored, say Set Theory, Topology, Number Theory, etc. Is there any priority in studying these fields?

P.S. I'm quite intrigued by ALGEBRA.

Answer 1

The following is just my opinion, so take it with a grain of salt. However, I will warn you not to neglect the fundamental subjects you may not like so much. For example, I strongly prefer algebra to analysis, and consequently didn't internalize much analysis. It turns out, I'm gravitating towards subjects now that require a stronger analysis background than I have, and paying the price for not working as hard as I should have.

• Linear algebra is a must, and is my recommendation as one of the first things you study. It opens the doors to multivariable calculus, and it can serve as your first introduction into many important topics in algebra/group theory: Invertible matrices form a group, changes of basis transformations are conjugation, etc. That way, when you see abstract groups/rings for the first time, you'll have more objects that you can relate to and say, "Hey, that too was a group/ring, all along!". Further, linear algebra shows up just about everywhere. It's never too early to get started, and it has very elementary underpinnings (linear equations).

• Number theory has a special relationship with much of (abstract) algebra. A lot of reasoning about finite algebraic structures reduces to number-theoretic considerations, like divisibility. You'll meet things like modular arithmetic ("clock arithmetic"), which are examples of groups. In a sense, basic algebra and basic number theory go hand-in-hand. While basic number theoretic results are used to study groups, you don't need a course in number theory beforehand, and it will help you in basic number theory. Conversely, understanding number theory before learning abstract algebra will make certain parts of algebra much easier to learn.

• For most studies of mathematics, a rudimentary understanding of set theory is sufficient, and I certainly wouldn't read a whole book devoted to it. My university's "Fundamentals of Advanced Mathematics" was essentially a course in mathematical reasoning (understanding the kinds of proofs, and practice with rigorous proofs), and often the basics of set theory are covered in this setting. You can usually learn more advanced topics on a "need to know basis", and pick them up as necessary.

• Once you know the basics of set theory, general ("point-set") topology is within your grasp. However, it's generally viewed as a fairly advanced topic of study, despite having basic prerequisites (motivated heavily by analysis). You can look into it now, but I wouldn't worry too much. Topology is an extremely diverse field, with subtopics including differential topology and algebraic topology.

Answer 2

At this point, I would say that there are two goals:

1) Pick up the basic background that any mathematician would be expected to have. Exactly what that background would entail is undefined (and controversial), but you should probably have some familiarity with linear algebra, real analysis, complex analysis, topology, group and ring theory, number theory, combinatorics, and set theory. This is a long-term goal, of course, and I'm probably leaving some areas out.

2) Find what parts of math interest you the most. Math is a huge subject; if you're just finishing up high school, you've probably only scratched the surface. Take a bunch of different classes, read a bunch of different books and papers, and figure out what you're thrilled about, what you like, and what you're not particularly interested in. Any area of math is going to have prerequisites, but learn what you can now and learn the rest when you can.

That having been said, for the immediate future, I'd recommend you take a look at an introductory group theory book. I've never run across a particularly good book along those lines, but Artin's "Algebra" is OK (not great, but OK).

Answer 3

I will recommend linear Algebra by-G S Strong.