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I'm studying math by myself, and now I'm wondering what is the most appropriate for me to study. I think my interest goes to rather pure mathematics than applied math or math for engineers.

So far I finished studying Thomas' Calculus (12th Edition), and I'm basically done with reading the first half of Baby Rudin. In the fall semester, I'm thinking of studying the second half of that and also starting something else together with that. The followings are some options that I'm considering.

  1. Linear Algebra.
  2. Algebra (These two seem to be very important in mathematics)
  3. Topology (I kinda liked the chapter of the basic topology in Baby Rudin)
  4. Abstract Algebra (I found it very powerful in problem-solving while preparing for Putnam. However, I don't know if I can study it or not.)

    I also need some recommendations for textbooks. I kinda liked Rudin's style, the repetition of Definition, Theorem and proof. Though it is really tough, I like to fill in the gaps and read between the lines. Last but not least, I prefer old books like Rudin to new books.

Any advice and recommendation are welcome :)

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Linear algebra comes first. I don't know what you mean by 2., since you listed abstract algebra already in 4. – Cocopuffs Aug 18 '12 at 16:57
Linear algebra is a must know – Belgi Aug 18 '12 at 16:59
I disagree; at least not anymore important than other area. It is helpful to know about vector spaces, bases and linear transformations. You can learn these in an abstract algebra textbook; however, there are plenty of math you can do if you don't know all the tricks to solve system of equations by matrices or put matrices into rational canonical form. – William Aug 18 '12 at 17:05
@Cocopuffs I see. I thought algebra, abstract algebra, linear algebra are all different. – Tengu Aug 18 '12 at 17:10
I agree with Belgi. One needs to know at least some linear algebra for pretty much everything else in maths (i.e. a familiarity with matrices, linear maps, it can serve as a good first introduction to the concepts of taking the quotient of a space by a subspace, of a group, of how transforming a problem into new coordinates may make it trivial, the usefulness of general definitions such as finding the right definition of a vector space as opposed to restriciting to $\mathbb R^n$). Therefore linear algebra (and analysis) really are a great introduction to maths and mathematical thinking, imo. – Sam Aug 18 '12 at 19:02
up vote 1 down vote accepted

My favorites are:

for linear algebra: Axler's "Linear Algebra Done Right"

here is a link to an MIT class that uses it with problems, etc.

and for algebra the free Harvard video series by Benedict Gross:

This follows Artin's "Algebra" which I think gives a very intuitive presentation which is yet rigorous. But for material after that I would switch to "Dummit and Foote"

These are all highly regarded and are widely used. They can give you a good basis for further study.

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Thank you for your recommendations :) I did not know either of the links. but both seems to be very helpful. How long do you think it will take to finish Artin's Algebra and Axler's Linear Algebra Done Right? I can spend 3-4 whole days per week to study math, and I think I will study analysis, algebra and linear algebra. – Tengu Aug 19 '12 at 3:51
Both links were for one semester courses. But in that context, they are embedded with 3 -4 other classes. Just focusing on one at a time, I would say around three months. One thing to consider is you want to really absorb and retain the material, so it might be better to pace yourself accordingly. BTW I would do the linear algebra first. – Andrew Aug 19 '12 at 11:43
I'm thinking of studying 2-4 subjects in math in fall semester and pacing myself accordingly. I will probably study Analysis, Topology, and Linear Algebra for fall semester. – Tengu Aug 19 '12 at 13:59
Of course, it's up to you. And math is so thrilling, that there is a desire to have it all. But, for example,I've been thinking about a problem all day now. And even though I know the answer, I am still luxuriating in looking into the matter further for a more elegant solution. It's hard to do that with three courses at one time. Rather than predetermine your program, why not try one and see how it goes. Maybe linear algebra since it is somewhat different than analysis. – Andrew Aug 19 '12 at 17:40

Having read half of Rudin, you should have the mathematical maturity to begin learning algebra or topology. These two topics really don't require any knowledge of analysis, but some comfort in reading proofs is helpful.

I also really like the topology chapter from Rudin. I believe Rudin only does metric space topology. If you want to learn general topology, you should read Topology by Munkres. It is very well-written and clear. (Possibly my favorite math textbook!) Further along you can go into Algebraic Topology, manifolds, and differential topology. Algebraic topology studies various ways to distinguish between various shapes like Torus, Spheres, n-genus surfaces, projective space, etc usually up to homotopy equivalence. Hatcher has a notable book in Algebraic Topology. There are even some people who believe that you can dive right into Algebraic Topology with minimal general topology background.

Algebra studies generalization of familiar algebraic structures like groups, rings, modules, fields, etc. Abstract Algebra by Dummit and Foote is a very comprehensive book.

If you want to continue in analysis, you may eventually want to study functional analysis. You will need to know what are linear transformations and vector spaces are so some linear algebra is needed. General topology is quite important for functional analysis. When you study things like the weak and weak * topology, some of your intuition about limit points and closures fall apart. Some knowledge of general topology such as the countability axioms may help here.

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"Rudin only does metric space topology"...alternatively, learn what a net is and then prove most of the metric space theorems with nets in place of sequences! :) – Jon Bannon Aug 18 '12 at 17:41
@William Thank you for your recommendations! I checked $Topology$ by Munkres and $Abstract Algebra$ by Dummit and Foote. It seems that both are more than 500 pages, so I'm wondering how long students usually spend on each book. (I think I can spend 3 or 4 whole days per week to study math.) – Tengu Aug 19 '12 at 3:46
@Tengu The first half of Topology can be done in about a quarter (which is about 10 weeks). The second half is algebraic topology, but Hatcher is more complete for that. My entire year long (3 quarter) algebra class could not finish all of Dummit and Foote. Don't forget to do some of the exercises. – William Aug 19 '12 at 4:02
@William I see. Then for topology, I will try to finish $Topology$ first around in half a year and decide what to read next depending on my interest. Thank you for all your advice! – Tengu Aug 19 '12 at 4:20

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