Advice for study plan.

Question

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 :)

Answer 1

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.

Answer 2

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.

http://math.mit.edu/~trasched/18.700.f11/index.html

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

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

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.