Preparing for University Mathematics

Question

I will enroll to university next Summer in a rigorous B.Math course. I am currently in high school and I am more than comfortable with most of high school mathematics . I am looking for books that prepares one for a more rigorous course than the high school one .

Books I have completed so far

• Calculus -> Calculus by Spivak
• Algebra -> Algebra by Gelfand, Problem Solving Strategies by Arthur Engel , Polynomials by Barbeau , Inequalities by Venkatchala , Functional Equations by Venkatchala , Complex Numbers by Titu Andreescu
• Combinatorics -> Principles and Techniques in Combinatorics by Chen & Koh
• Number Theory -> Elementary Number Theory by David Burton
• Geometry -> Trigonometry by SL Loney , Co-ordinate Geometry by S.L. Loney , Euclidean Geometry by Birkhoff & Beatley
• Linear Algebra -> Linear Algebra by Titu Andreescu

I want to expand and build up on this existing knowledge for further courses and hopefully a research career in pure mathematics , please recommend me some books to work through both theory and problem books .

Answer 1

If you've mastered those books (particularly the proofs in Spivak) you are more than well prepared already for rigorous university mathematics. Pick one of those subjects that you particularly like and study something a little more advanced. You could study some abstract algebra.

Edit in response to comment.

Caveat first: I haven't taught abstract algebra or real analysis for years, so my thoughts may well be out of date.

I think reading Tao is a good idea. I wish it had been around years ago.

Herstein's algebra is an ancient ( book with the kind of rigor you seem to like. Fraleigh and Dummit-and-Foote seem to be the favorites nowadays. You can begin thinking about them here: How does Dummit and Foote's abstract algebra text compare to others? Herstein or Herstein?

Answer 2

Considering the fact that I am also a high schooler, I think I can give you some suggestions.

(1) Since you have done linear algebra, I would highly recommend studying vector/multivariable calculus. You will see how the results of one-dimensional calculus generalize in a powerful and intuitive way to higher dimensions. Specifically, I must suggest Susan Colley's "Vector Calculus". It bases the subject using the techniques of linear algebra which is the proper way to teach the subject. This treatment means that the formulae are more concise and are easier to generalize. Colley also emphasizes the geometric intuition behind the formulae she presents which is crucial in my opinion.

(2) Once you have done vector calculus, I would move on to differential geometry. If you use Colley's book, you will actually get a nice introduction to the subject in the last chapter of her book. In the said chapter, Colley introduces the notion of exterior calculus (wedge products, differential forms, etc.) in the context of arbitrary manifolds embedded in $\mathbb{R^n}$. If this piques your interest, I would self-study out of Barrett O'Neill's "Elementary Differential Geometry". The book is fairly manageable and his presentation of the basic notions of the subject (such as frame fields, connection forms, and the Gauss-Bonnet theorem to mention a few) strikes a good balance between rigor and intuition. However, I will warn you that his notation bogged me down somewhat so be on the lookout for that.

(3) At this point, you have some options. You could learn about tensors and their calculus (which generalizes the notions of vector calculus). Tensors are widely used in differential geometry (such as in the study of Riemannian manifolds which are basically manifolds equipped with a metric tensor). For this, I would recommend "Vectors, Tensors, and the Basic equations of Fluid Mechanics" by Rutherford Aris which introduces tensors in the context of fluid mechanics. For me at least, grounding abstract topics in something physical is helpful. This is what I personally did. I am about a quarter of the way through the book and I really am enjoying it.

On the other hand, you could learn some topology (specifically point-set topology) before continuing the study of differential geometry. I have a book called "Lecture Notes on Elementary Topology and Geometry" by I. M Singer and John A. Thorpe which starts off with point-set topology and moves on to a more rigorous study of differential geometry. I have not gone through it yet but it seems like a manageable and well-written textbook.

I really could not say much more honestly, but hopefully, some of my humble recommendations are helpful if you are a geometric thinker and are looking for a career in pure mathematics.

Answer 3

I should say first that learning university mathematics is quite different from learning contest math like preparing for the IMO. Your request for both theory books and problem books is perhaps slightly off the mark, because these are often combined other than exceptional circumstances.

Since you've completed Spivak's Calculus, I strongly recommend learning multivariate calculus first. One way is through the lecture series on MIT open courseware, although of course you can pick your own book as well. Once you're done with that, it would probably be a good idea to learn the basic topics which will come up everywhere at higher levels: these are typically real and complex analysis, topology, elementary number theory and (linear and abstract) algebra. Of course, if you have a specialised interest in e.g. Graph Theory, you can go for that as well.

In any case, if you've already completed the books you say you have, then I agree with Ethan Bolker that you shouldn't be worried about your mathematical "maturity" or ability to deal with rigour. Learning the topics I suggested above is really about teaching you the foundational things that will occur elsewhere in the future, not about learning rigour (although you will pick that up too, if you think you're lacking in it in some ways).

Answer 4

At that point, I would study Rudin's Real and Complex Analysis from cover to cover.