Self Teaching Analysis, Topology and Differential Geometry

Question

I posted some questions similar to this one not long ago but I think I phrased them wrong and as such got valuable input but not really an answer to my questions. If reposting a similar question like this is against the rules of this forum please tell me! I am still new to this site and eager to learn.

Basically I am in electrical engineering (3rd year) but I think I should have done an undergrad in math. I have a limited interest in application and am really much more interested in class when we do rigorous math. At this point I am considering either doing a masters degree in math or going into Control Theory (not control systems, but rather the mathematical theories behind nonlinear control which I guess is more applied math than engineering).

What I want to do now is go through textbooks on my own to make up for the butchery of math that happened in my engineering classes. The great thing is that I will have a year-long internship starting in May during which I will have time to dedicate to this. I want to emphasize that although there is a strong chance I will go into Control Theory, I want to cover rigorous math and detach myself almost entirely from engineering.

My real question/concern is what textbooks to use for each subject and what order to do it all in. Here is my current plan, I would very much like your input:

1. Real Analysis by Chapman Pugh (I'm already quite deep into it and loving it)
2. Topology by Munkres (Part I: General Topology)
3. Abstract Algebra by Dummit and Foote (Group Theory)
4. Topology by Munkres (Part II: Algebraic Topology)
5. Smooth Manifolds by John M. Lee

And then perhaps more of the Abstract Algebra textbook and/or an intro to Dynamical Systems, Chaos and Fractals.

Please note I do have some experience in proofs despite engineering. Some of it is due to computer science courses behind quite rigorous and the rest is self practice. For example I am finding Chapman Pugh quite accessible.

Answer 1

Is the point of this project to learn to think rigorously, or to learn some important and exciting math? You might say it is both, in which case I would recommend focusing on each project separately, to some extent.

Bottom line: you don't need to read all these books to learn to think rigorously, and understand proofs. One book on any topic is probably enough for that, as long as you read closely. (My favorite is Spivak's Calculus. Also chapter 0 of Munkres is awesome for this purpose! Perhaps Pugh is serving this function for you.)

After you do that for a while, start on the project of learning some exciting math. The point is that for this goal, your methods of reading will be pretty different. Before, when you were focusing on learning to think rigorously, you probably wanted to take a single text, and read it quite closely, understanding every sentence, and why it is written the way it is. Now, you want to focus on the material, not the presentation. (If you ever find that your ability to read proofs is lacking along the way, you can return to that goal for a while, as you see fit.)

Frankly, you'd be a masochist to read all the sections you mentioned straight through--they are encyclopedic and dry, and you'll lose the forest for the trees. Math should be a conversation between your curiosity and the many sources out there--and what kind of a conversation is it if you're just listening to the same person over and over for 5 chapters at a time? No author is so good that you want to listen to only them. I promise you, you will get less done if you try to attempt it this way, because you will get bored--perhaps without realizing you're getting bored.

Suppose you want to learn about what a manifold is. There are a ton of online notes from various classes that cover this material, and a ton of texts that you can get from the library as well. Go check out a few that look like they treat it in an interesting way. Bring them all home, and scan through them a bit, until you come to something that you'd consider an interesting result. Try to express this result in a single sentence. (For instance, "sometimes a continuous bijection can fail to be a homeomorphism, but if the domain is compact, this problem is avoided.") Then see if you can find how the various sources treat this, and find the one that helps you understand it the best. The good thing is that in this process, you will naturally be led to research other things that come into play ("What is a compact space?" "Why are homeomorphisms important?" "What is this whole business about open and closed sets anyway?" "Why on earth would this be the way that we define a topology?")

Write some proofs in your own words, so you can remember them as you move on. Then move on to the next thing that seems interesting to you.

If you feel like you don't know yet what should seem "interesting", you can pick theorems that the books treat as important. But always better if it's something that strikes your curiosity.

One objection to working this way might be that you feel you're getting an incomplete view of the subjects, because you're only focusing on one thing at a time, whereas if you read the group theory section of Dummit and Foote straight through, you'd get a complete picture. But the truth is, no one source is going to give you a complete picture. Not just because they don't cover everything, but because they cover too much. It's only "complete" if it exists as a coherent and meaningful picture in your mind, and I've yet to find a text that can achieve this on its own.

Afterword: A helpful trick in searching for online notes is to do a Google search like smooth manifolds filetype:pdf site:edu. To help you know which books to use from the library, ask various people, and above all pick the ones that seem interesting and friendly to you.

Answer 2

I think that Eric Auld's answer is fantastic, and I agree with most of what he said. I'm adding an answer just to throw in my own two cents.

Learning some algebra is good, but if you want to do control theory you might be better off initially spending less time on that and more time on learning some dynamics/ODEs. For a first dynamics introduction, "Nonlinear Dynamics and Chaos" by Strogatz is very good and a pleasure to read. For ODEs, I'm a fan of "Differential Equations, Dynamical Systems, and Linear Algebra" by Hirsch/Smale (the 1974 edition), and also the ODE text by V.I. Arnold.

Chapters 1-3 of Munkres is a good first exposure to topology but I think that Lee's topological manifolds book is especially useful background for differential topology/geometry. I also liked the algebraic topology introduction in Lee's book a little better than the one in Munkres.

Lee's smooth manifolds book is now my favorite on the subject, but I read other books first. You might also like the books by Boothby and Guillemin/Pollack.

I was in your (almost) exact same shoes 5 years ago: third year electrical engineering student, unhappy with rigor in engineering classes, realized I should have been a math major, reading mathematics books, etc. I am now an electrical engineering PhD student in control theory/dynamics. Feel free to PM me if you ever want to talk more!