What is the best approach for self-learning advanced math?

[df003]

Hello all,

I'm beginning a self-administered education in math and physics as substitution for a college curriculum, and am looking for some advice on what my general course of study should be.

My goal is to learn advanced physics and maths -- I'm most interested in topology and differential geometry, especially as these relate to physics.

I have bookmarked probably 50 books on Amazon (fortunately I have the money to spend on whatever books I need), sifted through all the great names in Mathematics (Rudin, Spivak, Munkres, Coxeter, et al) and their works, but I'm slightly at a loss for how I should plan to climb this immense mountain.

Right now I have well-absorbed high school knowledge of algebra, geometry and trigonometry, and pre-calculus (geometry was always my favorite... the teachers that could relate more advanced math geometrically made it stick for me).

I know that, understandably, Calculus is the default starting point, so I take as a given that I should get very acquainted with Calculus I-II material and perhaps review advanced algebra before starting anything else.

What's next? What 'plateaus' or 'levels' are there that are necessary to be completed before moving on to the higher maths? Here's some areas I'm interested in:

Linear Algebra
Abstract Algebra
Differential and Riemannian Geometry
Topology / Algebraic Topology
Real and Complex Analysis

...I'm sure there are many in between those, but they're slipping my mind. Again, my 'goal' (but I know there is never an end) is to be fluent in the math of topology and advanced geometry, especially of higher dimensions, to augment my physics studies with.

I am expecting a long and somewhat tedious path, but I'm ready. Any and all advice is appreciated. :)

P.S. I've already found all the most popular online course materials for college-level math classes. I'm more interested in hearing opinions about how best to approach learning the various subjects of Mathematics (i.e. what order), from a purely mathematical perspective.

 

[Stephen Tashi]

My advice is not to spend your time planning a long term curriculum of study and bookmarking books etc. Spending time imagining that you will progress through a pile of books is like planning what you will do when you win the lottery. It's possible you will master them, but unlikely.

If your ambition is study pure math and be able to read and write proofs, study mathematical logic a little. There's no need to study it for years. Study it until you understand elementary things like De Morgan's law, quantifiers, how to negate statements involving quantifiers. You can study logic from a book on logic written for students of philosophy that doesn't have much math in it. Don't be disappointed if introductory math courses don't develop mathematics with logical precision. They can't assume that students understand logic.

 

[micromass]

You can tackle linear and abstract algebra right now. That is, without seeing calculus. Be sure to pick a book that is easy enough for a first acquaintance, though...

For differential geometry, you need multivariable calculus. So calculus I-III is necessary. A bit of real analysis can't hurt too.
For Real analysis, you will need calculus I-II and you will need to be acquainted with proofs.
For topology, you will need to be acquainted with real analysis, since topology is a generalization of that.

So, some things you can do right now are calculus, abstract algebra and linear algebra. You can also start by reading a proof book.

 

[verty]

Hi. I think it is very difficult to learn something without having fun. So my advice would be to focus on the interesting stuff first and avoid doing things "the right way". The right way is just too difficult for 99% of us. So use any shortcuts you can.

One particular bit of advice: look at solutions to problems when you can't immediately find a strategy to solve them. It saves a lot of time.

 

[Fredrik]

Are you doing this as a hobby, or are you trying to give yourself an education that you hope you will be able to use for something else?

A few tips:

1. You need very little abstract algebra for the other topics. So little that you can consider not studying it at all until you actually come across a proof in a book that requires that you know some abstract algebra.

2. In spite of what I said in 1, it's worth the effort to take a day or two pretty soon, to study the definitions of terms like "group", "ring" and "isomorphism" in a book on abstract algebra.

3. Linear algebra is much more important than abstract algebra.

4. Topology is hard as #¤%&.

5. You don't need a lot of topology to study differential geometry. If you can live with not knowing how some of the terms that are mentioned in the definition of "smooth manifold" are defined, you can get by on almost nothing. (Basically you just need to know what "continuous" means).

6. You need to be very good at topology before you can even begin to study functional analysis. (It's not on your list, but I figured I'd mention it anyway). (That's general/"point set" topology, not algebraic).

7. It's too early to decide to study algebraic topology.

 

[df003]

Thank you for the advice, everybody.

Are you doing this as a hobby, or are you trying to give yourself an education that you hope you will be able to use for something else?

A few tips:

1. You need very little abstract algebra for the other topics. So little that you can consider not studying it at all until you actually come across a proof in a book that requires that you know some abstract algebra.

2. In spite of what I said in 1, it's worth the effort to take a day or two pretty soon, to study the definitions of terms like "group", "ring" and "isomorphism" in a book on abstract algebra.

3. Linear algebra is much more important than abstract algebra.

4. Topology is hard as #¤%&.

5. You don't need a lot of topology to study differential geometry. If you can live with not knowing how some of the terms that are mentioned in the definition of "smooth manifold" are defined, you can get by on almost nothing. (Basically you just need to know what "continuous" means).

6. You need to be very good at topology before you can even begin to study functional analysis. (It's not on your list, but I figured I'd mention it anyway). (That's general/"point set" topology, not algebraic).

7. It's too early to decide to study algebraic topology.

I wouldn't call it a hobby, my goal is to be able to fluently read math and physics texts and use the information in my own work. At the very least, my goal is to have a firm understanding of advanced and theoretical physics, which I figure would be improved drastically with a good background in math.

I'm actually already familiar with those terms, as well as most of the basics of topology... it's that interesting to me. :tongue:

 

[micromass]

6. You need to be very good at topology before you can even begin to study functional analysis. (It's not on your list, but I figured I'd mention it anyway). (That's general/"point set" topology, not algebraic).

This is very true. But there are some "gentle" introductions available that require no topology. For example, the Kreyszig text only requires no knowledge of topology.
If you're serious about it though, then topology is a must!

 

[df003]

This is very true. But there are some "gentle" introductions available that require no topology. For example, the Kreyszig text only requires no knowledge of topology.
If you're serious about it though, then topology is a must!

Just to be clear, I wasn't expecting to learn topology and functional analysis right off the bat -- I think I'm definitely most interested in those areas, but I know I couldn't possibly delve into them without a good amount of preliminary study. It is something to aim for.

 

[chiro]

To me, the hardest part about math is the mathematical maturity.

Learning advanced math for the first time is in some ways like learning a new language. The symbols on the page are completely foreign, and there is no intuition of what those symbols mean. You struggle to put together a coherent mathematical sentence (or paragraph, page, etc) but slowly, you start to make sense of the language in the same way that at some point speaking in a foreign language becomes more natural.

Once you have enough "mathematical maturity", it will become a lot easier to grasp new concepts and navigate through them in your head. Things will just make sense and you will wonder how you couldn't have grasped something so simple.

In terms of what I think is important, I think its important to understand common structures, how they can be decomposed or transformed to something else, and more importantly an intuitive idea of what they represent.

The different decompositions will give you insight to how you should decode the intuitive structure of some representation. For example any orthogonal decomposition gives you a kind of "geometric" intuition for some model with respect to some basis. This might be a simple geometric object in standard Cartesian geometry, it might be in a Hilbert space type scenario like a Fourier decomposition or some wavelet decomposition, it might be something like a prime decomposition, or it could some set based decomposition by representing something in terms of its disjoint set representation.

Good luck with your learning, and if you have trouble, this forum is a great place to get help from the more experienced members.