Learning Topology for Physicists: A Realistic Timeframe?

  • Thread starter muppet
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If so, I definitely recommend Sharpe's book too, as it has a lot of exercises and it's a decent overview of the subject.
  • #1
I've decided that I seriously need to get some topology under my belt, and I've started to tackle the book by Munkres systematically. I only really hope to cover chapters 1-4 and 9 (executive summary for those not famililar with the book: foundational math/ set theory, topological spaces and continuous maps, connectedness and compactness, countability and separation axioms, and a brief intro to algebraic topology- path homotopies, the fundamental group and covering spaces).

I'm trying to work my way through all the corresponding exercises- something I never did enough of as an undergraduate, and I think particularly necessary as I'm studying alone. I'm slightly concerned however that material that looks like it should be easy is taking me a while; the exercises at the end of section 3 have taken me pretty much the whole afternoon. Can anyone suggest a realistic timeframe for me to attempt to master this material? I both hold unrealistic expectations of how long something will take me and dawdle/lose focus/ daydream in equal measure, so I'm really uncertain as to how quickly I should be able to progress through the text.

I'm particularly concerned about the timescale as I'm a theoretical physics PhD student in the UK. This means that a) I only have 3 and a half years to produce a thesis and any study I undertake, so my research should really take priority, and b)my real motivation is not to study topology for its own sake, but to be able to systematically tackle the mathematical literature on differential geometry- in particular, the books by John M Lee on smooth and Riemannian manifolds, and the text by Sharpe with a view to getting a good grasp on gauge theory. (I'd also like to know a hell of a lot more about physics than I presently do :rolleyes:).

As an aside, I do however appreciate that topology can be more directly useful as well, so I'd like to get a reasonable grasp of it as a subject rather than just be able to quote metrisation theorems etc. So any suggestions as to other topics in topology (general or algebraic) that are useful (or even just particularly interesting!) would also be gratefully received.

Thanks in advance.
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  • #2
Well, if you find that reading and doing the exercises are taking you too much time (btw, if I recall correctly this textbook is composed of a lecture series of Munkres which took him two semesters, so don't worry if it takes you too much time) to do, there are always solutions which are spreaded in the web (it's not pedagogically sound, but learning fast such a material isn't sound as well, but you have other priorities).
  • #3

I'm a Math self-learner and some Set Theory problems take a whole week if not longer as I can only devote like 5 hours a week.

I don't rely on the web because I want to learn by my own. Another point, it's really hard to find solutions for many problems on the web.

I'm going through Set Theory and Logic by Robert Stoll ...

  • #4
Obviously it's not easy to find solutions to every exercise there is, so you also look for similar exercises and try to make a connection between you exercise and of the solution (that's part of learning math btw).

Sometime because in some subject there's so much literature (as in set theory), if you keep searching you might find something similar to your problem.

But don't forget to do it after you tried your best.
  • #5
Thanx mate,

I do look at other exercises as you said and it really helps.

It has been like 3 years since I started studying Set Theory, of course three years because I have a Job and barely can devote some time for it. I guess the discontinued learning hours are my biggest problem as I have to recall where I was last time I worked on this problem or that.

Currently working on the Axiom of Choice... Woooo, it makes my brain much like scrambled eggs :P
  • #6
Last summer, I also self-studied topology from Munkres. Instead of doing every exercise, which seems excessive, I followed the syllabus and homework assignments at this page:


It took me about 2 weeks doing it for about 12 hours per day to go though all of the homework, doing all of the problems. So, quite a significant investment of time, but I feel like cramming it in all day every day helped me learn it much faster. It worked out pretty well, since I was able to skip undergraduate topology and enroll in the graduate level course during the fall.
  • #7
It looks like I can't edit anymore, but I didn't notice that you mentioned in your post that you wanted to study Lee's books on smooth manifolds. That is precisely the textbook I used in the class I took after studying from Munkres, so this program seemed like good preparation.

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