6
$\begingroup$

I like problem solving. In fact, that is the reason I wanted to study mathematics; This is a field where I could learn the underlying logic of the results rather than just learning ideas even the foremost experts did not necessarily understand.

The issue is that mathematics for the last few semesters have been becoming increasingly abstract, and the distance between what I am learning and actual problem solving seems larger than ever before.

However, I have also read about how the fields I am struggling with are being used in puzzles (a la puzzles Martin Gardner would write about). For instance, Numberphile has three videos on a problem called "Pebbling a Chessboard" in whih the mathematician says that functional analysis is used to prove a result. Also, Euler's solution to the Seven Bridges of Königsberg problem is the first result of topology.

Are there any books through which one can learn topology through problem solving?

$\endgroup$
  • $\begingroup$ Yeah, that's pretty broad. Maybe you can tell us a bit more about your mathematics background. You say, for instance, that mathematics for the last few semesters has become more abstract. What was the last course that you found straightforward, and which courses have you found too abstract? $\endgroup$ – Brian Tung Aug 13 '15 at 18:26
  • $\begingroup$ Well, I had a course on real analysis, and it covered some measure theory. That was very abstract. Also, abstract algebra was too abstract as well (especially group actions). Even linear algebra can get too abstract (e.g. when I am learning about decompositions without learning why I need to know them). But, yeah, I will split this into two questions. This one on topology, and the second on functional analysis. $\endgroup$ – Avatrin Aug 13 '15 at 18:39
  • $\begingroup$ Hmm...Maybe others will have some suggestions, but sometimes the problem is that the more abstract the mathematics, the more unlikely it is to admit of a simple-to-state, tangible problem. I will give it some thought. $\endgroup$ – Brian Tung Aug 13 '15 at 18:42
  • $\begingroup$ Group theory has applications to various physical puzzles. For example, an $N\times N\times N$ Rubik's cube is essentially a subgroup of $S_{6N^2}$ and can be solved that way (and in fact that's what I do); group theory also tells you whether a cube is solvable after being disassembled and randomly reassembled. I don't know of any books about such applications, however; I think I've seen some things written on the topic, but only in periodicals. $\endgroup$ – David K Aug 13 '15 at 22:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.