So I'm a Junior in college, swapped to Math to avoid the suicidal thoughts that education classes were beginning to provoke. I took elementary number theory last semester (the class itself went up to order and quadratic residues, I finished the book [primality testing and primitive roots] myself), and enjoyed every minute of it. At the moment, though, I'm stuck in some rather boring courses and would like to continue studying independently. I've messed around with factoring, pisano periods, and basic abstract algebra, but it's been a bit messy trying to work out where I can make progress (spent an entire evening trying to get Galois groups).
Can anyone suggest a good site, or at least a good topic to continue study on?
Solo study  where to go from here?
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Re: Solo study  where to go from here?
This advice is assuming that your number theory class was very proof oriented and that it is your only proof oriented class, you've taken linear algebra or familiar with at least basic properties of matrices (matrix multiplication, addition, and determinants), and you've taken up to multivariable and vector calculus and differential equations. If you haven't taken any of those classes, they may not be a bad place to start, depending upon your definition of "not boring".
Abstract algebra is a great place to start, especially because you've already taken number theory you can almost skip the "preliminaries" chapter that most algebra books contain. The most highly recommended textbooks (from what I've seen) are usually Dummit and Foote's Abstract Algebra and Gallian's Contemporary Abstract Algebra. I own and have gone through both a little bit and they are very good. Dummit and Foote's book covers a lot more material (it is 918 pages of content after all), but some people like reading Gallian's book more. Both books cover Galois theory, granted almost all algebra books do.
Complex variables is another possible route. I don't know much about it, but my number theory professor said with my background (only proof based class I've officially taken being number theory) that I would not have a problem. I've heard it described as "calculus with complex numbers but cooler."
Analysis/advanced calculus may be a viable route, but it may be very difficult to independently study.
Combinatorics and graph theory may be another viable option. Once again, I don't know too much about it.
Abstract algebra is a great place to start, especially because you've already taken number theory you can almost skip the "preliminaries" chapter that most algebra books contain. The most highly recommended textbooks (from what I've seen) are usually Dummit and Foote's Abstract Algebra and Gallian's Contemporary Abstract Algebra. I own and have gone through both a little bit and they are very good. Dummit and Foote's book covers a lot more material (it is 918 pages of content after all), but some people like reading Gallian's book more. Both books cover Galois theory, granted almost all algebra books do.
Complex variables is another possible route. I don't know much about it, but my number theory professor said with my background (only proof based class I've officially taken being number theory) that I would not have a problem. I've heard it described as "calculus with complex numbers but cooler."
Analysis/advanced calculus may be a viable route, but it may be very difficult to independently study.
Combinatorics and graph theory may be another viable option. Once again, I don't know too much about it.
Re: Solo study  where to go from here?
Graph theory is a good one if you want to do some independent research. One thing I've looked at recently is the Firefighter Problem. It is a relatively uncovered problem, so there is a lot of work that can be done with it, but it's basic enough where you can give anyone a good idea of what you are doing without going too in depth into the math behind it.
Re: Solo study  where to go from here?
mdyrud wrote:Graph theory is a good one if you want to do some independent research. One thing I've looked at recently is the Firefighter Problem. It is a relatively uncovered problem, so there is a lot of work that can be done with it, but it's basic enough where you can give anyone a good idea of what you are doing without going too in depth into the math behind it.
Definitely. There are lots of unsolved problems in graph theory and combinatorics that are not necessarily difficult, but just because new problems are produced all of the time. However, I can't say that this is what I had in mind when I suggested this (I basically looked at the math courses at my university that did not have 400 level prerequisites). In light of mdyrud's post, graph theory could very well be a worthwhile subject to study.
Re: Solo study  where to go from here?
To offer some starting points, my introduction to the Firefighter Problem was the paper "Catching the Fire on Grids" by Patricia Fogarty. If you Google it, it is the first result. She covers the basics of the problem in a few different cases. It's really easy to think of a variation you can look at. My research group looked at regular directed triangular graphs, and got some neat results, which we will be presenting tomorrow at a conference. If you don't feel like branching out into something entirely new, there are a few problems in the paper that you could try and find. One of them is with the Halllike Theorem for triangular (she calls them hexagonal) grids. If you can spot it, it means that you understand the topic very well.
Re: Solo study  where to go from here?
I saw complex analysis mentioned above, and thought I'd comment on it. I'm currently in a Theory of Complex Variables class (my first really proof based class. I'm taking it as a freshman while most others are juniors/seniors and have taken real analysis already). I think that it is amazingly interesting. I should warn you that the summary I give my friends of what we do is "Use some theorem named after Cauchy to show that an integral is zero." That's a little bit of an exaggeration, but there is a lot of it. Also, there are many proofs that are very counterintuitive and I'm not sure I would have understood doing self study.
Sorry if this post was slightly incoherent. It's 2 in the morning and I should really be asleep right now.
Sorry if this post was slightly incoherent. It's 2 in the morning and I should really be asleep right now.
cjmcjmcjmcjm wrote:If it can't be done in an 80x24 terminal, it's not worth doing

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Re: Solo study  where to go from here?
I've never met a math person who doesn't like Complex Analysis. "Calculus with complex numbers but cooler" is an apt description. The reason things are so much cooler is that it turns out that complexdifferentiable functions are much more well behaved than realdifferentiable functions. They follow very surprising, elegant patterns.
Re: Solo study  where to go from here?
Thanks for the suggestions  you weren't kidding about graph theory being relatively obscure, the results for 'firefighter problem' on wikipedia included the International Association of Black Professional Firefighters and Rescue Me. To the complex analysis post  I'm actually not a huge fan of Calculus, or at least not of Calculus classes  I tend to be the sort that sets up an integral and goes "Okay, we can evaluate this if we care about the value, so we're done!".
I did eventually find a site on the firefighter problem, and it's quite interesting. What are its applications, if any?
I did eventually find a site on the firefighter problem, and it's quite interesting. What are its applications, if any?
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