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I don't find the math taught at my school too challenging so out of personal interest I teach myself other fields of math. I was wondering if you could recommend some areas with a bit more complexity?

Current fields of knowledge:

  • Differential equations
  • Matrix Math
  • Complex Analysis
  • Real Analysis
  • Number Theory
  • Group Theory
  • Ring Theory
  • Multivariable Calculus
  • Trigonometry
  • Abstract Algebra
  • Game Theory

Thank You!

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By "current fields of knowledge," do you mean things you already know or things you think you'd like to know? (Most of those subjects are endless, by the way.) –  Brett Frankel Apr 30 '12 at 0:45
    
Non-euclidean geometry can be an eye-opener, and it cuts across many of your current fields of knowledge. Marta Sved's whimsical (Lewis-Carroll-inspired) "Journey into Geometries" is a readable introduction, while Marvin Greenburg's "Euclidean and Non-Euclidean Geometries" is more comprehensive (and expensive), as it's a college text. –  Blue Apr 30 '12 at 0:48
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I would point out that there is always more real analysis, complex analysis (especially several variables), abstract algebra and number theory to learn. Your list doesn't have much in the way of discrete math: perhaps you'd enjoy combinatorics and/or graph theory? –  Brett Frankel Apr 30 '12 at 0:53
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On a different vein, you may want to pick up a copy of Munkres's Topology. It's about as well-written as a book can get, and the second half gets into fundamental groups and cutting-and-pasting arguments, which are quite fun and very geometric. –  Brett Frankel Apr 30 '12 at 0:58
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It worries me you are studying Complex Analysis and Multivariate Calculus when this troubles you. I recommend you set your basis strong before moving on to more complex theory. –  Pedro Tamaroff Apr 30 '12 at 6:32

2 Answers 2

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First of all, kudos on attempting to broaden your horizons beyond the pretty coarse mathematics taught at most high schools.

By current fields of knowledge, I assume that you mean the subjects you're already somewhat familiar with. I would like to emphasize the sheer depth of all of these subjects, they're like an endless abyss which sucks you in with no end in sight. One that does indeed possess knowledge of these topics is painfully aware of how little he knows and how every one of those topics just unwinds faster and faster, with every new concept absorbed. This is extremely true of abstract algebra topics, like ring and group theory ( and many other things ). One could actually study every one of these topics for a lifetime and still feel overwhelmed in the grand scheme of things. That is good, that is normal. That is actually why we love mathematics.

If you wish to explore some fascinating subjects, I would like to defer back to Mr. Frankel in the comments, non-Euclidean geometry is a fascinating entry which gives an entirely new perspective to many things, just by toying around the definition of parallel lines and seeing what comes out of "supposing otherwise".

I must also defer to Mr. Tamaroff's comment - Studying math is like playing scales on a guitar - everyone wants to get really fast into it and consequently make a lot of mistakes along the way, instead of taking the time to fully amortize themselves, with small and careful steps, and benefit the most. I'd mostly suggest exploring something that really interests you and doesn't have dependencies in fields you are not versed in. Also, when you start something, see it through to a very good level. If your attention span gets torn off every 5 minutes, that will damage your chances in the long run.

Take it slow, truly understand all the notions, from the ground up, think about the intuition behind the definition and how it generalizes, cross reference and analyze... And most of all, enjoy your studies.

One of the things you could do is take the topics taught at school and expand upon them, but with real rigour, analysis and effort to comprehend all the implications. And as you go along, some things will strike you as interesting, you will make connections with the data in your head and find out a field that really catches your curiosity.

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Check out the website art of problem solving. Google it, sign up for an account. Go directly to the fourms and start solving some interesting problems. As far as books go, I know, but haven't used, that Dover produces some relatively cheap editions of older text books in a variety of areas. I would advise you to check them out. Since they're cheaper, you could buy a bunch of them and find out what you're interested in that way.

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