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I am a 16 year old high school student with too much free time. For the past two years I have been teaching myself university level maths in my free time. I have studied the basic first year subjects such as multivariable calculus, linear algebra and ordinary differential equations. My question is what should I study next?

I have tried more pure subjects such as real analysis, abstract algebra and number theory and additionally more applied subjects such as nonlinear dynamics, chaos theory and topology, but they either didn't spark my interest or were too difficult. I have even written a few papers that never really worked out.I posted one here I feel as if I need to regain my curiosity for mathematics and look for something beyond what academics present at seminars for children (referring to topology). Therefore, what is a subject that is seldom presented to an undergraduate but still is an valid field in mathematics?

hwood87
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  • This has been asked and answered here https://math.stackexchange.com/questions/533000/suggestions-for-high-school – xxxxxxxxx Nov 26 '20 at 06:57
  • And here https://math.stackexchange.com/questions/2315621/what-is-something-substantial-a-high-schooler-do-outside-of-school-in-mathematic – xxxxxxxxx Nov 26 '20 at 06:58
  • @MorganRodgers Did you read the second paragraph and question? The links you provide do not answer this nor are they the same question. – hwood87 Nov 26 '20 at 07:00
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    +1 for clearly demonstrated wisdom re "...but they either didn't spark my interest or were too difficult...". It's good to balance fun with practicality. You might enjoy probability/combinatorics which shouldn't be too difficult, and which you might enjoy. Beyond that, you may ask yourself whether you are going to want to keep private electronic math notebook(s). If so, LaTex is worth considering. Also, how do you plan to make a living. Do you enjoy computer programming, which is often a natural fit for mathematicians who want to make money? – user2661923 Nov 26 '20 at 07:07
  • Keep in mind that some books are much more interesting and have much better writing than others, so you should look at many different books. Get a big picture view and see which areas seem most exciting. Have you tried reading The Feynman Lectures on Physics? The No-Nonsense physics books, and Teach Yourself Physics (by the same author), are also great. It's not a bad idea to dive into programming while you're at it. I loved working through the book The Elements of Computing Systems, where you make a computer from scratch (starting with nand gates). Biology has become extremely interesting. – littleO Nov 26 '20 at 07:22
  • You said you studied multivariable analysis. If you haven't gotten to complex analysis I think it is a natural complement. – user619894 Nov 26 '20 at 07:34
  • @user619894 *multivariable calculus – hwood87 Nov 26 '20 at 07:41
  • I think you need to study set theory and logic first. Then you need to really analyze real analysis then study topology. Otherwise it's going to be difficult. I think numerical analysis is easy and interesting. You can give programming a chance too. And have you tried to study probability? It's a fun subject. – Hoda Bibo Nov 26 '20 at 07:56
  • @hwood87 I accept the correction, but my comment stands. – user619894 Nov 26 '20 at 10:52

3 Answers3

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I think you've already studied all the basic stuff common to science under-graduation courses. I think you should think about your interests now and from that move to more deeper and specific subjects. If you just want to have fun with mathematics, give another chance to set topology (did you really study set topology?), it would be your first rigorous math subject and then move to real analysis. Try different books, the fact is hard for you it means you can learn a lot from that (I don't think is too hard from your background, ask questions here).

user42912
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At the risk of swamping you with too much, you may want to look through past issues of the following.

Pi Mu Epsilon Journal

Quantum

Mathematical Spectrum

The Pentagon

Also, see the books in the New Mathematical Library series and the many translations of Russian "school level" books.

Dave L. Renfro
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Are you good in programming? Maybe you could learn Matlab and alike if you're interested in numerical mathematics too

NirvanicUniverse
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    I'd vote for Python over Matlab, due to the fact that Python is a much more marketable skill and is much more popular in the world of data science / machine learning (not to mention that Python is free). I believe Python is currently the most popular programming language. I also think it'd be a good idea to learn C, because it's a lower level language that arguably gives a better idea how a computer works and how computer memory works. – littleO Nov 26 '20 at 18:48
  • @littleO: great idea +1 – TShiong Dec 29 '24 at 23:44