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I hope you are having a wonderful day. My main question is how/what are the best ways for a high school student to go about working on, or at least attempting to do mathematical research.

I am currently a senior in high school, and have a working understanding of integral and differential calculus (Im in AP calc BC right now if that helps) and all preceding subjects. I have already been dabbling in self-study of vector/multivariate calc, basic differential equations, and basic linear algebra, but I would not yet trust myself to work with these concepts yet. I can look at work and understand it, but am not sufficiently skilled yet to tackle these problems myself. I could study these more thoroughly if it is essential, and I am honestly very interested in all of these subjects.. but I digress

I am really just asking all of you (much more advanced mathematicians than I will probably ever be) what your experience researching math tells you would be the most effective way for me to start. My career plan is to major in physics, get a PhD, and work in some form of theoretical or research physics.

Thank you for any time or thoughts you can share with me. It really means a lot, and I apologize for any inconvenience.

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    $\begingroup$ At a HS level, nice, useful and challenging research can be to try to cope some subject in undergraduate or, attempting to reach higher, even in advanced undergraduate mathematics. Anything not covered in usual HS curriculum can work: Fibonacci series, meaning of eigenvalues/eigenvectors in linear algebra, something about group theory or Galois theory, etc. To aspire to more than this one should, I think, study basic mathematics (i.e., university undergraduate level) first. $\endgroup$ – DonAntonio Jan 4 '18 at 23:33
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    $\begingroup$ I've been involved a research project at my university this past year in dynamical system. To really do a serious project and to dig deep and utilize all the resources to their full potential in the topic you are doing research in, you would need a guidance hand like a professor specialised in the topic. Otherwise another option is to pick a topic you like (e.g Linear Algebra) and start reading textbooks written on the topic and explore! $\endgroup$ – Itsnhantransitive Jan 4 '18 at 23:33
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    $\begingroup$ Unfortunately, I don't think there's much research in the realm of "calcluus" that you can get involved in until you've studied a few more course's worth of real/complex/numerical analysis and linear algebra. One thing I will say, however, is that I wish I knew about "Abstract Algebra" when I was in HS. Those two words may not sound like much, but it's this vast, wondrous world that you somehow never hear about in HS. Here's a start: youtube.com/watch?v=IP7nW_hKB7I. It doesn't take the most amount of knowledge in Abstract Algebra to find your way to a (very difficult) open problem. $\endgroup$ – AlkaKadri Jan 4 '18 at 23:57
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    $\begingroup$ If you are intending to purse higher studies, as you state, there will be an appropriate time for you to do proper research then: now's not the time for that. Perhaps by research you mean reading? I'm not sure. You could start looking at introductory undergraduate texts on linear algebra and calculus, for example, Calculus by Spivak, Multivariate Mathematics by Shifrin and Linear Algebra by Lang. $\endgroup$ – Pedro Tamaroff Jan 5 '18 at 1:03
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    $\begingroup$ It is wonderful to see such enthusiasm/interest and I wish you success as you pursue your career goals. But, ran across this quote recently, and, coming from the wealthiest man in the world (who went to Princeton to study physics), is of interest here: “Mediocre theoretical physicists make no progress. They spend all their time understanding other people's progress” - Jeff Bezos said of his prospects in the field. See en.wikipedia.org/wiki/Jeff_Bezos So, a really big world out there! Happy New Year and Good Luck! $\endgroup$ – CopyPasteIt Jan 5 '18 at 2:59
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Good for you!! One way to begin is to look around you for questions that you find interesting. This will give you practice

  1. modeling situations mathematically,
  2. looking for and playing around with new problems you like,
  3. trying to solve problems where you don't know exactly what tools you're "supposed" to be using, but instead you have a bunch of things you've learned that you can try applying,
  4. expressing your thoughts and conclusions carefully, and looking to prove that they're correct.

An example problem might be: you notice that water bottles fill up faster at the top where they round off, which seems interesting, so you think about it and decide to model it in a simpler way that you can probably solve: you have a 2D triangle (narrower at the top) filling up with water at a constant rate---how does the height of the water level change as it's filled? Does it fill faster toward the top, and if so by how much? What about other triangles, other shapes, or if the triangle is tilted over (seems harder)?

Or you might see a documentary about people walking across hot coals unharmed, and ask how you might model the problem physically to see how they do it.

Or you might learn that osmium is the densest naturally-occurring element, and try to get better intuition for density, so you decide to imagine a 1cm cube of osmium and a 1cm cube of some other metal like aluminum: if you balanced them on a seesaw, how far apart would you have to put the aluminum cube from the osmium cube? how long would the seesaw have to be? what happens if you increase the size of the cubes?


This is my way of thinking about getting into mathematics, with a selection of more physics-based problems. For you, you may have other problems you think are interesting: they may be more abstract, more like games, more practical, more based on engineering, and so on---playing around will help you discover what kinds of problems you like.

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  • $\begingroup$ This actually is extremely helpful; I plan to continue my studies and, on the side, work on these problems you talked about. I love what you said about problems that I don't know exactly what I will use to solve them, things I can attack from different angles. Thank you! $\endgroup$ – Mike H Jan 5 '18 at 2:23

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